A320323 Numbers whose product of prime indices (A003963) is a perfect power and where each prime index has the same number of prime factors, counted with multiplicity.

7, 9, 19, 23, 25, 27, 49, 53, 81, 97, 103, 121, 125, 131, 151, 161, 169, 225, 227, 243, 289, 311, 343, 361, 419, 529, 541, 625, 661, 679, 691, 719, 729, 827, 841, 961, 1009, 1089, 1127, 1159, 1183, 1193, 1321, 1331, 1369, 1427, 1543, 1589, 1619, 1681, 1849

Offset: 1

Author

Gus Wiseman, Oct 10 2018

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The terms together with their corresponding multiset multisystems (A302242):
    7: {{1,1}}
    9: {{1},{1}}
   19: {{1,1,1}}
   23: {{2,2}}
   25: {{2},{2}}
   27: {{1},{1},{1}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   81: {{1},{1},{1},{1}}
   97: {{3,3}}
  103: {{2,2,2}}
  121: {{3},{3}}
  125: {{2},{2},{2}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  225: {{1},{1},{2},{2}}
		

Crossrefs

Cf. A000720, A001222, A003963, A056239, A064573, A112798, A302242, A305551, A306017, A319056, A319066, A319071, A320324, A320325.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[GCD@@FactorInteger[Times@@primeMS[#]][[All,2]]>1,SameQ@@PrimeOmega/@primeMS[#]]&]

PARI

is(n) = my (f=factor(n), pi=apply(primepi, f[,1]~)); #Set(apply(bigomega, pi))==1 && ispower(prod(i=1, #pi, pi[i]^f[i,2])) \\ Rémy Sigrist, Oct 11 2018