Reduced Bernoulli Denominators and Carmichael Numbers

[Tomasz Ordowski]

Hello! 

The first primes p satisfying

lambda(Den(B_{p-1})/p) = p - 1 

are

2, 13, 31, 37, 61, 67, 113, 127, ... 

Recall that, for n > 1, the divisibility

n | Den(B_{n-1})

holds exactly when n is either a prime or a Carmichael number. Indeed, by the von Staudt–Clausen theorem this is equivalent, for composite n, to Korselt's criterion.

There is a related, more general condition

lambda(Den(B_{n-1})) = n - 1.

This condition holds for every prime n and for many composite n, but according to Pomerance, the set of such n has asymptotic density zero.

For a prime p, the factor p can therefore be removed from the Bernoulli denominator, leading to the first condition. Similarly below... 

Question: Are there any Carmichael numbers k satisfying 

lambda(Den(B_{k-1})/k) = k - 1 ?

More generally, does this condition have any interesting connection with the structure of Carmichael numbers?

Best,

Tom Ordo

[Amiram Eldar]

Up to 1000 the primes that satisfy lambda(Den(B_{p-1})/p) = p - 1 are:

2, 13, 31, 37, 61, 67, 113, 127, 139, 157, 181, 199, 211, 241, 277, 
281, 307, 331, 337, 349, 397, 401, 409, 421, 433, 461, 463, 499, 521, 
523, 541, 547, 561, 571, 577, 593, 601, 613, 617, 631, 661, 673, 691, 
701, 733, 739, 751, 757, 761, 769, 787, 811, 821, 829, 859, 877, 881, 
911, 937, 953, 967, 991, 997

I checked the first 30 Carmichael numbers and they all satisfy lambda(Den(B_{k-1})/k) = k - 1 except for 1105 and 63973.

[Tomasz Ordowski]

Ami, many thanks! 
More exceptions, please.
Cf. https://oeis.org/A317210 

See also https://oeis.org/A248614 

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[Tomasz Ordowski]

PS. Bernoulli-primitive primes: 

13, 31, 37, 61, 67, 113, 127, 139, 157, 181, 199, 211, 241, 277, 281, 307, 331, 337, 349, 397, 401, 409, 421, 433, 461, 463, 499, 521, 523, 541, 547, 571, ... 

Theorem. For an odd prime p, let D = Den(B_{p−1})/p. Then lambda(D) = p − 1 if and only if D is not the denominator of any Bernoulli number B_k with k < p − 1.

Proof. By von Staudt–Clausen, D is squarefree and q | D iff q − 1 | p − 1. Hence the least k for which D is a Bernoulli denominator is lambda(D).

The same holds for Carmichael numbers n: by Korselt's criterion, n is squarefree and q − 1 | n − 1 for every q | n. Thus, with D = Den(B_{n−1})/n, lambda(D) = n − 1 iff D is not the denominator of any B_k with k < n − 1.

I suggest calling these Bernoulli-primitive primes and Bernoulli-primitive Carmichael numbers.

What proportion of primes and Carmichael numbers are Bernoulli-primitive in this sense?

It seems that, unlike the Bernoulli-primitive primes, almost all Carmichael numbers (except 1105, 63973, ...) are primitive in this sense (more exceptions needed). 

[Amiram Eldar]

I checked the first 5204 Carmichael numbers and found these 24 exceptions:
1105, 63973, 825265, 1909001, 2628073, 3224065, 19384289, 23382529, 182356993, 684106401, 1419339691, 1674309385, 2602378721, 5615659951, 8152623721, 19904698081, 36901698733, 40004546113, 53282340865, 55723044637, 79623874561, 116748967297, 236359158267, 277535591965

[Tomasz Ordowski]

Ami, 

your data confirms that my condition 

reduces the proportion of primes more 

than the proportion of Carmichael numbers. 

Thanks for these large exceptions!

Tom 

 

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