A213386 -id:A213386 - cp's OEIS frontend

A213420 Smallest number k such that the sum of prime factors of k (counted with multiplicity) is n times a square > 1.

4, 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, 2587

Offset: 1

Michel Lagneau, Jun 11 2012

nonn

Smallest k such that sopfr(k) = n*q^2.

a(n) = A213386(n), except for n = 1, 105, 173, 213, 227, 287, …

			a(105) = 3764 because 3764 = 2^2 * 941 and  the sum of prime factors  (counted with multiplicity) is 4 + 941 = 945 = 105*9 where 9 is a square.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A213386, A212401.

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

A211144 Smallest number k such that the sum of the distinct prime divisors of k equals n times a nontrivial integer power.

Original entry on oeis.org

14, 15, 35, 39, 51, 95, 115, 87, 155, 111, 123, 215, 235, 159, 371, 183, 302, 335, 219, 471, 395, 415, 267, 623, 291, 303, 482, 327, 339, 791, 554, 1255, 635, 655, 411, 695, 662, 447, 698, 471, 734, 815, 835, 519, 1211, 543, 842, 1895, 579, 591, 914, 2167

Offset: 1

Michel Lagneau, Jun 27 2012

nonn

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

a(n) = A213386(n) except for n = 20, 32, 48, ...

			a(20) = 471 = 3*157, since the sum of the distinct prime divisors is 160 = 20*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008472, A213386.

with (numtheory):
sopf:= proc(n) option remember;
add(i, i=factorset(n))
end:
a:= proc(n) local k, q;
      for k while irem(sopf(k), n, 'q')>0 or
      igcd (map(i->i[2], ifactors(q)[2])[])<2 do od; k
    end:
seq (a(n), n=1..100);

A213401 Smallest number k such that the sum of divisors of k equals n times a square > 1.

Original entry on oeis.org

3, 7, 6, 22, 19, 14, 12, 21, 22, 27, 43, 33, 198, 28, 24, 66, 67, 30, 98, 57, 44, 197, 367, 42, 343, 63, 85, 91, 463, 54, 48, 93, 86, 202, 76, 66, 511, 111, 99, 120, 163, 60, 1285, 129, 88, 274, 751, 105, 364, 199, 134, 198, 211, 102, 763, 84, 147, 346, 1765

Offset: 1

Michel Lagneau, Jun 10 2012

nonn

Smallest k such that sigma(k) = n*q^2.

			a(8) = 21 because the sum of the divisors of 21 is 1 + 3 + 7 +21 = 32 = 8*4 where 4 is a square.

Michel Lagneau, Table of n, a(n) for n = 1..1500

Cf. A000203, A090688, A213386.

a:= proc(n) local k, p;
for k from 1 while irem(sigma(k), n, 'p')>0 or
sqrt(p)<>floor(sqrt(p)) or p=1 do od; k
end:
seq (a(n), n=1..100);

snk[n_]:=Module[{k=2,c},c=Sqrt[DivisorSigma[1,k]/n];While[!IntegerQ[c] || c==1,k++;c=Sqrt[DivisorSigma[1,k]/n]];k]; Array[snk,60] (* Harvey P. Dale, Aug 27 2013 *)

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

A213420 Smallest number k such that the sum of prime factors of k (counted with multiplicity) is n times a square > 1.

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Maple

A211144 Smallest number k such that the sum of the distinct prime divisors of k equals n times a nontrivial integer power.

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Maple

A213401 Smallest number k such that the sum of divisors of k equals n times a square > 1.

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Maple

Mathematica

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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