A387405 -id:A387405 - cp's OEIS frontend

A387406 Numbers k such that sigma(A253560(k)) / A253560(k) is equal to (sigma(k)+1) / k, where A253560(k) = k multiplied by its largest prime factor.

6, 18, 28, 54, 117, 162, 196, 486, 496, 775, 1372, 1458, 1521, 4374, 8128, 9604, 13122, 15376, 19773, 24025, 39366, 67228, 88723, 118098, 257049, 354294, 470596, 476656, 744775, 796797, 1032256, 1062882, 2896363, 3188646, 3294172, 3341637, 6725201, 9565938, 12326221, 14776336, 23059204, 23088025, 25774633, 27237961

Offset: 1

Antti Karttunen, Aug 30 2025

Terms k for which sigma(k/A053585(k)) = A006530(k). This further entails that A001221(k) = 2 [See A023194].

Antti Karttunen, Table of n, a(n) for n = 1..58 (larger b-file needed)
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)

Cf. A000203, A001221, A006530, A023194, A053585, A253560, A387405.

Subsequences: A000396 (even terms only), A240991 (conjectured, if true, then A000396 has only even terms).

fk[k_]:=k*FactorInteger[k][[-1,1]];Select[Range[10^6],DivisorSigma[1,fk[#]]/fk[#]==(DivisorSigma[1,#]+1)/#&] (* James C. McMahon, Aug 31 2025 *)

A253560(n) = if (n==1, 1, n*vecmax(factor(n)[, 1]));
isA387406(n) = { my(x=A253560(n)); ((sigma(x)/x) == ((sigma(n)+1)/n)); };

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A387406 Numbers k such that sigma(A253560(k)) / A253560(k) is equal to (sigma(k)+1) / k, where A253560(k) = k multiplied by its largest prime factor.

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