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