A194196 Numbers k such that the sum of the divisors of k and the sum of the distinct prime divisors of k are both a square.

1, 66, 94, 1092, 1146, 1416, 1491, 1782, 2130, 2159, 2805, 3012, 3531, 4836, 8736, 9065, 9911, 12532, 13156, 15960, 16194, 24096, 25866, 27652, 29316, 29484, 30942, 34162, 34782, 34860, 37736, 37884, 38232, 38688, 40257, 41331, 48204, 51460, 54162, 54411

Offset: 1

Author

Michel Lagneau, Aug 18 2011

Keywords

• nonn

Examples

			94 is in the sequence because the distinct prime divisors are {2,47} -> sum = 7^2, and the divisors are {1,2,47,94} -> sum = 12^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Crossrefs

Cf. A000203, A006532, A008472, A164722.

Programs

Maple

isA006532 :=proc(n) issqr(numtheory[sigma](n)) ; end proc:
A008472 := proc(n) add(d, d=numtheory[factorset](n)) ; end proc:
isA164722 :=proc(n) issqr(A008472(n)) ; end proc:
for n from 1 to 50000 do if isA006532(n) and isA164722(n) then printf("%d,",n); end if; end do; # R. J. Mathar, Aug 18 2011

PARI

isok(k) = my(f=factor(k)); issquare(sigma(f)) && issquare(vecsum(f[,1])); \\ Michel Marcus, Dec 05 2020

Formula

{A006532 intersection A164722}.