A240991 - OEIS

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A240991

Integers m such that A240923(m) = 1.

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

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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 they cannot have more than two distinct prime divisors. - Michel Marcus, Sep 08 2020

LINKS

Michel Marcus, Table of n, a(n) for n = 1..51

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

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 *)