A213420 -id:A213420 - cp's OEIS frontend

A213386 Smallest number k such that the sum of the distinct prime divisors of k equals n times a square > 1.

14, 15, 35, 39, 51, 95, 115, 87, 155, 111, 123, 215, 235, 159, 371, 183, 302, 335, 219, 511, 395, 415, 267, 623, 291, 303, 482, 327, 339, 791, 554, 1415, 635, 655, 411, 695, 662, 447, 698, 471, 734, 815, 835, 519, 1211, 543, 842, 1991, 579, 591, 914, 2167

Offset: 1

Michel Lagneau, Jun 10 2012

nonn

Smallest k such that sopf(k) = n*q where q is a square.

			a(55) = 2631 because 2631 = 3*877 and 3 + 877 = 880 = 55*16 where 16 is a square.

Michel Lagneau, Table of n, a(n) for n = 1..1000

Cf. A008492, A213420.

with (numtheory):
sopf:= proc(n) option remember;
add(i, i=factorset(n))
end:
a:= proc(n) local k, p;
for k from 2 while irem(sopf(k), n, 'p')>0 or
sqrt(p)<>floor(sqrt(p)) or p=1 do od; k
end:
seq (a(n), n=1..100);

A213931 Smallest number k such that the sum of divisors of k equals n times a nontrivial integer power.

Original entry on oeis.org

3, 7, 6, 21, 19, 14, 12, 21, 22, 27, 43, 33, 63, 28, 24, 66, 67, 30, 98, 57, 44, 129, 367, 42, 199, 63, 85, 84, 463, 54, 48, 93, 86, 201, 76, 66, 219, 111, 99, 120, 163, 60, 1285, 129, 88, 274, 751, 105, 156, 199, 134, 198, 211, 102, 327, 84, 147, 346, 1765

Offset: 1

Michel Lagneau, Jun 25 2012

nonn

Smallest k such that sigma(k) = n * m^q where m, q >= 2.

			a(34) = 201 because sigma(201) = 272 = 34*2^3.

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A000203, A001597, A213401, A213386, A213420.

with(numtheory):
a:= proc(n) local k, q;
      for k while irem(sigma(k), n, 'q')>0 or
      igcd(map(i->i[2], ifactors(q)[2])[])<2 do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 26 2012

a[n_] := Module[{k, q, r}, For[k = 1, {q, r} = QuotientRemainder[ DivisorSigma[1, k], n]; r>0 || GCD @@ FactorInteger[q][[All, 2]]<2, k++]; k];
Array[a, 100] (* Jean-François Alcover, Nov 21 2020, after Alois P. Heinz *)

a(n)=my(k);while(sigma(k++)%n || !ispower(sigma(k)/n), ); k \\ Charles R Greathouse IV, Jun 26 2012

cp's OEIS Frontend

A213386 Smallest number k such that the sum of the distinct prime divisors of k equals n times a square > 1.

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