Father Primes: Happy or Not

[Harry Neel]

On the evening of April 25, 2026, an email was sent to SeqFans from myself that was not complete.  Please accept my most sincere mea culpa. It was the result of an error that occurred during my typing. (If anyone knows what the keystrokes are that causes this issue, and know a solution to correct it, please let me know. (And please do not tell me to stop typing, or to use a sledgehammer.)

With respect to Father Primes, the question is simple. "Is the term Father Prime recognized mathematically?" If not, then what is the proper name for the terms that many call Father Primes. (I know there are many approved sequences in the OEIS.

Thanks, and again, many apologies.

Harry E. Neel

[Ruud H.G. van Tol]

On 2026-04-28 00:16, Harry Neel wrote:
> [...] With respect to Father Primes, the question is simple. "Is the

> term Father Prime recognized mathematically?" If not, then what is the
> proper name for the terms that many call Father Primes. (I know there
> are many approved sequences in the OEIS.

> [...]

Hello Harry,

See https://oeis.org/search?q=%22father%20prime%22&sort=number
and read the comment in A136071, which sends you to A136026.

So in OEIS-context, there is a definition.

When searching on internet, there are also pages like
https://www.trottermath.net/numthry/fatherprimes.html
https://teherba.org/trottermath.net/numthry/fatherprimes.html
with their own definition.

-- Ruud

[Harry Neel]

Hello Ruud

Thanks for the information. Even though I have looked at A136026 a number of times, I was not aware that the definition there would be called a definition in context.

The information in Trottermath concerning father primes and another site concerning happy numbers has led me into looking at the intersection, so to speak, of the two.

Clarification is good.

Regards, and thanks again.

Harry

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[David Radcliffe]

The original name for A136026 was "a(n) = smallest father prime of order n." The entry defined a father prime of order n as a prime of the form (2n+1)p + 2n, where p is also prime. The sequence was renamed in 2010, and the definition of "father prime" was removed. However, other entries use the term "father prime" and refer to A136026 for its definition. This is confusing.

The quickest fix is to restore the definition of father prime in A136026. Otherwise, the other sequences should be renamed.

A136071: Father primes of order 2.    ===>   Primes of the form 5p + 4 with p prime.

A136072: Father primes of order 3.    ===>   Primes of the form 7p + 6 with p prime.

A136073: Father primes of order 4.    ===>   Primes of the form 9p + 8 with p prime.

A136074: Father primes of order 5.    ===>   Primes of the form 11p + 10 with p prime.

A136075: Father primes of order 6.    ===>   Primes of the form 13p + 12 with p prime.

A136076: Father primes of order 7.    ===>   Primes of the form 15p + 14 with p prime.

A136077: Father primes of order 8.    ===>   Primes of the form 17p + 16 with p prime.

A136078: Father primes of order 9.    ===>   Primes of the form 19p + 18 with p prime.

A136079: Father primes of order 10.  ===>   Primes of the form 21p + 20 with p prime.

A136080: Father primes of order 11.  ===>   Primes of the form 23p + 22 with p prime.

To me, the term "father prime" seems obscure, so I would recommend renaming the other sequences.

- David

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[Sean A. Irvine]

I have changed the names of A136071-A136080 in accordance with David's suggestion.

Those systematic names are definitely preferable to fanciful terminology.

Sean.

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[Harry Neel]

Folks,

Thanks for the information and the history. Actually I always thought it odd that so many sequences referred to another sequence for a definition when, at least currently, each sequence should have a definition. Once I asked an editor if the Name and the Definition was the same thing and was told that it was.

Anyway, thanks to all.

Harry Neel

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