cp's OEIS Frontend

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).

%I A378662 #29 Dec 12 2024 15:14:22
%S A378662 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,
%T A378662 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,
%U A378662 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
%N 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).
%C A378662 Number of terms of A341614 that divide n.
%C A378662 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.
%H A378662 Antti Karttunen, <a href="/A378662/b378662.txt">Table of n, a(n) for n = 1..100000</a>
%H A378662 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%H A378662 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A378662 a(n) = Sum_{d|n} A341612(d).
%F A378662 a(n) = A337345(n) - A080224(n).
%F A378662 a(n) = A080225(n) + A378663(n).
%t A378662 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 *)
%o A378662 (PARI)
%o A378662 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378662 A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
%o A378662 A378662(n) = sumdiv(n,d,A341612(d));
%Y A378662 Inverse Möbius transform of A341612.
%Y A378662 Cf. A000203, A003961, A005101, A080224, A080225, A337345, A341614, A378663, A378664.
%Y A378662 Cf. A246281 (positions of 0's), A246282 (of terms > 0).
%Y A378662 Cf. also A337372, A378658.
%K A378662 nonn
%O A378662 1,8
%A A378662 _Antti Karttunen_, Dec 06 2024