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

1,1

Comments

Each prime factor must appear exactly once in the sum.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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.

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

Prog

(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

Crossrefs

Sequence in context: A326848 A328957 A030230 * A089352 A086486 A071139

Adjacent sequences: A366911 A366912 A366913 * A366915 A366916 A366917

Keyword

nonn

Author

Tanmaya Mohanty, Oct 27 2023

Extensions

Terms a(43) and beyond from Andrew Howroyd, Oct 27 2023

Status

approved