This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A317510 #43 Jun 22 2025 23:36:54
%S A317510 7,15,31,39,55,71,111,119,151,175,231,239,255,311,319,335,375,391,479,
%T A317510 559,575,591,655,679,791,855,871,879,911,959,991,1015,1079,1215,1271,
%U A317510 1351,1359,1375,1399,1471,1631,1639,1719,1879,1919,1935,1975,1999,2015,2079,2111,2135,2311,2415,2519,2535,2575,2631
%N A317510 Numbers (4p+1)/3 where p is a Sophie Germain prime p > 3.
%C A317510 It appears that this is a subsequence of A179882.
%C A317510 Define a set of consecutive positive odd numbers {1,......, (A077065(n)-1)} with n >= 3 and skip the number A077065(n)/2. Then the contraharmonic mean of that set gives the sequence. For example: ContraharmonicMean[{1, 3, 7, 9}] = 7, ContraharmonicMean[{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}] = 15, ContraharmonicMean[{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37,39, 41, 43, 45}] = 31. - _Hilko Koning_, Aug 28 2018
%C A317510 Let p be a Sophie Germain prime and define h = 2p + 1 a safe prime. Then the contraharmonic mean of the totatives of h is given by: CHM(h) = (Sum_{1 <= k < h, gcd(k, h) = 1} k^2) / (Sum_{1 <= k < h, gcd(k, h) = 1} k). Since h is prime, all integers k = 1, 2, ... , h - 1 are coprime to h. Then, CHM(h) = ((h - 1) * h * (2h-1) / 6) / ((h - 1) * h / 2). Thus CHM = (2h-1) / 3 = (4p+1) / 3. These values are integers precisely when p == 2 mod 3, which holds for all Sophie Germain primes, p >= 5. The resulting values for the sequence A317510, which is therefore a subsequence of A179882. - _Hilko Koning_, Jun 17 2025
%H A317510 Muniru A Asiru, <a href="/A317510/b317510.txt">Table of n, a(n) for n = 1..5000</a>
%t A317510 lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (4 p + 1)/3]], {p, 5, 2*10^3}]; lst
%t A317510 4 (Select[Prime@Range[3, 300], PrimeQ[2 # + 1] &] + 1)/3 - 1 (* _Robert G. Wilson v_, Jul 30 2018 *)
%o A317510 (PARI) lista(nn) = {forprime (p=5, nn, if (isprime(2*p+1), print1((4*p+1)/3, ", ")););} \\ _Michel Marcus_, Aug 27 2018
%o A317510 (GAP) a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a,(4*p+1)/3); fi; od; a; # _Muniru A Asiru_, Aug 28 2018
%Y A317510 Cf. A005384, A179882.
%Y A317510 Subsequence of A004767, and of A004771.
%K A317510 nonn
%O A317510 1,1
%A A317510 _Hilko Koning_, Jul 30 2018