A323528 Numbers whose sum of prime indices is a perfect square.

1, 2, 7, 9, 10, 12, 16, 23, 38, 51, 53, 65, 68, 77, 78, 94, 97, 98, 99, 104, 105, 110, 125, 126, 129, 132, 135, 140, 150, 151, 162, 168, 172, 176, 178, 180, 200, 205, 216, 224, 227, 240, 246, 249, 259, 288, 298, 311, 320, 328, 332, 333, 341, 361, 370, 377, 384

Offset

1,2

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. The sum of prime indices of n is A056239(n).

Also Heinz numbers of integer partitions of perfect squares, where the Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

10 is in the sequence because 10 = 2*5 = prime(1)*prime(3) and 1 + 3 = 4 is a square.

Maple

select(k-> issqr(add(numtheory[pi](i[1])*i[2], i=ifactors(k)[2])), [$1..400])[]; # Alois P. Heinz, Jan 22 2019

Mathematica

Select[Range[100], IntegerQ[Sqrt[Sum[PrimePi[f[[1]]]*f[[2]], {f, FactorInteger[#]}]]]&]

Prog

(PARI) isok(n) = {my(f=factor(n)); issquare(sum(k=1, #f~, primepi(f[k, 1])*f[k, 2])); } \\ Michel Marcus, Jan 18 2019

Crossrefs

Complement of A323527.

Cf. A000290, A001248, A026478, A056239, A112798, A323520, A323521, A323525, A323526.

Sequence in context: A201892 A352447 A376600 * A073074 A034796 A047526

Adjacent sequences: A323525 A323526 A323527 * A323529 A323530 A323531

Keyword

nonn

Author

Gus Wiseman, Jan 17 2019

Status

approved