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A284229
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a(n) is the least k such that A073802(k) = n.
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0
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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
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OFFSET
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1,2
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COMMENTS
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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 d(gcd(k,sigma(k)) = n. - Giovanni Resta, Mar 23 2017
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LINKS
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(13), a(17), a(19) and from a(22) to a(34) from Giovanni Resta, Mar 23 2017
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STATUS
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approved
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