A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

1, 2, 3, 4, 6, 7, 10, 25, 26, 28, 33, 46, 55, 57, 69, 91, 93, 496, 1034, 1054, 1558, 2211, 2626, 4825, 8128, 11222, 12046, 12639, 28225, 32043, 68727, 89575, 970225, 1392386, 2245557, 8550146, 12371554, 16322559, 22799825, 33550336, 48980427, 51326726, 55037217, 60406599, 68258725, 142901438, 325422273, 342534446

Offset: 1

Author

Antti Karttunen, Dec 12 2024

Keywords

• nonn

Comments

Numbers k such that A252748(k) divides A286385(k).

Conjecture: Apart from a(5)=6, this is a subsequence of A319630, i.e., for all terms k<>6, gcd(k, A003961(k)) = 1. See also A372562, A372566.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..69 (terms below 10^11; terms 1..60 from Antti Karttunen)
• Index entries for sequences related to prime indices in the factorization of n.
• Index entries for sequences related to sigma(n).

Crossrefs

Cf. A000203, A003961, A252748, A286385, A319630, A372562, A372566.

Positions of 0's in A378981.

Subsequence of A263837.

Subsequences: A000396, A048674, A348514, A326134, A349753 (odd terms of this sequence).

Cf. also A378983.

Programs

Mathematica

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[k_] := Module[{fct = FactorInteger[k], m, s}, s = Times @@ f1 @@@ fct; m = Times @@ f2 @@@ fct; Divisible[m - s, m - 2*k]]; q[1] = True; Select[Range[10^5], q] (* Amiram Eldar, Dec 19 2024 *)

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = { my(u=A003961(n)); ((u-sigma(n))%((2*n)-u)); };
isA378980(n) = !A378981(n);