A002496 -id:A002496 - cp's OEIS frontend

A331941 Hardy-Littlewood constant for the polynomial x^2 + 1.

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

Offset: 0

Hugo Pfoertner, Feb 02 2020

			0.686406731409123004556096348363509434089166550627879778968117...

Henri Cohen, Number Theory, Vol II: Analytic and Modern Tools, Springer (Graduate Texts in Mathematics 240), 2007.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 85.

Karim Belabas and Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998). [pdf copy, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).

Cf. A002496, A005574, A083844, A199401, A206709, A221712.

\\ See Belabas, Cohen link. Run as HardyLittlewood2(x^2+1)/2 after setting the required precision.

Equals (1/2)*Product_{p=primes} (1 - Kronecker(-4,p)/(p - 1)).

Equals A199401/2.

A084712 Smallest prime of the form (2n)^k + 1, or 0 if no such number exists.

Original entry on oeis.org

3, 5, 7, 0, 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, 0, 1336337, 37

Offset: 1

Amarnath Murthy, Jun 10 2003

It has not been proved that a(19), a(25), a(31), a(34), a(43) and a(46) are 0; if these values do exist, they have > 4000 digits. The other zeros are definite. - David Wasserman, Jan 03 2005
a((p-1)/2) = p for primes p > 2, or a(n) = 2n+1 for n = (p-1)/2. All other positive a(n) belong to A002496 = primes of form m^2 + 1. Corresponding positive exponents k are powers of 2. They are listed in A079706. - Alexander Adamchuk, Sep 17 2006
Because k must be a power of 2, numbers of the form (2n)^k+1 are called generalized Fermat numbers with base 2n. These numbers, like the regular Fermat numbers, are seldom prime. I checked n=19, 25, 31, 34, 43, 46 with k up to 2^16 without finding any primes. - T. D. Noe, May 13 2008
Comments from N. J. A. Sloane, Jan 27 2024: (Start)
As pointed out by Max Alekseyev, the previous version violated the OEIS rules, since a(19) has not been confirmed. I therefore removed the terms starting at a(19).
The previous DATA line read:
3, 5, 7, 0, 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, 0, 1336337, 37, 0, 41, 43, 197352587024076973231046657, 47, 5308417, 0, 53, 2917, 3137, 59, 61, 0, 0, 67, 0, 71, 73, 5477, 1238846438084943599707227160577, 79, 40960001, 83, 7057, 0, 89
The old b-file has been changed to an a-file.
(End)

			a(7) = 197 = 14^2 + 1 as 14 + 1 = 15 is not a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..500 (currently known terms, may contain 0s in place of unknown terms)

Cf. A084713, A005097.

Table[k=1; While[p=1+(2n)^k; k<1024 && !PrimeQ[p], k=2k]; If[k==1024, 0, p], {n,44}] (* T. D. Noe, May 13 2008 *)

More terms from David Wasserman, Jan 03 2005

Edited by N. J. A. Sloane, Jan 27 2024 at the suggestion of Max Alekseyev

A134407 Numbers n such that n^2 + 1 is composite.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 18, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

Jani Melik, Jan 18 2008

			a(1)=3, because 3^2 + 1 = 10 is composite,
a(2)=5, because 5^2 + 1 = 26 is composite,
a(3)=7, because 7^2 + 1 = 50 is composite.

Cf. A002496, A002808, A005574, A018252.

ts_fn2:=proc(n) local i,tren,ans; ans:=[ ]: for i from 1 to n do tren := i^(2)+1: if (isprime(tren) = false) then ans:=[ op(ans), i ]: fi od: RETURN(ans) end: ts_fn2(200);

Select[Range@100,!PrimeQ[#^2+1]&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

is(n)=!isprime(n^2+1) \\ Charles R Greathouse IV, Sep 15 2014

a(n) ~ n. - Charles R Greathouse IV, Sep 15 2014

A206328 Primes of the form n^2+1 such that (n+2)^2+1 is also prime.

Original entry on oeis.org

5, 17, 197, 577, 2917, 15377, 41617, 147457, 215297, 401957, 414737, 509797, 1196837, 1308737, 1378277, 1547537, 1623077, 1726597, 1887877, 2446097, 2604997, 2802277, 2835857, 3857297, 4218917, 4343057, 4384837, 5779217, 6022117, 6421157, 7096897, 8031557

Offset: 1

Michel Lagneau, Feb 06 2012

Primes corresponding to A096012 and subset of A002496.
For n > 1, a(n) ==7 (mod 10) because n ==4 (mod 10).
Conjecture: this sequence is infinite.

			For n = 4, n^2 + 1 = 17 is prime and  (n+2)^2 + 1 = 37 is also prime => 17 is in the sequence.

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

Cf. A096012, A002496.

for n from 1 to 4000 do: x:=n^2+1:y:=(n+2)^2+1:if type(x,prime)=true and type(y,prime)=true then printf(`%d, `,x): else fi:od:

Select[Partition[Range[3000]^2+1,3,1],AllTrue[{#[[1]],#[[3]]},PrimeQ]&][[All,1]] (* Harvey P. Dale, Jan 16 2023 *)

A260120 Least integer k > 0 such that (prime(kn)-1)^2 = prime(jn)-1 for some j > 0.

Original entry on oeis.org

1, 2, 14, 1, 12, 9, 30, 198, 69, 83, 66, 132, 44, 15, 4, 99, 71, 88, 339, 230, 10, 33, 167, 66, 42, 22, 126, 442, 318, 1185, 29, 289, 37, 174, 157, 44, 146, 301, 171, 403, 2, 5, 26, 699, 573, 144, 338, 33, 2032, 1212, 404, 11, 135, 267, 380, 221, 447, 159, 898, 1397

Offset: 1

Zhi-Wei Sun, Jul 17 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, if a,b,c and m are integers with a > 0, gcd(a,b,c-m) = 1 and c == (a+b+1)*(m+1) (mod 2) such that b^2-4a*(c-m) is not a square and gcd(a*m-b,b^2+b-a*c-1) is not divisible by 3, then for any positive integer n there are two elements x and y of the set {prime(k*n)+m: k = 1,2,3,...} with a*x^2+b*x+c = y.

This implies the conjecture in A259731.

			a(3) = 14 since (prime(14*3)-1)^2 = 180^2 = prime(3477)-1 = prime(1159*3)-1.
a(63) = 5162 since (prime(5162*63)-1)^2 = 4642456^2 = 21552397711936 = prime(726521033763)-1 = prime(11532079901*63)-1.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A002496, A259731, A260082, A260121.

P[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[aa];k=k+1; If[P[n,(Prime[k*n]-1)^2+1],Goto[bb]];Goto[aa];Label[bb];Print[n, " ", k];Continue,{n,1,60}]

A056898 a(n) = smallest number m such that m^2+n is prime.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(8) = 3 since 3^2+8 = 17 which is prime.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000040, A002496, A056892, A056893, A056894, A056895, A056896, A056897.

A056898(n) = { my(m=0); while(!isprime((m*m)+n),m++); (m); }; \\ Antti Karttunen, Mar 04 2018

a(n) = sqrt(A056896(n)-n) = sqrt(A056897(n)).

For p a prime: a(p) = 0 (and a(p-1) = 1 if p<>3).

A056909 Primes of the form k^2+6.

Original entry on oeis.org

7, 31, 127, 367, 631, 967, 1231, 3727, 4231, 6247, 7927, 8287, 11887, 17167, 21031, 22807, 30631, 34231, 39607, 48847, 72367, 108247, 109567, 126031, 160807, 185767, 198031, 231367, 235231, 261127, 265231, 279847, 290527, 323767, 354031, 366031, 373327, 421207

Offset: 1

Henry Bottomley, Jul 07 2000

a(n) mod 120 = 7 or 31 for all n.

			a(2)=127 since 11^2+6=127 which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..2400

Cf. A002496, A056899, A049423, A005473, A056905.

[a: n in [0..700] | IsPrime(a) where a is n^2+6]; // Vincenzo Librandi, Nov 30 2011

Intersection[Table[n^2+6,{n,0,10^2}],Prime[Range[9*10^3]]] (* or *) For[i=6,i<=6,a={};Do[If[PrimeQ[n^2+i],AppendTo[a,n^2+i]],{n,0,100}];Print["n^2+",i,",",a];i++ ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Table[n^2+6,{n,0,7000}],PrimeQ] (* Vincenzo Librandi, Nov 30 2011 *)

list(lim)=my(v=List(),t); forstep(k=1,sqrtint(lim\1-6),2, if(isprime(t=k^2+6), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Nov 06 2024

a(n) = 36*A056910(n)^2 + 12*A056910(n) + 7.

a(n) >> n^2 log n. - Charles R Greathouse IV, Nov 06 2024

A062325 Numbers k for which phi(prime(k)) is a square.

Original entry on oeis.org

1, 3, 7, 12, 26, 45, 55, 79, 106, 123, 211, 252, 422, 446, 595, 723, 907, 1019, 1101, 1448, 1595, 1687, 1797, 1849, 1949, 2058, 2393, 2516, 2703, 2819, 3146, 3339, 3477, 3626, 4353, 4437, 4590, 5153, 5398, 5653, 5836, 6276, 6543, 6736, 6911, 7207, 7695

Offset: 1

Jason Earls, Jul 05 2001

Also A002496 indexed by A000040.

			79 is in the sequence because the 79th prime is 401 and phi(401) is 400 = 20^2.
595 is in the sequence because the 595th prime is 4357 and phi(4357) is 4356 = 66^2.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A001912, A002496, A005574, A062325, A090693.

Flatten[Position[Table[IntegerQ[Sqrt[Prime[w]-1]], {w, 1, 25000}], True]]
Flatten[Position[EulerPhi[Prime[Range[8000]]],?(IntegerQ[Sqrt[#]]&)]] (* _Harvey P. Dale, Apr 23 2014 *)

for(n=1,1600, if(issquare(eulerphi(prime(n))),print(n)))

{ default(primelimit, 2*10^8); n=m=0; forprime (p=2, 2*10^8, m++; if (issquare(eulerphi(p)), write("b062325.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

a(n) = A000720(A002496(n)).

A000040(a(n)) = A002496(n).

More terms from Labos Elemer, Jul 09 2001

A083849 a(n) is the largest prime of the form x^2 + 1 <= 2^n.

Original entry on oeis.org

2, 2, 5, 5, 17, 37, 101, 197, 401, 677, 1601, 3137, 8101, 15877, 32401, 62501, 122501, 246017, 512657, 1020101, 2073601, 4137157, 8386817, 16695397, 33339077, 66977857, 133772357, 268304401, 536663557, 1073610757, 2146098277

Offset: 1

Harry J. Smith, May 05 2003

nonn

It is conjectured that this sequence is increasing, but this has never been proved.

It is easily shown that all terms greater than 5 end in 1 or 7.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Landau's Problems.

Cf. A005574, A002496, A083844, A083845, A083846, A083847, A083848.

a(n) = my(last = 2^n+1); while ((p = precprime(last-1)) && (! issquare(p-1)), last = p;); p \\ Michel Marcus, Jun 14 2013

a(n)=my(k=sqrtint(2^n-1)); while(!isprime(k^2+1), k--); k^2+1 \\ Charles R Greathouse IV, Nov 29 2013

A090693 Positive numbers n such that n^2 - 2n + 2 is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 15, 17, 21, 25, 27, 37, 41, 55, 57, 67, 75, 85, 91, 95, 111, 117, 121, 125, 127, 131, 135, 147, 151, 157, 161, 171, 177, 181, 185, 205, 207, 211, 225, 231, 237, 241, 251, 257, 261, 265, 271, 281, 285, 301, 307, 315, 327, 341, 351, 385, 387, 397

Offset: 1

Giovanni Teofilatto, Dec 19 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

A002496 gives primes, A062325 gives prime index. Cf. A001912.

A005574(n+1) + 1.

a={};Do[If[PrimeQ[n^2-2n+2],AppendTo[a,n]],{n,1000}];a (* Peter J. C. Moses, Apr 02 2013 *)
Select[Range[400],PrimeQ[#^2-2#+2]&] (* Harvey P. Dale, May 10 2013 *)

# Python 3.2 or higher required.
from itertools import accumulate
from sympy import isprime
A090693_list = [i for i,n in enumerate(accumulate(range(10**5),lambda x,y:x+2*y-3)) if i > 0 and isprime(n+2)] # Chai Wah Wu, Sep 23 2014

a(n) = A005574(n)+1.

Corrected and extended by Ray Chandler, Dec 28 2003

Definition corrected by Chai Wah Wu, Sep 23 2014

cp's OEIS Frontend

A331941 Hardy-Littlewood constant for the polynomial x^2 + 1.

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A084712 Smallest prime of the form (2n)^k + 1, or 0 if no such number exists.

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A134407 Numbers n such that n^2 + 1 is composite.

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A206328 Primes of the form n^2+1 such that (n+2)^2+1 is also prime.

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A260120 Least integer k > 0 such that (prime(k*n)-1)^2 = prime(j*n)-1 for some j > 0.

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A056898 a(n) = smallest number m such that m^2+n is prime.

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A056909 Primes of the form k^2+6.

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A062325 Numbers k for which phi(prime(k)) is a square.

A260120 Least integer k > 0 such that (prime(kn)-1)^2 = prime(jn)-1 for some j > 0.