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

Offset: 0

Author

Jean-Marc Rebert, Nov 09 2022

Keywords

• nonn
• more

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

Examples

			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.
		

Links

• BOINC, Amicable Numbers
• HandWiki, List of perfect numbers
• David Moews, A list of currently known aliquot cycles of length greater than 2

Crossrefs

Cf. A347770, A000396, A001065, A002025, A259180, A262625.

Cf. A090748, A358415.

Programs

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++)
a(n,TT=28)=my(p2n=2^n); forstep(m=1, +oo, 2, if(cycle(p2n*m,TT), return(m)))

Extensions

a(0), a(15)-a(20) from Jean-Marc Rebert, Nov 17 2022