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A053050 a(n) = smallest integer m such that Sum_{k=1..m} prime(k) is divisible by n.

%I A053050 #36 Jun 02 2024 13:47:18
%S A053050 1,1,10,5,2,57,5,11,20,3,8,97,49,5,57,11,4,113,23,9,40,17,23,99,9,49,
%T A053050 26,5,7,57,39,11,76,13,180,119,29,23,119,11,6,305,10,17,242,23,39,119,
%U A053050 40,9,179,49,25,187,17,115,70,7,30,103,151,39,40,171,131,175,38,37,52,209,19
%N A053050 a(n) = smallest integer m such that Sum_{k=1..m} prime(k) is divisible by n.
%C A053050 It follows from a theorem of Daniel Shiu that m always exists. See A111287 for details. - _N. J. A. Sloane_, Nov 05 2005
%D A053050 Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000
%H A053050 Alois P. Heinz, <a href="/A053050/b053050.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from T. D. Noe)
%H A053050 D. K. L. Shiu, <a href="http://dx.doi.org/10.1112/S0024610799007863">Strings of Congruent Primes</a>, J. Lond. Math. Soc. 61 (2) (2000) 359-373 [<a href="http://www.ams.org/mathscinet-getitem?mr=1760689">MR1760689</a>]
%F A053050 A007504(a(n))/n = A308749(n). - _Alois P. Heinz_, Jun 21 2019
%p A053050 read transforms; M:=1000; p0:=[seq(ithprime(i),i=1..M)]; q0:=PSUM(p0); w:=[]; for n from 1 to M do p:=n; hit := 0; for i from 1 to M do if q0[i] mod p = 0 then w:=[op(w),i]; hit:=1; break; fi; od: if hit = 0 then break; fi; od: w;
%t A053050 Transpose[With[{aprs=Thread[{Range[500],Accumulate[Prime[Range[ 500]]]}]}, Flatten[Table[ Select[ aprs,Divisible[Last[#],n]&,1],{n,80}],1]]][[1]] (* _Harvey P. Dale_, Dec 14 2011 *)
%o A053050 (Haskell)
%o A053050 a053050 n = head [k | (k, x) <- zip [1..] a007504_list, mod x n == 0]
%o A053050 -- _Reinhard Zumkeller_, Oct 04 2015, Feb 10 2012
%Y A053050 Cf. A007504, A111287, A002034, A011772.
%Y A053050 Cf. A045985, A308749.
%K A053050 easy,nice,nonn
%O A053050 1,3
%A A053050 _Felice Russo_, Feb 25 2000
%E A053050 More terms from _N. J. A. Sloane_, Nov 05 2005