A240923 -id:A240923 - cp's OEIS frontend

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

Michel Marcus, Aug 06 2014

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

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

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.

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

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 *)

A240990 Smallest integer m such that A240923(m) = n.

Original entry on oeis.org

1, 6, 15, 10, 14, 72, 22, 26, 42, 34, 38, 588, 46, 240, 78, 58, 62, 1456, 92, 74, 114, 82, 86, 1764, 70, 760, 640, 106, 195, 792, 108, 122, 186, 172, 134, 432, 110, 146, 222, 816, 158, 1656, 130, 396, 258, 178, 411, 3000, 154, 194, 342, 202, 206, 11904, 170

Offset: 0

Michel Marcus, Aug 06 2014

nonn

			sigma(6)/6 = 2/1 = (sigma(1)+1)/1, where sigma is A000203, so A240923(6)=1, and it is the smallest such integer, so a(1)=6.

Cf. A000203, A017665, A017666, A240923.

a240923(n) = my(ab = sigma(n)/n); numerator(ab) - sigma(denominator(ab));
a(n) = my(k = 1); while(a240923(k) != n, k++); k;

A272008 a(n) is the numerator of the fractional part of sigma(n)/n, where sigma(n) is the sum of the divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 4, 4, 1, 1, 1, 5, 3, 15, 1, 1, 1, 1, 11, 7, 1, 1, 6, 8, 13, 0, 1, 2, 1, 31, 5, 10, 13, 19, 1, 11, 17, 1, 1, 2, 1, 10, 11, 13, 1, 7, 8, 43, 7, 23, 1, 2, 17, 1, 23, 16, 1, 4, 1, 17, 41, 63, 19, 2, 1, 29, 9, 2, 1, 17, 1, 20, 49, 16, 19, 2, 1, 13, 40

Offset: 1

Michel Marcus, May 10 2016

a(n) = 0 when n is a multiply-perfect number (A007691).

a(n) = 1 when n is a prime or if n belongs to A215012.

			The sum of divisors of 4 is 7; its abundancy is 7/4 = 1 + 3/4 so a(4) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A007691, A017665, A017666, A108775, A215012, A240923, A243473.

f[n_] := Numerator[FractionalPart[DivisorSigma[1, n]/n]]; Array[f, 81] (* Robert G. Wilson v, Nov 24 2016 *)

a(n) = my(ab = sigma(n)/n); numerator(ab) % denominator(ab);

a(n) = A017665(n) mod A017666(n).

cp's OEIS Frontend

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

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A240990 Smallest integer m such that A240923(m) = n.

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A272008 a(n) is the numerator of the fractional part of sigma(n)/n, where sigma(n) is the sum of the divisors of n.

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