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

25, 841, 28561, 32959081, 1119638521, 1985636569351347658201, 3051519929713402294221039791281, 4689566069222821420312720463003656425961, 183840368926047361112315395593676258316051401, 17020879736268069268391497343746883355223007561030622302744641179601

Offset: 1

Author

Joerg Arndt, Dec 11 2014

Keywords

• nonn

Comments

The corresponding n are a subsequence of A001333; see example.

Examples

			The first few terms correspond to
7^2 + 1 = 2 * 5^2 = 2 * 25,
41^2 + 1 = 2 * 29^2 = 2 * 841,
239^2 + 1 = 2 * 13^4 = 2 * 28561,
8119^2 + 1 = 2 * 5741^2 = 2 * 32959081,
47321^2 + 1 = 2 * 33461^2 = 2 * 1119638521,
63018038201^2+1 = 2 * 44560482149^2 = 2 * 1985636569351347658201.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..19
• Joerg Arndt, Arctan relations for Pi (the author's starting point for this sequence).

Crossrefs

Cf. A027861 (primes of the form (n^2+1)/2), A001333, A008844 (primes and composites with k=2).

Programs

Mathematica

With[{r=Range[100]},Select[((ChebyshevT[r,I]/I^r)^2+1)/2,!PrimeQ[#]&&PrimePowerQ[#]&]] (* Paolo Xausa, Nov 13 2023, after Joerg Arndt *)

PARI

forstep(n=1,10^9,2, t=(n^2+1)/2; if( !isprime(t) && isprimepower(t), print1(t,", ")));

PARI

/* much more efficient: */
{b(n) = polchebyshev(n, 1, I) / I^n}
for(n=1,10^3,t=(b(n)^2+1)/2;if(!isprime(t)&&isprimepower(t),print1(t,", ")));