A366914 Numbers expressible as the sum of their distinct prime factors raised to a natural exponent.
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 42, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 102, 103, 107, 109, 113, 121, 125, 127, 128, 131, 132, 137, 139, 140, 149, 150, 151, 157, 163, 167
Offset: 1
Keywords
Examples
30 is a term because its distinct prime factors are 2, 3 and 5, and 30 = 2^1 + 3^1 + 5^2 = 2^4 + 3^2 + 5^1. 42 is a term because its distinct prime factors are 2, 3 and 7, and 42 = 2^3 + 3^3 + 7^1 = 2^5 + 3^1 + 7^1. 60 is a term because its distinct prime factors are 2, 3 and 5, and 60 = 2^5 + 3^1 + 5^2.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
filter:= proc(n) local P,S,p,i; P:= numtheory:-factorset(n); S:= mul(add(x^(p^i),i=1..floor(log[p](n))),p=P); coeff(S,x,n) > 0 end proc: select(filter, [$1..1000]); # Robert Israel, Dec 27 2023
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PARI
isok(n)={my(f=factor(n)[,1], m=n-vecsum(f)); polcoef(prod(k=1, #f, my(c=f[k]); sum(j=1, logint(m+c, c), x^(c^j-c)) , 1 + O(x*x^m)), m)} \\ Andrew Howroyd, Oct 27 2023
Extensions
Terms a(43) and beyond from Andrew Howroyd, Oct 27 2023
Comments