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

1,2

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

Links

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

Formula

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

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

Example

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.

Mathematica

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

Prog

(PARI) isOK(n) = local(fac, i); fac = factor(n); issquare(sum(i=1, matsize(fac)[1], fac[i, 1])); \\ Michel Marcus, Mar 19 2013
(Magma) [n:n in [1..600]| IsPower(&+PrimeDivisors(n), 2)]; // Marius A. Burtea, Jun 12 2019

Crossrefs

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

Sequence in context: A143204 A276525 A118904 * A162020 A230310 A044854

Adjacent sequences: A164719 A164720 A164721 * A164723 A164724 A164725

Keyword

easy,nonn

Author

Jonathan Vos Post, Aug 23 2009

Extensions

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

Status

approved