A358415 -id:A358415 - cp's OEIS frontend

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

Jean-Marc Rebert, Nov 09 2022

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

			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.

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

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

Cf. A090748, A358415.

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)))

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

A358546 Least odd number m such that m mod 3 > 0 and m*3^n is an amicable number, and -1 if no such number exists.

Original entry on oeis.org

5480828320492525, 4865, 7735, 455, 131285, 849355, 11689795, 286385, 187047685, 104255, 32851039955, 2085985, 47942199242945, 189296520259, 349700961302721360788238344333849, 580068028631, 50392682631679406080371010751466781

Offset: 0

Jean-Marc Rebert, Nov 21 2022

If a(n) > -1 then a(n)*3^n is the least amicable number k such that A007949(k) = n.

			a(1) = 4865, because 4865 is an odd number and 4865 % 3 > 0 and 4865 * 3 = 14595 is an amicable number, and no lesser number has this property.

BOINC, Amicable Numbers

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

Cf. A090748, A358415, A358320, A358022.

sigmap(k)=if(k,sigma(k)-k,0)
cycle(k, TT=2)=my(x=k, T=1); while(x>0&&T<=TT, x=sigmap(x); if(x==k, return(T)); T++)
a(n, TT=2)=my(p3n=3^n); forstep(m=1, +oo, 2, if(m%3&&cycle(p3n*m, TT)==2, return(m)))

cp's OEIS Frontend

A358320 Least odd number m such that m*2^n is a perfect, amicable or sociable number, and -1 if no such number exists.

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