A347261 Numbers k for which sigma(k)/k = 32/11.

924, 2970, 16368, 18018, 268224, 1107161088, 283465678848, 4535476813824, 76092819268618420224, 87729047721804447573604856326476791808

Offset: 1

Author

Timothy L. Tiffin, Aug 24 2021

Keywords

• nonn
• more

Comments

This sequence will contain terms of the form 33*P, where P is a perfect number (A000396) not divisible by 3 or 11. Proof: sigma(33*P)/(33*P) = sigma(33)*sigma(P)/(33*P) = 48*(2*P)/(33*P) = 32/11. QED

Terms ending in "24" or "x8" (where x is an even digit) have this form. Example: a(1) = 33*A000396(2), a(3) = 33*A000396(3), and a(n) = 33*A000396(n-1) for 5, 6, 7, 8.

Conjecture: a(n) = 33*A341623(n) for n >= 1. Motivation: If no term of A341623 is divisible by 11 (which appears to be the case), then sigma(33*A341623(n))/(33*A341623(n)) = sigma(11)*sigma(3*A341623(n))/(33*A341623(n)) = 12*(8*A341623(n))/(33*A341623(n)) = 32/11. Does this sequence, though, contain any additional terms that are not generated by A341623?

Examples

			2970 is a term, since sigma(2970)/2970 = 8640/2970 = 32/11.
18018 is a term, since sigma(18018)/18018 = 52416/18018 = 32/11.
		

Links

• G. P. Michon, Multiperfect Numbers and Hemiperfect Numbers
• Walter Nissen, Abundancy: Some Resources (preliminary version 4)
• Walter Nissen, Primitive Friendly Pairs with friends < 2^34 with denom < 20000

Crossrefs

Cf. A000203, A000396, A341623.

Subsequence of A005101.

Programs

Mathematica

Select[Range[5*10^8], DivisorSigma[1, #]/# == 32/11 &]
Do[If[DivisorSigma[1, k]/k == 32/11, Print[k]], {k, 5*10^8}]

Extensions

a(9)-a(10) from Michel Marcus, Aug 26 2021