A361683 - OEIS

%I #48 Mar 22 2023 15:17:55

%S 4,64,4,7168,4,606528,4,64,4,4194304,4

%N 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.

%C a(13) <= 31525197391593472. - _David A. Corneth_, Mar 20 2023

%C From _Thomas Scheuerle_, Mar 22 2023: (Start)

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

%C 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)

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

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

%F From _Thomas Scheuerle_, Mar 22 2023: (Start)

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

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

%t 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 *)

%o (PARI) isok(k, n) = my(f=factor(k), nd=numdiv(f)); (sigma(f) % nd) && !(sigma(f,n) % nd);

%o a(n) = my(k=1); while (!isok(k,n), k++); k; \\ _Michel Marcus_, Mar 20 2023

%Y Cf. A003601, A020486, A046839, A046840.

%Y Cf. A000005, A000203, A001157, A001158.

%Y Cf. A020639, A010709 (bisection).

%K nonn,more

%O 2,1

%A _Mohammed Yaseen_, Mar 20 2023