A272527 Numbers k such that prime(k) - 2 is the average of four consecutive odd squares.

9, 14, 20, 28, 36, 56, 67, 94, 124, 155, 173, 192, 213, 230, 253, 344, 395, 475, 504, 534, 596, 725, 759, 795, 1230, 1359, 1449, 1549, 1596, 1647, 1688, 1745, 1798, 2005, 2119, 2164, 2335, 2395, 2457, 2759, 2885, 2952, 3340, 3627, 3696, 3835, 3909, 3987, 4438

Offset

1,1

Comments

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,...}.

The sequence A173960 and the subsequence prime(a(n)) - 2 appear in a diagonal straight line in the Ulam spiral (see the illustration).

Links

Table of n, a(n) for n=1..49.

Michel Lagneau, Illustration

Example

a(1) = 9 because prime(9) - 2 = 23 - 2 = 21, and (1^2 + 3^2 + 5^2 + 7^2)/4 = 21;
a(2) = 14 because prime(14) - 2 = 43 - 2 = 41, and (3^2 + 5^2 + 7^2 + 9^2)/4 = 41.

Maple

for n from 9 to 1000 do:
p:=ithprime(n)-2:
for m from 1 by 2 to p do:
  s:=(m^2+(m+2)^2+(m+4)^2+(m+6)^2)/4:
  if s=p then printf(`%d, `, n):else fi:
od:
od:

Mathematica

PrimePi@ Select[(#^2 + (# + 2)^2 + (# + 4)^2 + (# + 6)^2)/4 &@ Range@ 210 + 2, PrimeQ] (* Michael De Vlieger, May 02 2016 *)

Crossrefs

Cf. A040976, A173960.

Sequence in context: A034703 A006624 A184218 * A272308 A186778 A070552

Adjacent sequences: A272524 A272525 A272526 * A272528 A272529 A272530

Keyword

nonn

Author

Michel Lagneau, May 02 2016

Status

approved