A378662 Number of divisors d of n such that sigma(d) <= 2*d < A003961(d), where A003961 is fully multiplicative with a(p) = nextprime(p).

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 0, 0, 3, 0, 0, 2, 3, 0, 3, 0, 4, 0, 0, 1, 3, 0, 0, 1, 3, 0, 3, 0, 2, 3, 0, 0, 4, 1, 2, 0, 2, 0, 3, 0, 4, 1, 0, 0, 4, 0, 0, 3, 5, 0, 1, 0, 2, 1, 3, 0, 4, 0, 0, 2, 2, 0, 2, 0, 4, 3, 0, 0, 5, 0, 0, 0, 3, 0, 5, 1, 2, 0, 0, 0, 5, 0, 3, 2, 3, 0, 1, 0, 3, 4

Offset

1,8

Comments

Number of terms of A341614 that divide n.

Claim: a(n) > 0 if and only if A003961(n) > 2*n [i.e., n is in A246282]. That a(n) must be zero when n is in A246281 is obvious, as is also that a(n) > 0 when n is a term of A341614 [as then A378664(n) = n], but that a(n) > 0 for all abundant numbers (A005101) is slightly less clear. So the claim boils down to this: All abundant numbers have at least one (by necessity a proper) divisor d|n such that it is in A341614, i.e., sigma(d) <= 2*d < A003961(d), i.e., that for abundant numbers n, A337345(n) is always strictly greater than A080224(n). Equivalently, of the all nonabundant divisors d of an abundant number, at least one is primeshift-abundant, i.e., A003961(d) > 2*d. This has been proved Dec 11 2024 by Jianing Song in A337372. The claim given in A378658 also follows from that proof.

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences related to prime indices in the factorization of n

Index entries for sequences related to sigma(n)

Formula

a(n) = Sum_{d|n} A341612(d).

a(n) = A337345(n) - A080224(n).

a(n) = A080225(n) + A378663(n).

Mathematica

Table[Length@ Select[Divisors[n], DivisorSigma[1, #] <= 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &], {n, 105}] (* Michael De Vlieger, Dec 06 2024 *)

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378662(n) = sumdiv(n, d, A341612(d));

Crossrefs

Inverse Möbius transform of A341612.

Cf. A000203, A003961, A005101, A080224, A080225, A337345, A341614, A378663, A378664.

Cf. A246281 (positions of 0's), A246282 (of terms > 0).

Cf. also A337372, A378658.

Sequence in context: A276806 A308427 A384103 * A252736 A351416 A253559

Adjacent sequences: A378659 A378660 A378661 * A378663 A378664 A378665

Keyword

nonn

Author

Antti Karttunen, Dec 06 2024

Status

approved