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

%I A284229 #33 Nov 21 2024 22:25:53
%S A284229 1,10,12,6,336,24,5952,168,792,496,666624,270,10924032,6720,7344,120,
%T A284229 3757637632,4284,45091651584,2160,79488,1820672,11544784011264,672,
%U A284229 298080,29331456,106200,13440,53620880789471232,10080,1501384662105194496,6552,7022592,7515275264
%N A284229 a(n) is the least k such that A073802(k) = n.
%C A284229 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).
%C A284229 Alternative definition: Least k such that tau(gcd(k,sigma(k))) = n. - _Giovanni Resta_, Mar 23 2017
%e A284229 The divisors of 12 are 1, 2, 3, 4, 6, 12 and sigma(12) = 28. Then:
%e A284229 1) 28 / 1 = 28;
%e A284229 2) 28 / 2 = 14;
%e A284229 3) 28 / 4 = 7;
%e A284229 and 12 is the least number to have this property. Therefore a(3) = 12.
%p A284229 with(numtheory): P:=proc(q) local k,n; for k from 1 to q do
%p A284229 for n from 1 to q do if tau(gcd(n,sigma(n)))=k then
%p A284229 print(n); break; fi; od; od; end: P(10^9);
%t A284229 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 *)
%t A284229 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 *)
%o A284229 (PARI) nb(n) = my(s = sigma(n)); sumdiv(n, d, (s % d) == 0);
%o A284229 a(n) = k=1; while(nb(k) != n, k++); k; \\ _Michel Marcus_, Mar 24 2017
%Y A284229 Cf. A000203, A003624, A073802.
%K A284229 nonn
%O A284229 1,2
%A A284229 _Paolo P. Lava_, Mar 23 2017
%E A284229 a(13), a(17), a(19) and from a(22) to a(34) from _Giovanni Resta_, Mar 23 2017
%E A284229 Name proposed by _Michel Marcus_, Mar 24 2017