A165346 Numbers such that the sum of the distinct prime factors is a fourth power.

1, 39, 55, 66, 117, 132, 158, 198, 264, 275, 316, 351, 396, 507, 528, 594, 605, 632, 726, 792, 1053, 1056, 1095, 1188, 1255, 1264, 1375, 1452, 1491, 1506, 1521, 1584, 1782, 2112, 2130, 2178, 2211, 2376, 2528, 2904, 3012, 3025, 3111, 3159, 3168, 3285, 3363

Offset: 1

Author

Jonathan Vos Post, Sep 15 2009

Keywords

• nonn

Comments

This is the 4th row of the infinite array A(k,n) = n-th positive integer such that the sum of the distinct prime factors is of the form j^k for integers j, k. The 2nd row is A164722 (hence the current sequence is a proper subset of A164722). The 3rd row is A164788. The smallest integers whose sum of distinct prime factors is 4^4 are {1255, 1506, 3012, ...}. The smallest integers whose sum of distinct prime factors is 5^4 are {9255, 21455, ...}. The smallest integers whose sum of distinct prime factors is 6^4 are {6455, 7746, ...}. The smallest integers whose sum of distinct prime factors is 7^4 are {4798, 9596, ...}.

Examples

			a(2) = 39, because 39 = 3*13, and 3+13 = 16 = 2^4.
a(7) = 158, because 158 = 2*79, and 2+79 = 81 = 3^4.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000583, A008472, A164722, A164788.

Programs

Maple

A008472 := proc(n) add( p, p = numtheory[factorset](n)) ; end: isA000583 := proc(n) iroot(n,4,'exct') ; exct ; end: A165346 := proc(n) if n = 1 then 1; else for a from procname(n-1)+1 do if isA000583(A008472(a)) then RETURN(a); fi; od: fi; end: seq(A165346(n),n=1..80) ; # R. J. Mathar, Sep 20 2009

Mathematica

a165346[n_] := Select[Range@n, IntegerQ[Power[Plus @@ Transpose[FactorInteger[#]][[1]], 1/4]] &]; a165346[3400] (* Michael De Vlieger, Jan 06 2015 *)

PARI

isok(n) = my(f=factor(n)); ispower(vecsum(f[,1]),4); \\ Michel Marcus, Jan 06 2015

Formula

{n such that A008472(n) = k^4 for k an integer}.

{n such that A008472(n) is in A000583}.

Extensions

More terms from R. J. Mathar, Sep 20 2009