A378664 Greatest divisor d of n such that sigma(d) <= 2*d < A003961(d), or 1 if no such divisor exists, where A003961 is fully multiplicative with a(p) = nextprime(p).

1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 6, 1, 14, 15, 16, 1, 9, 1, 10, 21, 1, 1, 8, 1, 1, 27, 28, 1, 15, 1, 32, 1, 1, 35, 9, 1, 1, 39, 10, 1, 21, 1, 44, 45, 1, 1, 16, 49, 50, 1, 52, 1, 27, 1, 28, 57, 1, 1, 15, 1, 1, 63, 64, 1, 6, 1, 68, 69, 35, 1, 9, 1, 1, 75, 76, 1, 39, 1, 16, 81, 1, 1, 28, 1, 1, 1, 44, 1, 45, 91, 92

Offset: 1

Author

Antti Karttunen, Dec 06 2024

Keywords

• nonn

Comments

Largest term of {1} U A341614 that divides n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• 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, A337372, A341612, A341614, A378662, A378665, A378736 [= a(A005101(n))], A378737, A378738, A378739, A378740, A378741, A378742.

Positions of fixed points (where a(n)=n) is given by {1} U A341614.

Cf. A246281 (positions of 1's), A246282 (of terms > 0), A005101 (of terms that are neither 1 nor fixed points).

Programs

Mathematica

Table[If[Length[#] == 0, 1, Max[#]] &@ Select[Divisors[n], DivisorSigma[1, #] <= 2 # < (Times @@ Map[Power @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi[p] + 1], e}] - Boole[# == 1]) &], {n, 92}] (* Michael De Vlieger, Dec 06 2024 *)

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)));
A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };

Formula

a(n) <= A378665(n).