A164722 Numbers whose sum of distinct prime factors is a square.

1, 14, 28, 39, 46, 55, 56, 66, 92, 94, 98, 112, 117, 132, 155, 158, 183, 184, 186, 188, 196, 198, 203, 224, 255, 264, 275, 290, 291, 295, 299, 316, 323, 334, 351, 354, 368, 372, 376, 392, 396, 446, 448, 455, 506, 507, 528, 546, 549, 558, 579, 580, 583, 594

Offset: 1

Author

Jonathan Vos Post, Aug 23 2009

Keywords

• easy
• nonn

Comments

This is to A008472 as A051448 is to A001414. It does seem that for any given k there should be a maximum n such that the sum of the prime factors of n = k^2, and a (perhaps different) maximum n such that the sum of distinct prime factors on n = k^2.

If k >= 3 and p = k^2 - 2 is prime (see A028870) then 2 * p is the term. - Marius A. Burtea, Jun 12 2019

Examples

			a(7) = 66 because 66 = 2 * 3 * 11 has sum of distinct prime factors 2 + 3 + 11 = 16 = 4^2. 8748 = 2^2 * 3^7 is the largest number whose prime factors (with multiplicity) add to 25 = 5^2, but it is not in this sequence because the sum of distinct prime factors of 8748 is 2 + 3 = 5, which is not a square.
		

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..14587 (terms up to 10^6)

Crossrefs

Cf. A000290, A001414, A008472, A028870, A051448.

Programs

Magma

[n:n in [1..600]| IsPower(&+PrimeDivisors(n), 2)]; // Marius A. Burtea, Jun 12 2019

Mathematica

Select[Range[600],IntegerQ[Sqrt[Total[Transpose[FactorInteger[#]] [[1]]]]]&] (* Harvey P. Dale, Mar 05 2014 *)

PARI

isOK(n) = local(fac, i); fac = factor(n); issquare(sum(i=1, matsize(fac)[1], fac[i, 1])); \\ Michel Marcus, Mar 19 2013

Formula

{n such that A008472(n) = k^2 for k an integer}.

{n such that A008472(n) is in A000290}.

Extensions

More terms (including missing terms 56, 183, and 196) from Jon E. Schoenfield, May 27 2010