A343300 -id:A343300 - cp's OEIS frontend

A344023 Numbers of the form p_1^1 + p_2^2 + ... + p_k^k where p_1 < p_2 < ... < p_k are distinct primes.

0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 27, 28, 29, 31, 37, 41, 43, 47, 51, 52, 53, 54, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 123, 124, 126, 127, 128, 131, 136, 137, 139, 149, 151, 157, 163, 167, 171, 172, 173, 174, 176, 179, 180, 181, 191, 193, 197, 199

Offset: 1

Bernard Schott, May 07 2021

nonn

Also, ordered distinct values taken by terms of A343300.

Primes form the subsequence corresponding to k = 1.

			0 is a term  because it is the empty sum.
11 is a term because 11 = 11^1 is prime and also 11 = 2^1 + 3^2.
52 is a term because 3^1 + 7^2 = 52.
1382 is a term because 2^1 + 7^2 + 11^3 = 13^1 +37^2 = 1382.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A027748, A343300, A344030.

f(n) = my(fn=factor(n)); sum(k=1, #fn~, fn[k, 1]^k); \\ A343300
lista(nn) = my(p=precprime(nn)); select(x->(x <=p), Set(vector(p, k, f(k)))); \\ Michel Marcus, May 08 2021

A344030 Composite numbers with distinct prime factors {p1, p2, ..., pk} in ascending order where p1^1 + p2^2 + ...+ pk^k is prime.

Original entry on oeis.org

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

Giorgos Kalogeropoulos, May 07 2021

nonn

			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.

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

Cf. A027748, A343300.

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

Select[Range@1000,!PrimeQ@#&&PrimeQ@Total[(a=First/@FactorInteger[#])^Range@Length[a]]&]

isok(c) = if (!isprime(c), my(f=factor(c)); isprime(sum(k=1, #f~, f[k,1]^k))); \\ Michel Marcus, May 07 2021

cp's OEIS Frontend

A344023 Numbers of the form p_1^1 + p_2^2 + ... + p_k^k where p_1 < p_2 < ... < p_k are distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI