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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.
2
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
1,1
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.
EXAMPLE
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}}
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[#]]&]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 10 2018
STATUS
approved