A343979 - OEIS

%I #55 Aug 08 2021 11:30:05

%S 5615659951,36901698733,55723044637,776733036121,2752403727511,

%T 7725145165297,14475486778537,15723055492417,22824071195485,

%U 29325910221631,54669159894469,62086332981241,125685944708905,180225455689481,298620660945331,335333122310629,426814989321721

%N 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.

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

%C 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).

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

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

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

%C 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.

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

%t 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]) &]

%o (PARI)

%o A002322(n) = lcm(znstar(n)[2]); \\ From A002322

%o A173614(n) = lcm(apply(p->p-1, factor(n)[, 1]));

%o A346467(n) = if(1==n,n,my(m=1); fordiv(n-1,d,if(isprime(1+d),m = lcm(m,d))); (m));

%o isA343979(n) = ((n>1) && !isprime(n) && (!((n-1)%A002322(n))) && A173614(n)==A346467(n)); \\ _Antti Karttunen_, Jul 22 2021

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

%K nonn

%O 1,1

%A _Amiram Eldar_ and _Thomas Ordowski_, May 06 2021