A240991 Integers m such that A240923(m) = 1, where A240923(n) = numerator(sigma(n)/n) - sigma(denominator(sigma(n)/n)).

6, 18, 28, 117, 162, 196, 496, 775, 1458, 8128, 9604, 13122, 15376, 19773, 24025, 88723, 118098, 257049, 470596, 744775, 796797, 1032256, 1062882, 2896363, 6725201, 9565938, 12326221, 14776336, 23059204, 25774633, 27237961, 33550336, 43441281, 63455131

Offset

1,1

Comments

Perfect numbers (A000396) are a subsequence, since they satisfy sigma(m)/m = 2/1 = (sigma(1)+ 1)/1, that is of the form (sigma(d)+1)/d, with sigma being A000203.

Similarly, k-multiperfect numbers satisfy A240923(m) = k-1.

The analogous sequence of integers such that A240923(m) = 0 is A014567.

Holdener et al. say that these numbers have a quasi-friendly divisor and prove that such quasi-friendly divisors cannot have more than two distinct prime divisors. - Michel Marcus, Sep 08 2020, clarified by Antti Karttunen, Aug 31 2025

a(68) > 3.2*10^11. - Giovanni Resta, Aug 30 2025

Question: Might it possible to prove that for all n, A001221(a(n)) = 2, e.g., by showing that this is a subsequence of A387406? In any case, that holds for 67 initial terms. - Antti Karttunen, Aug 31 2025

Links

Giovanni Resta, Table of n, a(n) for n = 1..67 (first 51 terms from Michel Marcus)

C. A. Holdener and J. A. Holdener, Characterizing Quasi-Friendly Divisors, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.4.

Index entries for sequences related to sigma(n)

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Maple

filter:= proc(n) uses numtheory; local r; r:= sigma(n)/n; numer(r) - sigma(denom(r)) = 1 end proc:
select(filter, [$1..10^5]); # Robert Israel, Aug 07 2014

Mathematica

a240923[n_Integer] :=
Numerator[DivisorSigma[1, n]/n] -
  DivisorSigma[1, Denominator[DivisorSigma[1, n]/n]];
a240991[n_Integer] := Flatten[Position[Thread[a240923[Range[n]]], 1]];
a240991[1000000] (* Michael De Vlieger, Aug 06 2014 *)

Crossrefs

Cf. A000203, A001221, A014567, A017665, A017666, A240923.

Cf. also A387405 (the least k whose quasi-friendly divisor a(n) is, or -1 if no such k exists).

Conjectured to be a subsequence of A387406.

Sequence in context: A225110 A387406 A077660 * A304050 A376028 A351220

Adjacent sequences: A240988 A240989 A240990 * A240992 A240993 A240994

Keyword

nonn

Author

Michel Marcus, Aug 06 2014

Status

approved