A335306 a(n) is the smallest composite number whose sum of distinct prime divisors is prime(n).

4, 9, 6, 10, 121, 22, 210, 34, 273, 399, 58, 435, 651, 82, 777, 903, 1645, 118, 885, 1281, 142, 1065, 1533, 1659, 1335, 3115, 202, 2037, 214, 2163, 3729, 6213, 2667, 274, 2919, 298, 2235, 4917, 3297, 3423, 5845, 358, 3801, 382, 7059, 394, 6501, 7385, 8229, 454, 4683

Offset

1,1

Comments

a(n) <= prime(n)^2 for all n, the equality applies to n = 1,2,5 since 2,3,11 are the only primes which cannot be expressed as the sum of distinct smaller primes. For n other than 1,2,5, a(n) is squarefree, and corresponds to the partition (q_1, q_2,....q_k) of n into distinct primes whose product is the least possible value compared with the product of all distinct prime partitions of n. The intersection of this sequence with A261023 corresponds to primes in A133225.

a(n) >= max(4,2*prime(n)-4) with equality if and only if n = 1 or n is in A107770. - Chai Wah Wu, Jun 01 2020

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

a(7) = 10 since (2,5) is the only prime partition of 7 into distinct smaller parts, and 2*5 = 10. a(11) = 11^2 = 121 because the prime partitions of 11 into smaller parts are: (2,2,7), (2,2,2,5), (2,2,2,2,3), (3,3,5), (2,3,3,3), none of which have only distinct primes.

Mathematica

a[n_] := Block[{k = 4, p = Prime@ n}, While[PrimeQ[k] || p != Total[First /@ FactorInteger[k]], k++]; k]; Array[a, 50] (* Giovanni Resta, May 31 2020 *)

Prog

(PARI) a(n) = {my(p=prime(n)); forcomposite(k=1, p^2, if (vecsum(factor(k)[, 1]) == p, return(k)); ); } \\ Michel Marcus, May 31 2020
(Python)
from sympy import prime, primefactors
def A335306(n):
    p = prime(n)
    for m in range(max(4, 2*p-4), p**2+1):
        if sum(primefactors(m)) == p:
            return m # Chai Wah Wu, Jun 01 2020

Crossrefs

Cf. A000040, A000586, A008472, A107770, A070215, A133225, A261023.

Sequence in context: A089390 A053667 A218072 * A261023 A171095 A075065

Adjacent sequences: A335303 A335304 A335305 * A335307 A335308 A335309

Keyword

nonn

Author

David James Sycamore, May 31 2020

Extensions

More terms from Michel Marcus, May 31 2020

Status

approved