A055514 Composite numbers that are the sum of consecutive prime numbers and are divisible by the first and last of these primes.

10, 39, 155, 371, 10225245560, 2935561623745, 454539357304421, 7228559051256366318, 1390718713078158117206

Offset: 1

Author

Jud McCranie, Jul 03 2000

Keywords

• nice
• nonn

Comments

Composite n such that n = p_1 + p_2 + ... + p_k where the p_i are consecutive primes and n is divisible by p_1 and p_k.

Problem proposed by Carlos Rivera, who found the first 4 terms.

No more terms below 10^22. - Michael Beight, Jul 22 2012

In subsequence A055233 the first and last term of the sum must also be its smallest and largest prime factor. Therefore a(5) (cf. first EXAMPLE) is not in that sequence, since it has smaller factors 2^3*5. - M. F. Hasler, Nov 21 2021

Examples

			503 + 509 + 521 + ... + 508213 = 10225245560, which is divisible by 503 and 508213. - _Manuel Valdivia_, Nov 17 2011
From _Michael Beight_, Jul 22 2012: (Start)
a(8) = 7228559051256366318 = 73 + ... + 18281691653;
a(9) = 1390718713078158117206 = 370794889 + ... + 267902967061. (End)
		

Links

• C. Rivera, Puzzle

Crossrefs

Subsequence of A050936.

Cf. A055233.

Programs

Mathematica

Module[{nn=200},Table[Total/@Select[Partition[Prime[Range[10000]],n,1],scpQ],{n,2,nn}]]//Flatten (* The program generates the first four terms of the sequence. *)
(* Harvey P. Dale, Oct 22 2022 *)

PARI

S=vector(N=50000); s=0; i=1; forprime(p=2,oo, S[i++]=s+=p; for(j=1,i-2, (s-S[j])%p || (s-S[j])%prime(j)|| print1(s-S[j]",")|| break)) \\ gives a(1..5), but too slow to go beyond. - M. F. Hasler, Nov 21 2021

Extensions

a(7) from Donovan Johnson, Jun 19 2008

a(8) and a(9) from Michael Beight, Jul 22 2012