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Numbers k such that the sum of the first k composite numbers is prime.
7

%I #16 Feb 09 2023 16:50:18

%S 5,14,17,20,35,36,37,43,47,48,53,54,63,64,68,73,74,75,86,101,106,127,

%T 142,149,154,159,208,209,214,221,231,234,250,254,258,259,272,283,302,

%U 304,329,332,346,352,374,398,417,424,439,440,445,458,471,550,551,556

%N Numbers k such that the sum of the first k composite numbers is prime.

%H Chai Wah Wu, <a href="/A053782/b053782.txt">Table of n, a(n) for n = 1..10000</a>

%t f[ n_Integer ] := Block[ {k = n + PrimePi[ n ] + 1}, While[ k - PrimePi[ k ] - 1 != n, k++ ]; k ]; s = 0; Do[ s = s + f[ n ]; If[ PrimeQ[ s ], Print[ n ] ], {n, 1, 1000} ]

%t With[{cn=Accumulate[Select[Range[1000],CompositeQ]]},Position[cn,_?PrimeQ]]// Flatten (* _Harvey P. Dale_, Feb 09 2023 *)

%o (Python)

%o from sympy import isprime

%o A053782_list, n, m, s = [], 1, 4, 4

%o while len(A053782_list) < 10000:

%o if isprime(s):

%o A053782_list.append(n)

%o m += 1

%o if isprime(m):

%o m += 1

%o n += 1

%o s += m # _Chai Wah Wu_, May 13 2018

%o (PARI) lista(nn) = {my(s = 0, nb = 0); forcomposite(c=1, nn, s += c; nb++; if (isprime(s), print1(nb, ", ")););} \\ _Michel Marcus_, May 13 2018

%Y Cf. A002808, A000040, A053872, A013919.

%K nonn

%O 1,1

%A _G. L. Honaker, Jr._, Mar 30 2000

%E More terms from _Robert G. Wilson v_, Mar 22 2001