A361683 a(n) is the least k such that tau(k) divides sigma_n(k) but not sigma(k), or -1 if no such k exists.

4, 64, 4, 7168, 4, 606528, 4, 64, 4, 4194304, 4

Offset: 2

Author

Mohammed Yaseen, Mar 20 2023

Keywords

• nonn
• more

Comments

a(13) <= 31525197391593472. - David A. Corneth, Mar 20 2023

From Thomas Scheuerle, Mar 22 2023: (Start)

a(17) <= 15211807202738752817960438464512 and a(19) <= 2^190*11.

Conjecture: a(n) is of the form 2^b*p1^c*p2^d*...*pk^j with b > 0 and A020639(n) divides b*(c+1)*(d+1)*...*(j+1). (p1, p2, ..., pk are distinct odd prime numbers). (End)

Crossrefs

Cf. A003601, A020486, A046839, A046840.

Cf. A000005, A000203, A001157, A001158.

Cf. A020639, A010709 (bisection).

Programs

Mathematica

a[n_] := Module[{k = 1, d}, While[Divisible[DivisorSigma[1, k], (d = DivisorSigma[0, k])] || !Divisible[DivisorSigma[n, k], d], k++]; k]; Array[a, 11, 2] (* Amiram Eldar, Mar 20 2023 *)

PARI

isok(k, n) = my(f=factor(k), nd=numdiv(f)); (sigma(f) % nd) && !(sigma(f,n) % nd);
a(n) = my(k=1); while (!isok(k,n), k++); k; \\ Michel Marcus, Mar 20 2023

Formula

a(2*m) = 4 for m >= 1.

a(6*m-3) = 64 for m >= 1.

From Thomas Scheuerle, Mar 22 2023: (Start)

a(m) <= a(A020639(m)) if a(A020639(m)) <> -1.

Conjecture: For primes q > p, a(q) > a(p). If true, we could replace "<=" with "=" in the above formula. (End)