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A272527 Numbers k such that prime(k) - 2 is the average of four consecutive odd squares. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 17 02:10 EDT 2019. Contains 325092 sequences. (Running on oeis4.)