A344030 Composite numbers with distinct prime factors {p1, p2, ..., pk} in ascending order where p1^1 + p2^2 + ...+ pk^k is prime.
4, 6, 8, 9, 12, 16, 18, 24, 25, 27, 32, 36, 48, 49, 54, 64, 72, 81, 96, 108, 121, 125, 128, 144, 162, 169, 192, 216, 243, 256, 288, 289, 324, 343, 361, 384, 390, 399, 432, 455, 465, 486, 512, 529, 570, 576, 595, 625, 627, 648, 690, 729, 768, 780, 841, 864, 903
Offset: 1
Keywords
Examples
24 has distinct prime factors {2, 3} and 2^1 + 3^2 = 11 is prime.
570 has distinct prime factors {2, 3, 5, 19} and 2^1 + 3^2 + 5^3 + 19^4 = 130457 is prime.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
filter:= proc(n) local F,i; if isprime(n) then return false fi; F:= sort(convert(numtheory:-factorset(n),list)); isprime(add(F[i]^i,i=1..nops(F))) end proc: select(filter, [$4..1000]); # Robert Israel, Apr 09 2024
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Mathematica
Select[Range@1000,!PrimeQ@#&&PrimeQ@Total[(a=First/@FactorInteger[#])^Range@Length[a]]&]
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PARI
isok(c) = if (!isprime(c), my(f=factor(c)); isprime(sum(k=1, #f~, f[k,1]^k))); \\ Michel Marcus, May 07 2021