A305026 Numbers k such that sopfr(k) = tau(k)^2.

39, 55, 354, 578, 1634, 1644, 6604, 8253, 9825, 12573, 13516, 14749, 15244, 16684, 18669, 18672, 19276, 19564, 21032, 22225, 25305, 28449, 29853, 31688, 33633, 35793, 41261, 41768, 41949, 42813, 48013, 50670, 54048, 59750, 60804, 63609, 63869, 65265, 78832

Offset: 1

Author

Parker Grootenhuis, May 23 2018

Keywords

• nonn

Comments

For numbers k that satisfy the condition, tau(k) will always be even because tau(k) is odd only if k is a square, but if k is a square then sopfr(k) is even (because every prime appears with an even exponent) and thus it cannot be equal to tau(k)^2 which is odd as tau(k). - Giovanni Resta, May 24 2018

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A001414, A078511.

Programs

Mathematica

Rest@ Select[Range[10^5], Total[Times @@@ FactorInteger@ #] == DivisorSigma[0, #]^2 &] (* Michael De Vlieger, May 27 2018 *)

PARI

isok(n) = my(f=factor(n)); sum(k=1,#f~,f[k,1]*f[k,2]) == numdiv(n)^2; \\ Michel Marcus, May 24 2018

Formula

k such that A001414(k) = A000005(k)^2.