I'd like to know how you interpret the definition of this sequence.

[Ramon Ortega]

The sum of the powers of ten of each term, individually, in the increasing sequence of positive integers, whose base-ten representation is obtained by concatenating powers of ten with non-negative integer exponents.

1, 10, 2, 100, 11, 11, 3, 1000, 101, 20, 12, 101, 12, 12, 4, 10000, 1001, 110, 102, 110, 21, 21, ...

[M F Hasler]

You mean: 

a(n) = sum of the powers of ten occurring in the binary expansion of n, considered as concatenation of the decimal representation of powers of ten.

Equivalently, 

a(n) = sum of 10^(d(k)-1), where d(k) are the first differences of the positions of nonzero bits in 2n+1.

For example, 

n=5 => 2n+1 = 11 = 2⁰ + 2¹ + 2³

Positions: 0, 1, 3

First differences : 1-0=1,  3-1=2. Therefore, a(5) = 10^0 + 10^1.

Using 2n+1 instead of n is an artefact which you can avoid by replacing d(k) with p(k)-p(k-1) where p(k) is the position of the k-th nonzero bit in n, and p(0) := -1.

(Then for n=5 = 2⁰ + 2², p(1)=0 and p(2) = 2.)

- Maximilian

On Sat, Apr 18, 2026, 21:23 Ramon Ortega <tumblr3...@gmail.com> wrote:

The sum of the powers of ten of each term, individually, in the increasing sequence of positive integers, whose base-ten representation is obtained by concatenating powers of ten with non-negative integer exponents.

1, 10, 2, 100, 11, 11, 3, 1000, 101, 20, 12, 101, 12, 12, 4, 10000, 1001, 110, 102, 110, 21, 21, ...

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[Ramon Ortega]

Exactly, Maximilian. Taking your advice, I've now thought of defining a(n) like this: Sum of the powers of ten of the nth positive integer formed by concatenating powers of ten

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[Rémy Etc]

For inspiration, you can take a look at A162439, which is a binary version of your sequence.

Best regards,

Rémy

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[Ramon Ortega]

Thank you very much for this reference, Rémy. That's exactly it. I just don't want my definition to be related to base two. In my definition, everything is done in base ten.

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[M F Hasler]

Then I understood it wrong, maybe.

It's not the binary expansion you are considering? How do you get 100 for n=4 ?

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[Ruud H.G. van Tol]

On 2026-04-19 13:12, Ramon Ortega wrote:
> [...] A162439 [...] I just don't want my definition to be related to

> base two. In my definition, everything is done in base ten.

The '+' would then most likely come in front of the digit 9.

(PARI)
? a(n) = eval(strjoin(strsplit(Str(n), "9"), "+9"));

? [ print([n, strjoin(strsplit(Str(n),"9"),"+9"), a(n)]) | n<-[0..200],
n!=a(n)];

[19, "1+9", 10]
[29, "2+9", 11]
[39, "3+9", 12]
[49, "4+9", 13]
[59, "5+9", 14]
[69, "6+9", 15]
[79, "7+9", 16]
[89, "8+9", 17]
[99, "+9+9", 18]
[109, "10+9", 19]
[119, "11+9", 20]
[129, "12+9", 21]
[139, "13+9", 22]
[149, "14+9", 23]
[159, "15+9", 24]
[169, "16+9", 25]
[179, "17+9", 26]
[189, "18+9", 27]
[190, "1+90", 91]
[191, "1+91", 92]
[192, "1+92", 93]
[193, "1+93", 94]
[194, "1+94", 95]
[195, "1+95", 96]
[196, "1+96", 97]
[197, "1+97", 98]
[198, "1+98", 99]
[199, "1+9+9", 19]

-- Ruud

[Ramon Ortega]

Sorry for my confusion in the previous post, Remi. Yes, you understood correctly. The sequence is defined from base two. That's why a(2) = 100, because 2₂ = 100 and 100 to the base ten is a power of ten.
Thank you for your attention!

[Ramon Ortega]

To avoid referring to binary numbers, I thought of redefining the sequence like this. Is this new interpretation correct?

a(n) = sum of the powers of ten of the nth positive integer in base ten, written as a concatenation of powers of ten in base ten.

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[Ramon Ortega]

What do you think of this new version, below?

a(n) is the sum of the powers of ten obtained by interpreting the nth number (in ascending order), which in base ten uses only 0 and 1, as a concatenation of powers of ten.

[Ramon Ortega]

I prefer this definition because it eliminates the need to refer to binary representation.

a(n) is the sum of the powers of ten obtained by interpreting the nth number (in ascending order), which in base ten uses only 0 and 1, as a concatenation of powers of ten.

[Ray Chandler]

Ramon,

Please look at A007088.  Your sequence sounds like a match or similar.  How does your sequence differ from it?  Did you try looking up the first few terms in the database?

Ray

Sent from AT&T Yahoo Mail on Android

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[M F Hasler]

On Mon, Apr 20, 2026 at 10:57 AM Ray Chandler <rayjch...@sbcglobal.net> wrote:

Ramon,

Please look at A007088.  Your sequence sounds like a match or similar.  How does your sequence differ from it?  Did you try looking up the first few terms in the database?

Ray

Ray, unless I got it wrong, he doesn't want the binary expansion,

but the sum of the powers of ten *seen in the binary expansion*.

For example, n=3 = 11 [base 2] => 1+1 = 2

n=5 = 101 [base 2] => 10+1 = 11

n=10 = 1010 [base 2] => 10+10 = 20

Ramon: In any case, you *MUST* say "the binary expansion/representation of n" somewhere in your definition

(or speak of "zero / nonzero bits" as I did in some of my proposals).

In all of our more recent proposals you dropped that.

- Maximilian

[Ramon Ortega]

Maximilian, I prefer this definition below because it eliminates the need to refer to binary representation.

a(n) is the sum of the powers of ten obtained by interpreting the nth number (in ascending order), which in base ten uses only 0 and 1, as a concatenation of powers of ten.

[Allan Wechsler]

I think all will agree that there is something a little funky about the definition of A007088, but it would probably be inappropriate to try to "fix" this ancient sequence at this late date. The data section of an entry is supposed to unambiguously specify a sequence of integers -- not of digit strings, not of abstract words of some sort. The notion that there is some kind of ambiguity about whether A007088(2) is this many [**********] or this many [**] is anathema to most modern contributors and users. It must be the former. It can't be the latter.

The first honest definition of this sequence is not in the title, but is rather the second interpretation in the comments, "numbers that are sums of distinct powers of 10". The lead description, "The binary numbers", is a philosophical disaster. Is 417 a binary number? If it is, isn't this sequence identical in content to A001477, "The nonnegative integers"? Neil felt compelled to add a caveat to the very first comment line, apologizing for the anomaly.

But taking this sequence at face value as an ordinary list of integers written in base 10, as every other OEIS sequence is, Ramon Ortega's description for his proposed sequence is pretty much spot on. Obviously one can interpret any element of A007088 as a concatenation of powers of ten, and it's certainly well-defined to add these up. This operation does make explicit reference to base ten, and the resulting sequence certainly deserves a "base" keyword, because it refers to the decimal concatenation operator. But nowhere does it need to mention binary numbers.

-- Allan

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[Ramon Ortega]

I tried to improve it this way. What do you think of this new version below?

a(n) is the sum of the largest powers of ten that appear, visually concatenated, in the decimal representation of the nth sum of distinct powers of ten.

[Ramon Ortega]

Which is the best definition of the sequence in your opinion: d1 or d2?

d1) a(n) is the sum of the largest powers of ten that appear, visually concatenated, in the decimal representation of the nth sum of distinct powers of ten.

d2) Let T_n be the nth positive integer with only the digits 0 and 1 in decimal. Then a(n) = ∑ 10^z, where for each digit 1 in T_n, z is the count of consecutive zeros immediately to its right (or z=0 if there are none).

[Ramon Ortega]

Do you think this sequel is interesting enough to be submitted to OEIS? Please comment if you've seen any interesting properties that might catch the attention of the OEIS editors.

[Ramon Ortega]

Which is the best definition of the sequence in your opinion: d1, d2 or d3?

d1) a(n) is the sum of the largest powers of ten that appear, visually concatenated, in the decimal representation of the nth sum of distinct powers of ten.

d2) Let T_n be the nth positive integer with only the digits 0 and 1 in decimal. Then a(n) = ∑ 10^z, where for each digit 1 in T_n, z is the count of consecutive zeros immediately to its right (or z=0 if there are none). 

d3) a(n) = sum of the powers of 10 whose exponents are the differences, decreased by 1, between consecutive exponents (in descending order) of the distinct powers of 2 that sum to n+1/2

[Md. Rad Sarar Anando]

d3 in my opinion, at least it states the main interest of the sequence. 

Regards,
Rad

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[Arthur O'Dwyer]

d3 is the only description that comes close to mentioning "binary representation"; the other two seem like non-starters.

Fundamentally, your sequence is about taking an integer (a mathematical number), writing it in binary notation, partitioning that string into several smaller strings, and then reading each of those smaller strings as if it were in decimal notation. The arbitrary choice of binary for step 1 and decimal for step 3 — the "type-punning" between strings written in these two arbitrary bases — makes this an uninteresting sequence.

Choosing binary instead of decimal for step 3 would give you this sequence, which strikes me as slightly more interesting but still I wouldn't have guessed it was a candidate for OEIS:

a(1) = 1 = 1

a(2) = 10 = 2

a(3) = 1+1 = 2

a(4) = 100 = 4

a(5) = 10+1 = 3

a(6) = 1+10 = 3

a(7) = 1+1+1 = 3

a(8) = 1000 = 8

a(9) = 100+1 = 5

[...]

Yet it turns out that this one is already in the OEIS, as A162439!

So your sequence could best be described analogously: "Write down the binary representation of n. Partition the string which is this binary representation by placing a '+' just left of every 1. Add the resulting base-10 numbers. a(n) = this sum."

Now, you have shown no understanding in this thread of concepts like "base-10" or "binary representation," so you really ought to go off and fully read some material starting with at least these Wikipedia entries:

https://en.wikipedia.org/wiki/Positional_notation

https://en.wikipedia.org/wiki/Decimal

https://en.wikipedia.org/wiki/Binary_number

before you touch this sequence again. Knowing this stuff will not only help you explain your ideas to other people, it will probably even give you different (better) ideas in the first place!

–Arthur

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[Ramon Ortega]

I really liked your observations, Arthur, but it's intentional that I don't make any reference to bases. Interpreting it as relationships between bases is also obviously relevant. However, it can also be interpreted simply in terms of the most basic arithmetic operations. That's my intention in always trying to improve the definition, but always in a simpler way.

d1) a(n) is the sum of the largest powers of ten that appear, visually concatenated, in the decimal representation of the nth sum of distinct powers of ten.

d2) Let T_n be the nth positive integer with only the digits 0 and 1 in decimal. Then a(n) = ∑ 10^z, where for each digit 1 in T_n, z is the count of consecutive zeros immediately to its right (or z=0 if there are none). 

d3)
a(n) = sum of the powers of 10 whose exponents are the differences, decreased by 1, between consecutive exponents (in descending order) of the distinct powers of 2 that sum to n+1/2

d5)For every non-negative integer n, let e₁ > e₂ > … > eₖ be the exponents of distinct powers of 2 whose sum is equal to n + ½; the resulting value is given by ∑ 10^(eⱼ − eⱼ₊₁ − 1), for j = 1 to k − 1.

d6) "Write down the binary representation of n. Partition the string which is this binary representation by placing a '+' just left of every 1. Add the resulting base-10 numbers. a(n) = this sum." ( Arthur O'Dwyer  )

[Ramon Ortega]

Is every infinite subsequence of a(n) periodic?

[L. Edson Jeffery]

Ramon, I agree with Arthur. Several others have also made suggestions regarding how you should proceed with the definition. Trying to avoid base 2 in the definition is a terrible mistake. You should study related sequences such as A007088 (the binary numbers) suggested by Maximilian Hasler, I think, and which you ignored.

Ed Jeffery 

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[Ramon Ortega]

Is every infinite subsequence of a(n) of the form b(n, 1) = n, b(n, k+1) = a(b(n,k)) periodic?

[Sean A. Irvine]

Enough is enough, Ramon Ortega has been removed from the list for an excessive number of posts.

Sean.

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