cp's OEIS Frontend

A171555 Numbers of the form prime(n)*(prime(n)-1)/4.

%I A171555 #25 Jun 27 2024 22:26:10
%S A171555 5,39,68,203,333,410,689,915,1314,1958,2328,2525,2943,3164,4658,5513,
%T A171555 6123,7439,8145,9264,9653,13053,13514,14460,16448,18023,19113,19670,
%U A171555 21389,24414,25043,28308,30363,31064,34689,37733,39303,40100,41718,44205,46764,50288
%N A171555 Numbers of the form prime(n)*(prime(n)-1)/4.
%C A171555 The halves of even numbers of the form p(p-1)/2 for p prime.
%C A171555 Sum of the quadratic residues of primes of the form 4k + 1.  For example, a(3)=68 because 17 is the 3rd prime of the form 4k + 1 and the quadratic residues of 17 are 1, 4, 9, 16, 8, 2, 15, 13 which sum to 68.  This sum is also the sum of the quadratic nonresidues.  Cf. A230077. - _Geoffrey Critzer_, May 07 2015
%D A171555 R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.21 p. 110.
%H A171555 Alois P. Heinz, <a href="/A171555/b171555.txt">Table of n, a(n) for n = 1..10000</a>
%H A171555 Aebi, Christian, and Grant Cairns. <a href="http://arxiv.org/abs/1512.00896">Sums of Quadratic residues and nonresidues</a>, arXiv preprint arXiv:1512.00896 (2015).
%t A171555 Table[Table[Mod[a^2, p], {a, 1, (p - 1)/2}] // Total, {p,
%t A171555 Select[Prime[Range[100]], Mod[#, 4] == 1 &]}] (* _Geoffrey Critzer_, May 07 2015 *)
%t A171555 Select[(# (#-1))/4&/@Prime[Range[100]],IntegerQ] (* _Harvey P. Dale_, Dec 24 2022 *)
%o A171555 (PARI) lista(nn) = forprime(p=2, nn, if ((p % 4)==1, print1(p*(p-1)/4, ", "))); \\ _Michel Marcus_, Mar 23 2016
%Y A171555 Cf. A005098, A007742, A008837.
%Y A171555 Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.
%K A171555 nonn
%O A171555 1,1
%A A171555 _Juri-Stepan Gerasimov_, Dec 11 2009
%E A171555 Corrected (16448 inserted, 25043 inserted) by _R. J. Mathar_, May 22 2010