OFFSET
1,2
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 d(gcd(k,sigma(k)) = n. - Giovanni Resta, Mar 23 2017
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Mar 23 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
STATUS
approved