

A321328


a(n) is the smallest number k such that k = (sigma(n*(sigma(k)k))  n*(sigma(k)k))/n.


1



6, 20, 14, 4, 10, 26, 1012, 8, 1442, 68, 376, 38, 1660, 14, 506, 574, 352, 117, 590, 22, 254, 1292, 460, 82, 26108, 416, 266, 10, 3496, 15, 124, 32, 470, 5176, 658, 362, 104696, 152, 19305, 51, 12782, 62, 618770, 232, 15561, 1136, 4136, 1006, 8588, 49166, 154, 13988
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OFFSET

1,1


COMMENTS

A sort of generalization of amicable numbers where x = n*(sigma(k)k), y = (sigma(x)x)/n = k and x >= y.
All the numbers that satisfy the equation for n=1 are listed in A206708.
a(n) = n for n = 4, 8, 14, 32, 128, 2366, 8193, 131072, etc.
In particular a(n) = n if n = 2^p where p is a Mersenne exponent (A000043).


LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..200


EXAMPLE

a(7) = 1012 because (sigma(7*(sigma(1012)1012))  7*(sigma(1012)1012))/7 = (sigma(7*1004)  7*1004)/7 = (141127028)/7 = 7084/7 = 1012 and this is the least number to have this property.


MAPLE

with(numtheory): P:=proc(q) local k, n; for n from 1 to q do
for k from 1 to q do if (sigma(n*(sigma(k)k))n*(sigma(k)k))/n=k
then print(k); break; fi; od; od; end: P(10^6);


MATHEMATICA

s[n_] := DivisorSigma[1, n]n; a[n_] := Module[{k=2}, While[k != s[n*s[k]]/n, k++]; k]; Array[a, 52] (* Amiram Eldar, Nov 06 2018 *)


PROG

(PARI) f(n, k) = {my(sk = sigma(k)k); iferr((sigma(n*sk)n*sk)/n, E, 0); }
a(n) = {my(k=1); while (k != f(n, k), k++); k; } \\ Michel Marcus, Nov 06 2018


CROSSREFS

Cf. A000040, A000043, A000203, A206708.
Sequence in context: A002566 A087998 A096823 * A007253 A096897 A063601
Adjacent sequences: A321325 A321326 A321327 * A321329 A321330 A321331


KEYWORD

nonn


AUTHOR

Paolo P. Lava, Nov 05 2018


STATUS

approved



