A346468 -id:A346468 - cp's OEIS frontend

A346467 a(n) is the least common multiple of the divisors d of n-1 such that d+1 is prime; a(1) = 1.

1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 20, 1, 22, 1, 12, 1, 2, 1, 28, 1, 30, 1, 16, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 44, 1, 46, 1, 48, 1, 10, 1, 52, 1, 18, 1, 28, 1, 58, 1, 60, 1, 2, 1, 16, 1, 66, 1, 4, 1, 70, 1, 72, 1, 2, 1, 4, 1, 78, 1, 80, 1, 82, 1, 84, 1, 2, 1, 88, 1, 90, 1, 92, 1, 2, 1, 96

Offset: 1

Antti Karttunen and Thomas Ordowski, Jul 22 2021

nonn

Original definition: a(n) is the least common multiple of p-1 computed over all primes p for which p-1 is a divisor of n-1; a(1) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002322, A027642, A173614, A346466, A346468, A346481.

Cf. also A343979.

f:= proc(n)
  if n::even then return 1 fi;
  ilcm(op(select(d -> isprime(d+1), numtheory:-divisors(n-1))));
end proc:
f(1):= 1:
map(f, [$1..200]); # Robert Israel, Aug 30 2021

{1}~Join~Array[CarmichaelLambda@ Denominator@ BernoulliB@ # &, 96] (* Michael De Vlieger, Jul 22 2021 *)

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

apply( {A346467(n)=if(n>1, lcm([d|d<-divisors(n-1),isprime(d+1)]), 1)}, [1..99]) \\ M. F. Hasler, Nov 23 2021

a(n) = A002322(A027642(n-1)).
a(n) = A346466(n) * A346481(n).
For n > 1, a(n) = (n-1) / A346468(n).
a(n) = LCM { d | n-1; d+1 is prime }, where "|" means "divides". - M. F. Hasler, Nov 23 2021

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.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar and Thomas Ordowski, May 06 2021

nonn

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.

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

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]) &]

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

cp's OEIS Frontend

A346467 a(n) is the least common multiple of the divisors d of n-1 such that d+1 is prime; a(1) = 1.

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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.

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