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%I #7 May 04 2016 19:45:27
%S 9,14,20,28,36,56,67,94,124,155,173,192,213,230,253,344,395,475,504,
%T 534,596,725,759,795,1230,1359,1449,1549,1596,1647,1688,1745,1798,
%U 2005,2119,2164,2335,2395,2457,2759,2885,2952,3340,3627,3696,3835,3909,3987,4438
%N Numbers k such that prime(k) - 2 is the average of four consecutive odd squares.
%C The numbers prime(k)- 2 are a subsequence of A173960 (averages of four consecutive odd squares, or numbers of form 4*m^2+8*m+9), and also subsequence of A040976 (numbers prime(n) - 2). So, a(n) are the indices k such prime(k) are of the form 4*m^2+8*m+11 with the corresponding m = {1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 15, 16, 17, 18, 19, 23, 25, 28,...}.
%C The sequence A173960 and the subsequence prime(a(n)) - 2 appear in a diagonal straight line in the Ulam spiral (see the illustration).
%H Michel Lagneau, <a href="/A272527/a272527.pdf">Illustration</a>
%e a(1) = 9 because prime(9) - 2 = 23 - 2 = 21, and (1^2 + 3^2 + 5^2 + 7^2)/4 = 21;
%e a(2) = 14 because prime(14) - 2 = 43 - 2 = 41, and (3^2 + 5^2 + 7^2 + 9^2)/4 = 41.
%p for n from 9 to 1000 do:
%p p:=ithprime(n)-2:
%p for m from 1 by 2 to p do:
%p s:=(m^2+(m+2)^2+(m+4)^2+(m+6)^2)/4:
%p if s=p then printf(`%d, `,n):else fi:
%p od:
%p od:
%t PrimePi@ Select[(#^2 + (# + 2)^2 + (# + 4)^2 + (# + 6)^2)/4 &@ Range@ 210 + 2, PrimeQ] (* _Michael De Vlieger_, May 02 2016 *)
%Y Cf. A040976, A173960.
%K nonn
%O 1,1
%A _Michel Lagneau_, May 02 2016
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