A002313 -id:A002313 - cp's OEIS frontend

A084165 Primes which are 1 mod m, where m is the index of the prime in sequence A002313 (Real primes with corresponding complex primes). The index m can be found in A084166 Primes which are -1 mod m can be found in sequence A084163.

5, 13, 17, 37, 89, 97, 181, 2689, 2969, 4621, 7457, 8081, 8161, 36709, 62701, 169489, 169709, 169753, 282809, 770101, 5763577, 9491101, 9491281, 9495121, 42544261, 115195501, 189689041, 189689653, 312315373, 312316409, 2294883817

Offset: 1

Sven Simon, May 17 2003

nonn

Real primes 2,5,13,17,29,37,... have a unique representation as sum of two squares. Values larger 2 are the primes p with p = 1 mod 4. This is sequence A002313. If p = x^2 + y^2, the corresponding complex prime is x+y*i First complex prime is 1+i with 2 as corresponding real prime, according to reference, page 1-2.

			89 is the 11th prime in sequence A002313, 11*8 = 88, so 89 = 1 mod 11

Handbook of First Complex Prime Numbers, Part1 + 2 Ervand Kogbetliantz and Alice Krikorian, Gordon and Breach, 1971

Cf. A002313, A084163, A084166.

Module[{nn=112*10^6,pr,len},pr=Select[Prime[Range[nn]],MemberQ[ {1,2},Mod[ #,4]]&];len=Length[pr];Select[Thread[{pr,Range[len]}],Mod[ #[[1]],#[[2]]] == 1&]][[All,1]] (* Harvey P. Dale, Aug 13 2020 *)

A379346 Number of integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 9, 16, 17, 20, 26, 36, 36, 40, 46, 47, 56, 67, 73, 86, 97, 107

Offset: 1

Andrew Howroyd, Dec 22 2024

See A223702 for additional information.

Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.

Row lengths of A223702.
First differences of A285283.
Cf. A002313, A177979, A185389.

A282970 Number of partitions of n into primes of form x^2 + y^2 (A002313).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 12, 12, 13, 14, 14, 17, 16, 19, 19, 21, 22, 23, 25, 27, 27, 30, 30, 34, 35, 37, 40, 41, 45, 46, 50, 52, 55, 58, 60, 65, 67, 71, 75, 78, 84, 86, 92, 97, 100, 108, 110, 118, 123, 127, 137, 139, 150, 154, 162

Offset: 0

Ilya Gutkovskiy, Feb 25 2017

nonn

Number of partitions of n into primes congruent to 1 or 2 mod 4.

			a(10) = 2 because we have [5, 5] and [2, 2, 2, 2, 2].

Index entries for related partition-counting sequences

Cf. A000607, A002313, A024941, A281273.

nmax = 82; CoefficientList[Series[Product[1/(1 - Boole[SquaresR[2, k] != 0 && PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Vec(prod(k=1, 82, (1/(1 - (isprime(k) && k%4<3)*x^k))) + O(x^83)) \\ Indranil Ghosh, Mar 15 2017

G.f.: Product_{k>=1} 1/(1 - x^A002313(k)).

A047818 a(n) is the least number m such that A002313(n)*m - 1 is a square.

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 2, 10, 2, 10, 13, 5, 1, 10, 2, 10, 13, 5, 37, 2, 34, 1, 50, 34, 17, 1, 25, 13, 10, 65, 2, 41, 65, 53, 5, 29, 34, 10, 1, 50, 2, 74, 10, 26, 5, 85, 106, 5, 25, 13, 1, 10, 26, 2, 61, 37, 34, 17, 5, 1, 26, 13, 170, 10, 2, 5, 130, 58, 125, 106, 73, 130, 50, 26, 170

Offset: 1

N. J. A. Sloane

nonn

A002313 has the 4k+1 and 4k+2 primes.

Related to Stormer numbers.

			a(3) = 2 because A002313(3)=13 and 13*2-1 = 5^2.

J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.

Cf. A002313, A002314, A005528.

a(n) = ((A002314(n-1))^2 + 1) / A002313(n).

Edited by Don Reble, Apr 13 2006

A084167 Numbers k in A002313 such that the count of primitive roots of k (A084168) are also in sequence A002313, where A002313 is the list of primes p having a unique representation as the sum of two squares, p = x^2 + y^2.

Original entry on oeis.org

1, 5, 17, 109, 157, 317, 389, 449, 709, 1201, 1237, 1249, 1429, 1621, 1801, 2341, 3001

Offset: 1

Sven Simon, May 17 2003

			449 = 20^2 + 7^2 has 192 primitive roots, of which 41 are prime, of which 20 are in A002313.

Cf. A002313, A084168.

Offset changed to 1 by Jinyuan Wang, Mar 09 2020

Name edited by Michel Marcus, Mar 09 2020

A244290 Smallest prime a(n) = x^2 + y^2 such that c^2 + d^2 = A002313(n) and cx + dy = 1, where c,d,x,y are integers.

Original entry on oeis.org

5, 2, 2, 53, 5, 173, 2, 17, 2, 29, 13, 5, 1697, 53, 2, 73, 13, 5, 37, 2, 389, 733, 2753, 89, 17, 1093, 773, 13, 397, 1789, 2, 41, 821, 53, 5, 29, 193, 281, 6257, 173, 2, 149, 593, 701, 5, 1289, 157, 5, 7993, 13, 2213, 449, 877, 2, 61, 37, 389, 17, 5, 24061

Offset: 1

Thomas Ordowski, Jun 27 2014

nonn

Let c^2 + d^2 = p be a prime, A002313(n). Then x^2 + y^2 = q is the smallest prime, a(n), such that cx + dy = 1 (Bézout's identity), where c,d,x,y are integers. We have pq = m^2 + 1 at m = cy - dx.
a(n) is the smallest prime q such that q*A002313(n)-1 is a square. - Thomas Ordowski, Sep 13 2015
Conjecture: a(n) < A002313(n)^2 for n > 1. - Thomas Ordowski, Dec 28 2017

			For prime 2 = 1^2 + 1^2 is 1*2 + 1*(-1) = 1 and 2^2 + (-1)^2 = 5 is prime, so a(1) = 5. For A002313(2) = 5 is vice versa so a(2) = 2.

Robert Israel, Table of n, a(n) for n = 1..2910

Cf. A002313, A002330, A002331.

N:= 10^6: # to get all a(n) before the first one > N
P:= select(isprime, [2,seq(4*i+1, i=1..floor((N-1)/4))]):
f:= proc(p) local i;
  for i from 1 to nops(P) do
   if issqr(p*P[i]-1) then return P[i] fi
od:
  -1
end proc:
for i from 1 to nops(P) do
  v:= f(P[i]);
if v = -1 then break fi;
A[i]:= v;
od:
seq(A[j],j=1..i-1); # Robert Israel, Sep 13 2015

\\ cs should contain terms from A002330
\\ ds should contain terms from A002331
a244290(cs, ds) = {
  vector(#cs, i,
    c=cs[i]; d=ds[i]; [u,v]=gcdext(c, d);
    x=u; y=v; while(!isprime(x^2+y^2), x+=d; y-=c); e=x^2+y^2;
    x=u; y=v; while(!isprime(x^2+y^2), x-=d; y+=c); f=x^2+y^2;
    min(e, f)
  )
} \\ Colin Barker, Jul 06 2014

More terms from Colin Barker, Jul 06 2014

A248746 a(n) is the index k of the greatest prime divisor A002313(k) of n^2 + 1.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 3, 7, 13, 9, 5, 4, 22, 15, 26, 5, 3, 20, 39, 4, 12, 8, 51, 31, 60, 10, 18, 41, 8, 6, 7, 14, 11, 54, 105, 16, 4, 65, 121, 5, 35, 6, 17, 83, 10, 4, 45, 97, 9, 106, 48, 29, 209, 11, 221, 3, 59, 133, 28, 138, 66, 38, 25, 155, 294, 43, 6, 174, 5

Offset: 1

Michel Lagneau, Oct 13 2014

nonn

a(n) is the number k such that A002313(k) = A014442(n).

			a(5)=3 because A002313(3)=13 and 5^2+1 = 2*13 with A002313(3)= A014442(5).

Michel Lagneau, Table of n, a(n) for n = 1..5000

Cf. A014442 (greatest prime divisor of n^2+1), A002313 (primes congruent to 1 or 2 modulo 4).

Cf. also A002522.

with(numtheory):T:=array(1..50000):T[1]:=2:kk:=1:nn:=10^5:
for i from 1 to nn do:
  p:=4*i+1:
  if type(p,prime)=true
  then
    kk:=kk+1:T[kk]:=p:
    else
    fi:
   od:
     for k from 1 to 5000 do:ii:=0:
      y:=factorset(k^2+1):n2:=nops(y):t:=y[n2]:
        for l from 1 to kk while(ii=0)do :
        if t=T[l]
         then
         printf(`%d, `,l):
         else
        fi:
     od:
    od:

A282971 Number of compositions (ordered partitions) of n into primes of form x^2 + y^2 (A002313).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 9, 11, 15, 18, 24, 29, 37, 48, 58, 78, 92, 124, 149, 195, 243, 308, 393, 490, 629, 786, 1004, 1263, 1603, 2024, 2564, 3239, 4106, 5184, 6571, 8301, 10508, 13298, 16807, 21296, 26895, 34082, 43060, 54528, 68952, 87245, 110392, 139622, 176696, 223484, 282798, 357731

Offset: 0

Ilya Gutkovskiy, Feb 25 2017

nonn

Number of compositions (ordered partitions) of n into primes congruent to 1 or 2 mod 4.

Conjecture: every number > 16 is the sum of at most 4 primes of form x^2 + y^2.

			a(12) = 4 because we have [5, 5, 2], [5, 2, 5], [2, 2, 5] and [2, 2, 2, 2, 2, 2].

Index entries for sequences related to compositions

Cf. A002124, A002313, A023360, A077608, A280917, A282906, A282970.

nmax = 60; CoefficientList[Series[1/(1 - Sum[Boole[SquaresR[2, k] != 0 && PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Vec(1/(1 - sum(k=1, 60, (isprime(k) && k%4<3)*x^k)) + O(x^61)) \\ Indranil Ghosh, Mar 15 2017

G.f.: 1/(1 - Sum_{k>=1} x^A002313(k)).

A379347 a(n) is the sum of all integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

Original entry on oeis.org

1, 12, 327, 391, 703, 20510, 5667, 661016, 507004, 644098, 24977604, 38394505, 2621510449, 465558141, 624692559, 63435958, 507041846, 8133206945, 70119049516045, 45102364892, 49035127231, 154823547391

Offset: 1

Andrew Howroyd, Dec 22 2024

See A223702 for additional information.

			a(2) = 12 = 2 + 3 + 7. The corresponding values for k^2 + 1 are 5, 10 and 50 each of whose greatest prime factor is 5 = A002313(2).

Row sums of A223702.

Cf. A002313, A185389, A379344.

A094249 Sequence A002313 is the sequence of primes p = aa + bb, starting 2,5,13,17,..., members p > 2 have p = 1 mod 4. In analogy to the definition of primorial primes use the primes of sequence A002313 to build the product, written here as cp#353+1 or cp#1609-1. If cp#n+1 or cp#n-1 is prime, then n is in the sequence. Using +1 or -1 to define the type of prime cp#n+-1 we get the sequence 1,1,1,1,-1,1,...

Original entry on oeis.org

53, 353, 433, 733, 1609, 7789

Offset: 0

Sven Simon, Apr 25 2004

nonn

Primes of type cp#n-1 are in sequence A002313 again, only one such prime was found, cp#1609-1 was certified with Primo. 1609 = 40*40 + 3*3 = 1600 + 9. Because of the starting value 2, cp#n+1 = 3 mod 4 for every possible n, so they are primes, but not primes of sequence A002313. The search limits were 32000 for primes cp#n+1 and 64000 for primes cp#n-1, having more than 10000 decimal digits in the range of 64000.

			a(0) = 53 with cp#53+1 = 2*5*13*17*29*37*41*53 + 1 = 5152900091

Cf. A002313.

cp's OEIS Frontend

A084165 Primes which are 1 mod m, where m is the index of the prime in sequence A002313 (Real primes with corresponding complex primes). The index m can be found in A084166 Primes which are -1 mod m can be found in sequence A084163.

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A379346 Number of integers of the form k^2 + 1 whose greatest prime factor is A002313(n), the n-th prime not congruent to 3 mod 4.

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A282970 Number of partitions of n into primes of form x^2 + y^2 (A002313).

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A047818 a(n) is the least number m such that A002313(n)*m - 1 is a square.

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A084167 Numbers k in A002313 such that the count of primitive roots of k (A084168) are also in sequence A002313, where A002313 is the list of primes p having a unique representation as the sum of two squares, p = x^2 + y^2.

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A244290 Smallest prime a(n) = x^2 + y^2 such that c^2 + d^2 = A002313(n) and c*x + d*y = 1, where c,d,x,y are integers.

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A248746 a(n) is the index k of the greatest prime divisor A002313(k) of n^2 + 1.

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A282971 Number of compositions (ordered partitions) of n into primes of form x^2 + y^2 (A002313).

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Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

A244290 Smallest prime a(n) = x^2 + y^2 such that c^2 + d^2 = A002313(n) and cx + dy = 1, where c,d,x,y are integers.