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%I #18 Sep 20 2023 16:49:11
%S 36,41,60,72,83,90,100,112,119,120,138,143,152,180,187,197,199,204,
%T 210,221,223,228,240,251,258,276,281,287,300,304,311,323,330,340,371,
%U 372,384,390,395,401,408,410,434,439,456,462,473,480,491,492,508,510,533
%N Integers expressible as the sum of (at least two) consecutive primes in at least 2 ways.
%H David A. Corneth, <a href="/A067372/b067372.txt">Table of n, a(n) for n = 1..10841</a> (terms <= 10^5, first 1000 terms from Donovan Johnson)
%H P. De Geest, <a href="http://www.worldofnumbers.com/em122.htm">WONplate 122</a>
%H C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_046.htm">Puzzle 46</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%F A084143(a(n)) > 1. - _Ray Chandler_, Sep 20 2023
%e 36 = (17 + 19) = (5 + 7 + 11 + 13) or (#2,17) (#4,5).
%t m=5!; lst={}; Do[p=Prime[a]; Do[p+=Prime[b]; If[p<Prime[m]*3+8,AppendTo[lst,p]],{b,a+1,m,1}],{a,m}]; lst1=Sort[lst]; lst={}; Do[If[lst1[[n]]==lst1[[n+1]],AppendTo[lst,lst1[[n]]]],{n,Length[lst1]-1}]; Union[lst] (* _Vladimir Joseph Stephan Orlovsky_, Aug 15 2009 *)
%o (PARI) upto(n) = {my(s = 0, pr = List([0]), l = List(), res = List()); forprime(p = 2, n + 100, s+=p; listput(pr, s) ); for(i = 3, #pr, for(j = 2, i-1, if(pr[i] - pr[i-j] <= n, listput(l, pr[i] - pr[i-j]) , next(2) ) ) ); listsort(l); for(i = 2, #l, if(l[i-1] == l[i], listput(res, l[i]) ) ); Set(res); } \\ _David A. Corneth_, Aug 22 2019
%Y Cf. A050936, A067372-A067381, A054997.
%K nonn,easy
%O 1,1
%A _Patrick De Geest_, Feb 04 2002
%E Offset corrected by _Donovan Johnson_, Nov 14 2013
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