A358320 - OEIS

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A358320

Least odd number m such that m*2^n is a perfect, amicable or sociable number, and -1 if no such number exists.

12285, 3, 7, 779, 31, 37, 127, 651, 2927269, 93, 25329329, 7230607, 8191, 66445153, 7613527, 18431675687, 131071, 264003743, 524287, 59592560831, 949755039781

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OFFSET

0,1

COMMENTS

For n in {1,2,4,6,12,16,18}, a(n)*2^n is a perfect number. See A090748.

For n in {0,3,5,8,10,11,13,14,15,17,19}, a(n)*2^n is an amicable number.

For n in {7,9} a(n)*2^n is a sociable number of order 28.

That is, h_k(m*2^n) = m*2^n for some k > 0, where h_{k+1}(n) = h_k(h(n)) and h(n) = A001065(n), the sum of aliquot parts of n. - Charles R Greathouse IV, Nov 09 2022

Least m such that m*2^n is in A347770. - Charles R Greathouse IV, Nov 09 2022

LINKS

Table of n, a(n) for n=0..20.

BOINC, Amicable Numbers

HandWiki, List of perfect numbers

David Moews, A list of currently known aliquot cycles of length greater than 2

EXAMPLE

a(1) = 3, because 3 is an odd number and 3 * 2^1 = 6 is a perfect number, and no lesser number has this property.

PROG

(PARI)

sigmap(n)=if(n<=1, return(0)); sigma(n)-n

cycle(n, TT=28)=my(x=n, T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==n, return(T)); T++)