A284229 a(n) is the least k such that A073802(k) = n.

1, 10, 12, 6, 336, 24, 5952, 168, 792, 496, 666624, 270, 10924032, 6720, 7344, 120, 3757637632, 4284, 45091651584, 2160, 79488, 1820672, 11544784011264, 672, 298080, 29331456, 106200, 13440, 53620880789471232, 10080, 1501384662105194496, 6552, 7022592, 7515275264

Offset: 1

Author

Paolo P. Lava, Mar 23 2017

Keywords

• nonn

Comments

Composite numbers that have just 1 as divisor that satisfies the condition for which sigma(k) / d_i is an integer are the Duffinian numbers (A003624).

Alternative definition: Least k such that tau(gcd(k,sigma(k))) = n. - Giovanni Resta, Mar 23 2017

Examples

			The divisors of 12 are 1, 2, 3, 4, 6, 12 and sigma(12) = 28. Then:
1) 28 / 1 = 28;
2) 28 / 2 = 14;
3) 28 / 4 = 7;
and 12 is the least number to have this property. Therefore a(3) = 12.
		

Crossrefs

Cf. A000203, A003624, A073802.

Programs

Maple

with(numtheory): P:=proc(q) local k,n; for k from 1 to q do
for n from 1 to q do if tau(gcd(n,sigma(n)))=k then
print(n); break; fi; od; od; end: P(10^9);

Mathematica

TakeWhile[#, # > 0 &] &@ Table[If[KeyExistsQ[#, n], First@ Lookup[#, n], -1], {n, Max@ Keys@ #}] &@ KeySort@ PositionIndex@ Table[DivisorSum[k, 1 &, IntegerQ[DivisorSigma[1, k]/#] &], {k, 10^6}] (* per Name, Version 10, or *)
TakeWhile[#, # > 0 &] &@ Table[If[KeyExistsQ[#, n], First@ Lookup[#, n], -1], {n, Max@ Keys@ #}] &@ KeySort@ PositionIndex@ Table[DivisorSigma[0, GCD[k, DivisorSigma[1, k]]], {k, 10^7}] (* faster, Version 10, Michael De Vlieger, Mar 24 2017 *)

PARI

nb(n) = my(s = sigma(n)); sumdiv(n, d, (s % d) == 0);
a(n) = k=1; while(nb(k) != n, k++); k; \\ Michel Marcus, Mar 24 2017

Extensions

a(13), a(17), a(19) and from a(22) to a(34) from Giovanni Resta, Mar 23 2017

Name proposed by Michel Marcus, Mar 24 2017