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A251925 Prime powers p^k (k>=2) of the form (n^2+1)/2.

%I A251925 #25 Nov 13 2023 09:20:02
%S A251925 25,841,28561,32959081,1119638521,1985636569351347658201,
%T A251925 3051519929713402294221039791281,
%U A251925 4689566069222821420312720463003656425961,183840368926047361112315395593676258316051401,17020879736268069268391497343746883355223007561030622302744641179601
%N A251925 Prime powers p^k (k>=2) of the form (n^2+1)/2.
%C A251925 The corresponding n are a subsequence of A001333; see example.
%H A251925 Paolo Xausa, <a href="/A251925/b251925.txt">Table of n, a(n) for n = 1..19</a>
%H A251925 Joerg Arndt, <a href="http://jjj.de/arctan/">Arctan relations for Pi</a> (the author's starting point for this sequence).
%e A251925 The first few terms correspond to
%e A251925 7^2 + 1 = 2 * 5^2 = 2 * 25,
%e A251925 41^2 + 1 = 2 * 29^2 = 2 * 841,
%e A251925 239^2 + 1 = 2 * 13^4 = 2 * 28561,
%e A251925 8119^2 + 1 = 2 * 5741^2 = 2 * 32959081,
%e A251925 47321^2 + 1 = 2 * 33461^2 = 2 * 1119638521,
%e A251925 63018038201^2+1 = 2 * 44560482149^2 = 2 * 1985636569351347658201.
%t A251925 With[{r=Range[100]},Select[((ChebyshevT[r,I]/I^r)^2+1)/2,!PrimeQ[#]&&PrimePowerQ[#]&]] (* _Paolo Xausa_, Nov 13 2023, after _Joerg Arndt_ *)
%o A251925 (PARI) forstep(n=1,10^9,2, t=(n^2+1)/2; if( !isprime(t) && isprimepower(t), print1(t,", ")));
%o A251925 (PARI) /* much more efficient: */
%o A251925 {b(n) = polchebyshev(n, 1, I) / I^n}
%o A251925 for(n=1,10^3,t=(b(n)^2+1)/2;if(!isprime(t)&&isprimepower(t),print1(t,", ")));
%Y A251925 Cf. A027861 (primes of the form (n^2+1)/2), A001333, A008844 (primes and composites with k=2).
%K A251925 nonn
%O A251925 1,1
%A A251925 _Joerg Arndt_, Dec 11 2014