A079545 -id:A079545 - cp's OEIS frontend

A079739 Primes of the form x^2 + y^2 + 2 (x,y nonnegative).

2, 3, 7, 11, 19, 31, 43, 47, 67, 83, 103, 127, 139, 151, 199, 223, 227, 263, 271, 283, 307, 367, 379, 443, 463, 479, 487, 523, 547, 571, 587, 607, 619, 631, 643, 659, 691, 727, 787, 811, 823, 859, 883, 907, 911, 967, 983, 1019, 1039, 1051, 1063, 1091, 1231

Offset: 1

N. J. A. Sloane, Feb 18 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A079545, A079740.

S := {}: for x from 0 to 100 do for y from 0 to 100 do S := S union {x^2+y^2+2} od:od:S := sort(convert(S, list)): for i from 1 to 500 do if isprime(S[i]) then printf(`%d,`,S[i]) fi:od: # James Sellers, Feb 25 2003

f[upto_]:=Module[{max=Ceiling[Sqrt[upto]]},Select[Select[ Union[Total[#]+2&/@(Tuples[Range[0,max],{2}]^2)], PrimeQ], #<=upto&]]; f[1250] (* Harvey P. Dale, Mar 19 2011 *)

More terms from James Sellers, Feb 25 2003

A079544 Primes of the form x^2 + y^2 + 1, x>0, y>0.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jan 23 2003

nonn

Sequence is known to be infinite due to a result of Linnik.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Aminu Alhaji Ibrahim, Sa’idu Isah Abubaka, Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties, Advances in Pure Mathematics, 2016, 6, 409-419.
Ju. V. Linnik. An asymptotic formula in an additive problem of Hardy and Littlewood (Russian). Izv. Akad. Nauk SSSR, ser. math., 24:629-706, 1960. Cited in Matomäki 2007.
Kaisa Matomäki, Prime numbers of the form p = m^2+n^2+1 in short intervals, Acta Arith. 128 (2007), pp. 193-200.
Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2+y^2+1, arXiv preprint arXiv:1708.08629 [math.NT], 2017.

Cf. A079545.

iMax=7!; a=Floor[Sqrt[iMax]]; lst={}; Do[Do[p=x^2+y^2+1; If[PrimeQ@p&&p<=iMax,AppendTo[lst,p]],{y,1,a}],{x,1,a}]; Union[lst] (* Vladimir Joseph Stephan Orlovsky, Aug 11 2009 *)

list(lim)=my(v=List(),t); lim\=1; for(x=1,sqrtint(lim-2), forstep(y=2-x%2,min(x,sqrtint(lim-x^2-1)), 2, if(isprime(t=x^2+y^2+1), listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 13 2012

A079740 Primes of the form x^2 + y^2 + 2 (x,y positive).

Original entry on oeis.org

7, 19, 31, 43, 47, 67, 103, 127, 139, 151, 199, 223, 227, 263, 271, 283, 307, 367, 379, 463, 479, 487, 523, 547, 571, 587, 607, 619, 631, 643, 659, 691, 727, 787, 811, 823, 859, 883, 907, 911, 967, 983, 1019, 1039, 1051, 1063, 1231, 1279, 1291, 1303, 1307

Offset: 1

N. J. A. Sloane, Feb 18 2003

Cf. A079545, A079739.

S := {}: for x from 1 to 100 do for y from 1 to 100 do S := S union {x^2+y^2+2} od:od:S := sort(convert(S, list)): for i from 1 to 500 do if isprime(S[i]) then printf(`%d,`,S[i]) fi:od: # James Sellers, Feb 25 2003

More terms from James Sellers, Feb 25 2003

A166687 Numbers of the form x^2 + y^2 + 1, x, y integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 11, 14, 17, 18, 19, 21, 26, 27, 30, 33, 35, 37, 38, 41, 42, 46, 50, 51, 53, 54, 59, 62, 65, 66, 69, 73, 74, 75, 81, 82, 83, 86, 90, 91, 98, 99, 101, 102, 105, 107, 110, 114, 117, 118, 122, 123, 126, 129, 131, 137, 138, 145, 146, 147, 149, 150, 154, 158, 161

Offset: 1

N. J. A. Sloane, Mar 05 2010

nonn

A001481 is the main entry for this sequence.

As Ng points out (Lemma 2.2), each prime divides some member of this sequence: 2 divides a(2) = 2, 3 divides a(3) = 3, 5 divides a(4) = 5, 7 divides a(9) = 14, etc. - Charles R Greathouse IV, Jan 04 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Jia Hong Ray Ng, Quarternions and the four square theorem, 2008 Summer VIGRE Program for Undergraduates
Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2 + y^2 + 1, Monatshefte für Mathematik vol. 189 (2019), pp. 715-733. arXiv:1708.08629 [math.NT]

Cf. A000408, A000378, A005767, A169580, A166265, A079545.

N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2+y^2+1,y=0..floor(sqrt(N-1-x^2))),x=0..floor(sqrt(N-1)))}:
sort(convert(S,list)); # Robert Israel, Jan 05 2016

Select[Range@ 162, Resolve[Exists[{x, y}, Reduce[# == x^2 + y^2 + 1, {x, y}, Integers]]] &] (* Michael De Vlieger, Jan 05 2016 *)

is(n)=my(f=factor(n-1)); for(i=1, #f~, if(f[i,1]%4==3 && f[i,2]%2, return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2016

list(lim)=my(v=List(),t); lim\=1; for(m=0,sqrtint(lim-1), t=m^2+1; for(n=0, min(sqrtint(lim-t),m), listput(v,t+n^2))); Set(v) \\ Charles R Greathouse IV, Jan 05 2016

A333909 Numbers k such that phi(k) is the sum of 2 squares, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 19, 20, 22, 24, 25, 27, 30, 32, 33, 34, 37, 38, 40, 41, 44, 48, 50, 51, 53, 54, 55, 57, 59, 60, 63, 64, 66, 68, 73, 74, 75, 76, 80, 82, 83, 85, 88, 91, 95, 96, 100, 101, 102, 106, 107, 108, 110, 111, 114, 117, 118, 120

Offset: 1

Amiram Eldar, Apr 09 2020

nonn

			1 is a term since phi(1) = 1 = 0^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, Values of arithmetical functions equal to a sum of two squares, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, alternative link.

Cf. A000010, A001481, A039770, A079545, A333910, A333911, A333912.

Select[Range[120], SquaresR[2, EulerPhi[#]] > 0 &]

from itertools import count, islice
from sympy import factorint, totient
def A333909_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(totient(n)).items()),count(1))
A333909_list = list(islice(A333909_gen(),30)) # Chai Wah Wu, Jun 27 2022

c1 * x/log(x)^(3/2) < N(x) < c2 * x/log(x)^(3/2), where N(x) is the number of terms <= x, and c1 and c2 are two positive constants (Banks et al., 2005).

A333911 Numbers k such that sigma(k) is the sum of 2 squares, where sigma is the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 7, 9, 10, 17, 19, 21, 22, 27, 30, 31, 40, 46, 51, 52, 55, 57, 58, 63, 66, 67, 70, 71, 73, 79, 81, 88, 89, 90, 93, 94, 97, 103, 106, 115, 118, 119, 120, 127, 133, 138, 145, 153, 154, 156, 163, 165, 170, 171, 174, 179, 184, 189, 190, 193, 198, 199, 201, 202

Offset: 1

Amiram Eldar, Apr 09 2020

nonn

			1 is a term since sigma(1) = 1 = 0^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, Values of arithmetical functions equal to a sum of two squares, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, alternative link.

Cf. A000203, A001481, A079545, A272405, A333909, A333910.

Select[Range[200], SquaresR[2, DivisorSigma[1, #]] > 0 &]

from itertools import count, islice
from collections import Counter
from sympy import factorint
def A333911_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in sum((Counter(factorint((p**(e+1)-1)//(p-1))) for p, e in factorint(n).items()),start=Counter()).items()),count(1))
A333911_list = list(islice(A333911_gen(),30)) # Chai Wah Wu, Jun 27 2022

c1 * x/log(x)^(3/2) < N(x) < c2 * x/log(x)^(3/2), where N(x) is the number of terms <= x, and c1 and c2 are two positive constants (Banks et al., 2005).

A194627 a(1)=1, a(n+1) = p(n)^2 + q(n)^2 + 1, where p(n) and q(n) are the number of prime and nonprime numbers respectively in the sequence so far.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 21, 30, 41, 46, 59, 66, 81, 98, 117, 138, 161, 186, 213, 242, 273, 306, 341, 378, 417, 458, 501, 546, 593, 602, 651, 702, 755, 810, 867, 926, 987, 1050, 1115, 1182, 1251, 1322, 1395, 1470, 1547, 1626, 1707, 1790, 1875, 1962, 2051, 2142

Offset: 1

Greg Knowles, Sep 15 2011

			For n=1, we have no primes and one nonprime (a(1)=1), so a(2)=0^2+1^2+1=2. Now we have one prime (a(2)=2) and one nonprime, so a(3)=1^2+1^2+1=3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A079545, A377791, A377882.

R:= 1: v:= 1: p:= 0: q:= 0:
for i from 2 to 100 do
  if isprime(v) then p:= p+1 else q:= q+1 fi;
  v:= p^2 + q^2 + 1;
  R:= R,v
od:
R; # Robert Israel, Nov 10 2024

t = {1}; Do[ps = Count[t, ?(PrimeQ[#] &)]; AppendTo[t, ps^2 + (n - ps - 1)^2 + 1], {n, 2, 100}]; t (* _T. D. Noe, Sep 15 2011 *)

p=q=0;for(n=1,50,print1(k=p^2+q^2+1", ");if(isprime(k),p++,q++)) \\ Charles R Greathouse IV, Sep 16 2011

A333910 Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 7, 10, 17, 18, 19, 20, 21, 22, 27, 30, 31, 36, 40, 44, 45, 46, 50, 51, 55, 57, 58, 60, 66, 67, 70, 71, 72, 73, 79, 80, 88, 89, 92, 93, 94, 97, 99, 100, 103, 106, 115, 116, 118, 119, 120, 126, 127, 132, 133, 138, 140, 144, 145, 150, 154, 160, 162, 163, 165

Offset: 1

Amiram Eldar, Apr 09 2020

nonn

			1 is a term since psi(1) = 1 = 0^2 + 1^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, Values of arithmetical functions equal to a sum of two squares, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, alternative link.

Cf. A001481, A001615, A079545, A333909, A333911.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[200], SquaresR[2, psi[#]] > 0 &]

from itertools import count, islice
from collections import Counter
from sympy import factorint
def A333910_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in sum((Counter(factorint(1+p))+Counter({p:e-1}) for p ,e in factorint(n).items()),start=Counter()).items()),count(1))
A333910_list = list(islice(A333910_gen(),30)) # Chai Wah Wu, Jun 27 2022

c1 * x/log(x)^(3/2) < N(x) < c2 * x/log(x)^(3/2), where N(x) is the number of terms <= x, and c1 and c2 are two positive constants (Banks et al., 2005).

A258839 Carmichael numbers whose prime factors all have the form p=1+x^2+y^2 for some x,y in Z.

Original entry on oeis.org

561, 162401, 410041, 488881, 656601, 2433601, 36765901, 109393201, 171454321, 176659201, 178837201, 189941761, 221884001, 288120421, 600892993, 618068881, 721244161, 931694401, 985052881, 1183104001, 1828377001, 1848112761, 1943951041, 2361232477, 2438403661

Offset: 1

Michel Marcus, Jun 12 2015

nonn

Banks & Freiberg show that this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..733 from Giovanni Resta)
William D. Banks and Tristan Freiberg, Carmichael numbers and the sieve, Journal of Number Theory, Vol. 165 (2016), pp. 15-29; arXiv preprint, arXiv:1506.03497 [math.NT], 2015.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997 (Carmichael numbers), A079545 (primes of the form x^2 + y^2 + 1).

has(n)=for(x=sqrtint(n\2),sqrtint(n-1), if(issquare(n-x^2-1), return(1)));0
Korselt(n,f=factor(n))=for(i=1,#f~,if(f[i, 2]>1||(n-1)%(f[i, 1]-1),return(0))); 1
is(n)=my(f); if(n%2==0||isprime(n)||!Korselt(n,f=factor(n))||n<9, return(0)); for(i=1,#f~,if(!has(f[i,1]), return(0))); 1 \\ Charles R Greathouse IV, Jun 12 2015

cp's OEIS Frontend

A079739 Primes of the form x^2 + y^2 + 2 (x,y nonnegative).

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A079544 Primes of the form x^2 + y^2 + 1, x>0, y>0.

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A079740 Primes of the form x^2 + y^2 + 2 (x,y positive).

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A166687 Numbers of the form x^2 + y^2 + 1, x, y integers.

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A333909 Numbers k such that phi(k) is the sum of 2 squares, where phi is the Euler totient function (A000010).

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A333911 Numbers k such that sigma(k) is the sum of 2 squares, where sigma is the sum of divisors function (A000203).

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A194627 a(1)=1, a(n+1) = p(n)^2 + q(n)^2 + 1, where p(n) and q(n) are the number of prime and nonprime numbers respectively in the sequence so far.

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A333910 Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).

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