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

Author

Michel Lagneau, May 02 2016

Keywords

• nonn

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).

Examples

			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.
		

Links

• Michel Lagneau, Illustration

Crossrefs

Cf. A040976, A173960.

Programs

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 *)