A343979 Composite numbers m such that lambda(m) = lambda(D_{m-1}), where lambda(n) is the Carmichael function of n (A002322) and D_k is the denominator (A027642) of Bernoulli number B_k.

5615659951, 36901698733, 55723044637, 776733036121, 2752403727511, 7725145165297, 14475486778537, 15723055492417, 22824071195485, 29325910221631, 54669159894469, 62086332981241, 125685944708905, 180225455689481, 298620660945331, 335333122310629, 426814989321721

Offset: 1

Author

Amiram Eldar and Thomas Ordowski, May 06 2021

Keywords

• nonn

Comments

Squarefree composites m such that LCM_{prime p|m} (p-1) = LCM_{prime p, p-1|m-1} (p-1).

Carmichael numbers m such that LCM_{prime p|m} (p-1) = LCM_{prime p, p-1|m-1} (p-1), i.e., with A173614(m) = A346467(m).

Carmichael numbers m such that their index (m-1)/lambda(m) = A346468(m), cf. A174590.

Carl Pomerance noted that, for k = 40826, Chernick's Carmichael number (6k+1)*(12k+1)*(18k+1) = 88189878776579929 satisfies this condition.

Theorem: lambda(m) | lambda(D_{m-1}) if and only if m | D_{m-1}.

Composites m such that lambda(m) | lambda(D_{m-1}) are all Carmichael numbers, defined as composites m such that lambda(m) | m-1, while lambda(D_{m-1}) | m-1 for every m.

Note that if p is prime, then lambda(p) = lambda(D_{p-1}) = p-1.

Crossrefs

Cf. A002322, A002997, A027642, A033502, A173614, A174590, A346467, A346468.

Programs

Mathematica

c = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; q[d_] := If[PrimeQ[d + 1], d, 1]; Select[c, LCM @@ (FactorInteger[#][[;; , 1]] - 1) == LCM @@ (q /@ Divisors[# - 1]) &]

PARI

A002322(n) = lcm(znstar(n)[2]); \\ From A002322
A173614(n) = lcm(apply(p->p-1, factor(n)[, 1]));
A346467(n) = if(1==n,n,my(m=1); fordiv(n-1,d,if(isprime(1+d),m = lcm(m,d))); (m));
isA343979(n) = ((n>1) && !isprime(n) && (!((n-1)%A002322(n))) && A173614(n)==A346467(n)); \\ Antti Karttunen, Jul 22 2021