A079544 -id:A079544 - cp's OEIS frontend

A079545 Primes of the form x^2 + y^2 + 1 with x,y >= 0.

2, 3, 5, 11, 17, 19, 37, 41, 53, 59, 73, 83, 101, 107, 131, 137, 149, 163, 179, 181, 197, 227, 233, 251, 257, 293, 307, 347, 389, 401, 443, 467, 491, 521, 523, 563, 577, 587, 593, 613, 641, 677, 739, 773, 809, 811, 821, 883

Offset: 1

N. J. A. Sloane, Jan 23 2003

nonn

Bredihin proves that this sequence is infinite. Motohashi improves the upper and lower bounds. - Charles R Greathouse IV, Sep 16 2011
Sun & Pan prove that there are arbitrarily long arithmetic progressions in this sequence. - Charles R Greathouse IV, Mar 03 2018
For this sequence in short intervals, see Wu and Matomäki; for its Goldbach problem, see Teräväinen. - Charles R Greathouse IV, Oct 10 2018

			17 = 0^2 + 4^2 + 1 is prime so in this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
B. M. Bredihin, Binary additive problems of indeterminate type II. Analogue of the problem of Hardy and Littlewood (in Russian). Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya 27 (1963), pp. 577-612.
M. N. Huxley and H. Iwaniec, Bombieri's theorem in short intervals, Mathematika 22 (1975), pp. 188-194.
Henryk Iwaniec, Primes of the type φ(x, y) + A where φ is a quadratic form, Acta Arithmetica 21 (1972), pp. 203-234.
Kaisa Matomäki, Prime numbers of the form p = m^2 + n^2 + 1 in short intervals, Acta Arithmetica 128 (2007), pp. 193-200.
Y. Motohashi, On the distribution of prime numbers which are of the form x^2 + y^2 + 1. Acta Arithmetica 16 (1969), pp. 351-364.
Y. Motohashi, On the distribution of prime numbers which are of the form x^2 + y^2 + 1. II", Acta Mathematica Academiae Scientiarum Hungaricae 22 (1971), pp. 207-210.
Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2 + y^2 + 1, arXiv:1708.08629 [math.NT], 2017.
Joni Teräväinen, The Goldbach problem for primes that are sums of two squares plus one, Mathematika 64 (2018), pp. 20-70. arXiv:1611.08585 [math.NT], 2016-2017.
J. Wu, Primes of the form p = 1 + m^2 + n^2 in short intervals, Proceedings of the American Mathematical Society 126 (1998), pp. 1-8.

Primes in A166687.

Cf. A079544, A079739, A079740.

Select[Select[Range[1000], SquaresR[2, #] != 0&]+1, PrimeQ] (* Jean-François Alcover, Aug 31 2018 *)

list(lim)={
    my(A,t,v=List([2]));
    forstep(a=2,sqrt(lim-1),2,
        A=a^2+1;
        forstep(b=0,min(a,sqrt(lim-A)),2,
            if(isprime(t=A+b^2),listput(v,t))
        )
    );
    forstep(a=1,sqrt(lim-2),2,
        A=a^2+1;
        forstep(b=1,min(a,sqrt(lim-A)),2,
            if(isprime(t=A+b^2),listput(v,t))
        )
    );
    vecsort(Vec(v),,8)
}; \\ Charles R Greathouse IV, Sep 16 2011

is(n)=for(x=sqrtint(n\2),sqrtint(n-1), if(issquare(n-x^2-1), return(isprime(n)))); 0 \\ Charles R Greathouse IV, Jun 12 2015

B=bnfinit('x^2+1);
is(n)=!!#bnfisintnorm(B,n-1) && isprime(n) \\ Charles R Greathouse IV, Jun 13 2015

Iwaniec proves that a(n) ≍ n (log n)^(3/2), that is, n (log n)^(3/2) << a(n) << n (log n)^(3/2). - Charles R Greathouse IV, Mar 06 2018

A159828 a(n) is smallest number m > 0 such that m^2 + n^2 + 1 is prime.

Original entry on oeis.org

1, 6, 1, 6, 9, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 27, 8, 9, 24, 1, 6, 21, 4, 69, 12, 3, 6, 21, 6, 3, 6, 1, 6, 9, 2, 9, 6, 1, 6, 15, 6, 9, 6, 1, 6, 27, 2, 3, 36, 9, 6, 3, 6, 15, 18, 1, 48, 3, 4, 9, 6, 7, 6, 15, 4, 21, 42, 5, 6, 3, 2, 69, 18, 5, 6, 3, 2, 9, 24, 1, 6, 3, 8, 9, 6, 11, 18, 15, 4, 3, 6, 7, 18

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009

a(2k-1) is odd, a(2k) is even.

There are infinitely many primes of the forms n^2 + m^2 and n^2 + m^2 + 1, but it is not known if the number of primes of the form n^2 + 1 is infinite; cf. comments in A002496, A002313, A079544.

			n = 1: 1^2 + 1^2 + 1 = 3 is prime, so a(1) = 1.
n = 2: 1^2 + 2^2 + 1 = 6, 2^2 + 2^2 + 1 = 9, 3^2 + 2^2 + 1 = 14, 4^2 + 2^2 + 1 = 21, 5^2 + 2^2 + 1 = 30 are composite, but 6^2 + 2^2 + 1 = 41 is prime, so a(2) = 6.
n = 27: 1^2 + 27^2 + 1 = 731 = 17*43, 2^2 + 27^2 + 1 = 734 = 2*367 are composite, but 3^2 + 27^2 + 1 = 739 is prime, so a(27) = 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A069003 (smallest d such that n^2+d^2 is prime), A002496 (primes of form n^2+1), A002313 (primes of form x^2+y^2), A079544 (primes of form x^2+y^2+1, x>0, y>0).

S:=[]; for n in [1..100] do q:=n^2+1; m:=1; while not IsPrime(m^2+q) do m+:=1; end while; Append(~S,m); end for; S; // Klaus Brockhaus, May 21 2009

snm[n_]:=Module[{c=n^2+1,x=NextPrime[n^2+1]},While[!IntegerQ[Sqrt[x-c]], x= NextPrime[x]];Sqrt[x-c]]; Array[snm,100] (* Harvey P. Dale, Sep 22 2018 *)

Edited and extended by Klaus Brockhaus, May 21 2009

A340778 Decimal expansion of (2 - e*log(2))/4.

Original entry on oeis.org

0, 2, 8, 9, 5, 7, 6, 5, 3, 6, 5, 9, 0, 6, 9, 9, 7, 2, 5, 2, 4, 4, 6, 0, 2, 2, 0, 7, 6, 8, 6, 4, 9, 6, 6, 6, 3, 2, 5, 6, 8, 3, 7, 3, 5, 8, 8, 6, 3, 1, 4, 6, 5, 3, 4, 7, 6, 8, 4, 8, 3, 8, 0, 0, 4, 9, 2, 8, 0, 4, 4, 8, 6, 2, 8, 8, 2, 8, 8, 4, 9, 4, 3, 8, 8, 0, 5, 8

Offset: 0

Stefano Spezia, Jan 21 2021

This constant appears in an asymptotic formula proved by Linnik in 1960 in an additive problem of Hardy-Littlewood (see Formula 1 in Dimitrov and Formula 0.3 in Linnik).

			0.02895765365906997252446022076864966632568373588631465...

Stoyan I. Dimitrov, The ternary Piatetski-Shapiro inequality with one prime of the form p = x^2 + y^2 + 1, arXiv:2011.03967 [math.NT], 2020.
Juriĭ Vladimirovič Linnik, An asymptotic formula in an additive problem of Hardy-Littlewood, Izv. Akad. Nauk SSSR Ser. Mat., 24 (1960), 629-706 (in Russian).

Cf. A001113, A002162, A079544, A276415, A283239.

Mathematica
```
First[RealDigits[N[(2-E*Log[2])/4,87]]]
```

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A079545 Primes of the form x^2 + y^2 + 1 with x,y >= 0.

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A159828 a(n) is smallest number m > 0 such that m^2 + n^2 + 1 is prime.

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A340778 Decimal expansion of (2 - e*log(2))/4.

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