A370139
Primes p such that the sums of three, five, and seven consecutive primes starting with p are prime.
Original entry on oeis.org
19, 29, 31, 53, 79, 379, 401, 839, 883, 1301, 1409, 1951, 1973, 2113, 2683, 2791, 2833, 3407, 3613, 3793, 3823, 4441, 4751, 4831, 5623, 5827, 6133, 6329, 7187, 7237, 7703, 8527, 9173, 10103, 10853, 11317, 12277, 13163, 13933, 14159, 14827, 15241, 15667
Offset: 1
379 is in the sequence because the seven consecutive primes starting with 379 are 379, 383, 389, 397, 401, 409, and 419, and (379+383+389)=1151, and (379+383+389+397+401)=1949, and (379+383+389+397+401+409+419)=2777, and 1151 and 1949 and 2777 are all primes.
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Select[Partition[Prime[Range[5000]],7,1],AllTrue[{Total[Take[#,3]],Total[Take[#,5]],Total[#]},PrimeQ]&][[;;,1]]
A187762
a(n) = smallest final prime in a chain of 2n+1 consecutive primes such that sum of the last 1, 3, 5, ..., 2n+1 terms in the sequence is also a prime.
Original entry on oeis.org
2, 11, 17, 47, 71, 157, 157, 167, 203569, 203569, 2803083484951
Offset: 0
Consider the chain of following consecutive prime numbers 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157
Take the sum of an odd number of primes out of this sequence starting at the end:
S(1) = 157
S(3) = 157 + 151 + 149 = 457
S(5) = 157 + 151 + 149 + 139 + 137 = 733
S(7) = 157 + 151 + 149 + 139 + 137 + 131 +127 = 991
S(9) = 157 + 151 + 149 + 139 + 137 + 131 +127 + 113 + 109 = 1213
S(11) = 157 + 151 + 149 + 139 + 137 + 131 +127 + 113 + 109 + 107 + 103 = 1423
All of these are prime numbers.
Currently a(10) is the last known term, a chain of 21 primes found after searching up to 4*10^13. The 21 consecutive primes are 2803083484321, 2803083484343, 2803083484349, 2803083484363, 2803083484391, 2803083484429, 2803083484499, 2803083484507, 2803083484633, 2803083484637, 2803083484639, 2803083484673, 2803083484697, 2803083484703, 2803083484763, 2803083484777, 2803083484781, 2803083484819, 2803083484921, 2803083484937, 2803083484951, where the sums S(21), S(19), S(17), S(15) . . . . to S(1): 58864753177133, 53258586208469, 47652419239757, 42046252270937, 36440085301931, 30833918332661, 25227751363349, 19621584393949, 14015417424409, 8409250454809, 2803083484951 respectively are also primes.
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(* This program is not convenient for n > 9 *) run[m_, n_] := Prime /@ Range[m + 2n, m, -1]; ok[ru_List] := (test = True; For[k = 1, k <= Length[ru], k = k+2, s = Total[ru[[1 ;; k]]]; If[! PrimeQ[s], test = False; Break[]]]; test); a[n_] := a[n] = Catch[For[m = 1, m <= 10^5, m++, r = run[m, n]; If[ok[r ], Throw[r[[1]]]]]]; Table[Print[a[n]]; a[n], {n, 0, 9}] (* Jean-François Alcover, Jan 08 2013 *)
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