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A361419 Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.

%I A361419 #16 Mar 12 2023 04:08:37
%S A361419 0,6,7,9,11,18,32,44,56,62,72,82,94,96,98,102,104,110,116,122,132,136,
%T A361419 138,146,150,152,178,180,182,210,222,226,230,236,238,242,248,252,264,
%U A361419 272,284,292,296,304,322,332,338,342,350,356,360,374,382,390,392,404
%N A361419 Numbers k such that there is a unique number m for which the sum of the aliquot infinitary divisors of m (A126168) is k.
%C A361419 Numbers k such that A331973(k) = 1.
%H A361419 Amiram Eldar, <a href="/A361419/b361419.txt">Table of n, a(n) for n = 1..10000</a>
%F A361419 a(n) = A126168(A361420(n)).
%t A361419 f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; is[1] = 0; is[n_] := Times @@ f @@@ FactorInteger[n] - n;
%t A361419 seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = is[k] + 1; If[i <= max, v[[i]]++], {k, 1, max^2}]; -1 + Position[v, 1] // Flatten];
%t A361419 seq[500]
%o A361419 (PARI) s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
%o A361419 lista(nmax) = {my(v = vector(nmax+1)); for(k=1, nmax^2, i = s(k) + 1; if(i <= nmax+1, v[i] += 1)); for(i = 1, nmax+1, if(v[i] == 1, print1(i-1, ", "))); }
%Y A361419 Cf. A126168, A324277, A331974, A331973, A361420.
%Y A361419 Similar sequences: A057709, A357324.
%K A361419 nonn
%O A361419 1,2
%A A361419 _Amiram Eldar_, Mar 11 2023