cp's OEIS Frontend

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

%I A055514 #46 Feb 22 2024 20:01:38
%S A055514 10,39,155,371,10225245560,2935561623745,454539357304421,
%T A055514 7228559051256366318,1390718713078158117206
%N A055514 Composite numbers that are the sum of consecutive prime numbers and are divisible by the first and last of these primes.
%C A055514 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.
%C A055514 Problem proposed by Carlos Rivera, who found the first 4 terms.
%C A055514 No more terms below 10^22. - _Michael Beight_, Jul 22 2012
%C A055514 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
%H A055514 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_098.htm">Puzzle</a>
%e A055514 503 + 509 + 521 + ... + 508213 = 10225245560, which is divisible by 503 and 508213. - _Manuel Valdivia_, Nov 17 2011
%e A055514 From _Michael Beight_, Jul 22 2012: (Start)
%e A055514 a(8) = 7228559051256366318 = 73 + ... + 18281691653;
%e A055514 a(9) = 1390718713078158117206 = 370794889 + ... + 267902967061. (End)
%t A055514 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. *)
%t A055514 (* _Harvey P. Dale_, Oct 22 2022 *)
%o A055514 (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
%Y A055514 Subsequence of A050936.
%Y A055514 Cf. A055233.
%K A055514 nice,nonn
%O A055514 1,1
%A A055514 _Jud McCranie_, Jul 03 2000
%E A055514 a(7) from _Donovan Johnson_, Jun 19 2008
%E A055514 a(8) and a(9) from _Michael Beight_, Jul 22 2012