A320323 - OEIS

%I #7 Oct 11 2018 10:09:13

%S 7,9,19,23,25,27,49,53,81,97,103,121,125,131,151,161,169,225,227,243,

%T 289,311,343,361,419,529,541,625,661,679,691,719,729,827,841,961,1009,

%U 1089,1127,1159,1183,1193,1321,1331,1369,1427,1543,1589,1619,1681,1849

%N 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.

%C 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.

%e The terms together with their corresponding multiset multisystems (A302242):

%e     7: {{1,1}}

%e     9: {{1},{1}}

%e    19: {{1,1,1}}

%e    23: {{2,2}}

%e    25: {{2},{2}}

%e    27: {{1},{1},{1}}

%e    49: {{1,1},{1,1}}

%e    53: {{1,1,1,1}}

%e    81: {{1},{1},{1},{1}}

%e    97: {{3,3}}

%e   103: {{2,2,2}}

%e   121: {{3},{3}}

%e   125: {{2},{2},{2}}

%e   131: {{1,1,1,1,1}}

%e   151: {{1,1,2,2}}

%e   161: {{1,1},{2,2}}

%e   169: {{1,2},{1,2}}

%e   225: {{1},{1},{2},{2}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],And[GCD@@FactorInteger[Times@@primeMS[#]][[All,2]]>1,SameQ@@PrimeOmega/@primeMS[#]]&]

%o (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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 10 2018