A320323 - OEIS

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

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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.

LINKS

Table of n, a(n) for n=1..51.

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

CROSSREFS

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

Sequence in context: A138749 A320700 A053803 * A032791 A046257 A074343

Adjacent sequences: A320320 A320321 A320322 * A320324 A320325 A320326

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 10 2018

STATUS

approved