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

Author

Gus Wiseman, Jan 17 2019

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. 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).

Examples

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

Links

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

Crossrefs

Complement of A323527.

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

Programs

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[#]}]]]&]

PARI

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