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

1,2

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, ...}.

Links

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

Formula

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

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000583, A008472, A164722, A164788.

Sequence in context: A013658 A227735 A319983 * A268083 A063480 A305026

Adjacent sequences: A165343 A165344 A165345 * A165347 A165348 A165349

Keyword

nonn

Author

Jonathan Vos Post, Sep 15 2009

Extensions

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

Status

approved