A214511 -id:A214511 - cp's OEIS frontend

A045636 Numbers of the form p^2 + q^2, with p and q primes.

8, 13, 18, 29, 34, 50, 53, 58, 74, 98, 125, 130, 146, 170, 173, 178, 194, 218, 242, 290, 293, 298, 314, 338, 365, 370, 386, 410, 458, 482, 530, 533, 538, 554, 578, 650, 698, 722, 818, 845, 850, 866, 890, 962, 965, 970, 986, 1010, 1058, 1082, 1130, 1202, 1250

Offset: 1

Felice Russo

A045698(a(n)) > 0. - Reinhard Zumkeller, Jul 29 2012

All terms greater than 8 are of the form 8k+2 or 8k+5 (A047617). - Giuseppe Melfi, Oct 06 2022

			18 belongs to the sequence because it can be written as 3^2 + 3^2.

A214723 is a subsequence. Complement: A214879.
Cf. A214511 (least number having n orderless representations as p^2 + q^2).
Cf. A047617.

import Data.List (findIndices)
a045636 n = a045636_list !! (n-1)
a045636_list = findIndices (> 0) a045698_list
-- Reinhard Zumkeller, Jul 29 2012

q=13; imax=Prime[q]^2; Select[Union[Flatten[Table[Prime[x]^2+Prime[y]^2, {x,q}, {y,x}]]], #<=imax&] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
With[{nn=60},Take[Union[Total/@(Tuples[Prime[Range[nn]],2]^2)],nn]] (* Harvey P. Dale, Jan 04 2014 *)

list(lim)=my(p1=vector(primepi(sqrt(lim-4)),i,prime(i)^2), t, p2=List()); for(i=1,#p1, for(j=i,#p1, t=p1[i]+p1[j];if(t>lim, break, listput(p2,t)))); vecsort(Vec(p2),,8) \\ Charles R Greathouse IV, Jun 21 2012

from sympy import primerange
def aupto(limit):
    primes = list(primerange(2, int((limit-4)**.5)+2))
    nums = [p*p + q*q for i, p in enumerate(primes) for q in primes[i:]]
    return sorted(set(k for k in nums if k <= limit))
print(aupto(1251)) # Michael S. Branicky, Aug 13 2021

A045698 Number of ways n can be written as the sum of two squares of primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Felice Russo

a(A214879(n)) = 0; a(A045636(n)) > 0; a(A214723(n)) = 1; a(A214511(n)) = n and a(m) < n for m < A214511(n). - Reinhard Zumkeller, Jul 29 2012

The smallest value of n such that a(n) = 2 is 338. (This helps distinguish it from the characteristic function of A045636.) - Wesley Ivan Hurt, Jun 13 2013

			For example, a(29) = 1 because 29 = 2^2 + 5^2. a(3) = 0 because there is no way to write 3 as sum of two squares of primes.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A001248, A010051, A037213.

a045698 n = length $ filter (\x -> x > 0 && a010051' x == 1) $
map (a037213 . (n -)) $
takeWhile (<= div n 2) a001248_list
-- Reinhard Zumkeller, Jul 29 2012

a(n)=my(s=0,q);forprime(p=2,sqrtint(n\2),if(issquare(n-p^2,&q)&&isprime(q),s++));s \\ Charles R Greathouse IV, Jun 04 2014

More terms from Erich Friedman

A214512 Least number having n orderless representations as p^2 + q^2 + r^2, where p, q, and r are primes.

Original entry on oeis.org

12, 219, 363, 699, 1179, 2019, 2259, 3891, 4059, 6459, 5379, 10899, 13179, 10659, 12579, 21819, 20979, 26859, 34419, 38379, 41019, 61299, 39459, 41811, 82131, 50379, 77451, 71379, 141099, 85491, 103971, 74571, 180411, 108339, 179739, 161139, 126819, 225099

Offset: 1

T. D. Noe, Jul 29 2012

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A025414, A025415, A214511, A214513.

nn = 10^6; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2 + ps[[k]]^2, {i, Length[ps]}, {j, i, Length[ps]}, {k, j, Length[ps]}]]; t = Select[t, # <= nn &]; t2 = Sort[Tally[t]]; u = Union[Transpose[t2][[2]]]; d = Complement[Range[u[[-1]]], u]; If[d == {}, nLim = u[[-1]], nLim = d[[1]]-1]; t3 = Table[Select[t2, #[[2]] == n &, 1][[1]], {n, nLim}]; Transpose[t3][[1]]

A214513 Least number having n orderless representations as p^2 + q^2 + r^2 + s^2, where p, q, r, and s are primes.

Original entry on oeis.org

16, 148, 196, 436, 388, 628, 868, 988, 1228, 1468, 1708, 2212, 2068, 2860, 2620, 2380, 3220, 3388, 3700, 4108, 3940, 4180, 5260, 4228, 5068, 4900, 5500, 6220, 6340, 7780, 5908, 5740, 6580, 7540, 8260, 7420, 8860, 9340, 11260, 10708, 9940, 9100, 10180, 12820

Offset: 1

T. D. Noe, Jul 29 2012

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A025416, A025417, A214511, A214512.

nn = 10^5; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2 + ps[[k]]^2 + ps[[l]]^2, {i, Length[ps]}, {j, i, Length[ps]}, {k, j, Length[ps]}, {l, k, Length[ps]}]]; t = Select[t, # <= nn &]; t2 = Sort[Tally[t]]; u = Union[Transpose[t2][[2]]]; d = Complement[Range[u[[-1]]], u]; If[d == {}, nLim = u[[-1]], nLim = d[[1]]-1]; t3 = Table[Select[t2, #[[2]] == n &, 1][[1]], {n, nLim}]; Transpose[t3][[1]]

A226539 Numbers which are the sum of two squared primes in exactly two ways (ignoring order).

Original entry on oeis.org

338, 410, 578, 650, 890, 1010, 1130, 1490, 1730, 1802, 1898, 1970, 2330, 2378, 2738, 3050, 3170, 3530, 3650, 3842, 3890, 4010, 4658, 4850, 5018, 5090, 5162, 5402, 5450, 5570, 5618, 5690, 5858, 6170, 6410, 6530, 6698, 7010, 7178, 7202, 7250, 7850, 7970, 8090

Offset: 1

Henk Koppelaar, Jun 10 2013

nonn

			338 = 7^2 + 17^2 = 13^2 + 13^2;
410 = 7^2 + 19^2 = 11^2 + 17^2.

Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A054735 (restricted to twin primes), A037073, A069496.
Cf. A045636 (sum of two squared primes: a superset).
Cf. A214511 (least number having n representations).
Cf. A226562 (restricted to sums decomposed in exactly three ways).

Prime2PairsSum := p -> select(x ->`if`(andmap(isprime, x),true,false), numtheory:-sum2sqr(p)):
for n from 2 to 10^6 do
  if nops(Prime2PairsSum(n)) = 2 then print(n, Prime2PairsSum(n)) fi;
od;

Select[Range@10000, Length[Select[ PowersRepresentations[#, 2, 2], And @@ PrimeQ[#] &]] == 2 &] (* Giovanni Resta, Jun 11 2013 *)

select( is_A226539(n)={#[0|t<-sum2sqr(n),isprime(t[1])&&isprime(t[2])]==2}, [1..10^4]) \\ For more efficiency, apply selection to A045636. See A133388 for sum2sqr(). - M. F. Hasler, Dec 12 2019

a(25)-a(44) from Giovanni Resta, Jun 11 2013

A226562 Numbers which are the sum of two squared primes in exactly three ways (ignoring order).

Original entry on oeis.org

2210, 3770, 5330, 6290, 12818, 16490, 18122, 19370, 24050, 24650, 26690, 32810, 33410, 34970, 36530, 39650, 39770, 44642, 45050, 45890, 49010, 50690, 51578, 57770, 59450, 61610, 63050, 66170, 67490, 72410, 73610, 74210, 80330, 85202, 86210, 86330, 88010

Offset: 1

Henk Koppelaar, Jun 11 2013

nonn

Suggestion: difference between successive terms is always at least 3. (With the known 115885 terms <10^9, the smallest difference is 24.) - Zak Seidov, Jun 12 2013

			2210 = 19^2 + 43^2 = 23^2 + 41^2 = 29^2 + 37^2;

Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.

Zak Seidov, Table of n, a(n) for n = 1..2464 (all terms up to 10^7).

Cf. A054735 (restricted to twin primes), A037073, A069496.
Cf. A045636 (sum of two squared primes), A226539.
Cf. A214511 (least number having n representations).
Cf. A226539 (restricted to sums decomposed in exactly three ways).

Prime2PairsSum := s -> select( x -> `if`(andmap(isprime, x), true, false), numtheory:-sum2sqr(s)):
for n from 2 to 10 do
if nops(Prime2PairsSum(n)) = 3 then print(n,Prime2PairsSum(n)) fi
od;

Select[Range@20000, Length[Select[ PowersRepresentations[#, 2, 2], And @@ PrimeQ[#] &]] == 3 &] (* Giovanni Resta, Jun 11 2013 *)

a(22)-a(37) from Giovanni Resta, Jun 11 2013

A242230 Primes p of the form p^2 + q + 1 where p < q are consecutive primes.

Original entry on oeis.org

61, 4561, 9511, 17299, 19471, 26737, 30109, 37447, 49957, 69439, 94561, 196699, 209311, 259603, 317539, 333517, 352249, 414097, 427069, 459013, 678157, 845491, 886429, 943819, 1027189, 1217719, 1410163, 1472587, 1647379, 2165323, 2200777, 2230549, 2603389

Offset: 1

K. D. Bajpai, May 08 2014

nonn

			a(1) = 61 = 7^2 + 11 + 1: 61 is prime, 7 and 11 are consecutive primes.
a(2) = 4561 = 67^2 + 71 + 1: 4561 is prime, 67 and 71 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..7040

Cf. A242231, A000040, A241945, A045636, A214723, A214511.

with(numtheory): A242230:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)+1); if  isprime(k) then RETURN (k); fi;end: seq(A242230 (),x=1..500);

A242230 = {}; Do[p = Prime[n]^2 + Prime[n + 1] + 1; If[PrimeQ[p], AppendTo[A242230, p]], {n, 500}]; A242230
Select[#[[1]]^2+#[[2]]+1&/@Partition[Prime[Range[300]],2,1],PrimeQ] (* Harvey P. Dale, Mar 28 2016 *)

A242231 Primes p of the form p^2 + q - 1 where p < q are consecutive primes.

Original entry on oeis.org

13, 31, 59, 307, 383, 557, 997, 1409, 1723, 3541, 5113, 5407, 6323, 6977, 8017, 10303, 19469, 52673, 94559, 109897, 151717, 158009, 187927, 193163, 249503, 274069, 326617, 361807, 383791, 419261, 427067, 546863, 573809, 592133, 636017, 684757, 735307, 738743

Offset: 1

K. D. Bajpai, May 08 2014

nonn

			a(1) = 13 = 3^2 + 5 - 1: 13 is prime, 3 and 5 are consecutive primes.
a(2) = 31 = 5^2 + 7 - 1: 31 is prime, 5 and 7 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..6900

Cf. A242230, A000040, A241945, A045636, A214723, A214511.

with(numtheory): A242231:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)-1);if  isprime(k) then RETURN (k); fi;end: seq(A242231 (),x=1..500);

A242231 = {}; Do[p = Prime[n]^2 + Prime[n + 1] - 1; If[PrimeQ[p], AppendTo[A242231, p]], {n, 500}]; A242231
Select[#[[1]]^2+#[[2]]-1&/@Partition[Prime[Range[250]],2,1],PrimeQ] (* Harvey P. Dale, Mar 05 2022 *)

A226599 Numbers which are the sum of two squared primes in exactly four ways (ignoring order).

Original entry on oeis.org

10370, 10730, 11570, 12410, 13130, 19610, 22490, 25010, 31610, 38090, 38930, 39338, 39962, 40970, 41810, 55250, 55970, 59330, 59930, 69530, 70850, 73730, 76850, 77090, 89570, 98090, 98930, 103298, 118898, 125450, 126290, 130730, 135218, 139490

Offset: 1

Henk Koppelaar, Jun 13 2013

nonn

It appears that all first differences are divisible by 24. - Zak Seidov, Jun 14 2013

			10370 = 13^2 + 101^2 = 31^2 + 97^2 = 59^2 + 83^2 = 71^2 + 73^2.
10730 = 11^2 + 103^2 = 23^2 + 101^2 = 53^2 + 89^2 = 67^2 + 79^2.

Stan Wagon, Mathematica in Action, Springer, 2000 (2nd ed.), Ch. 17.5, pp. 375-378.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A054735 (restricted to twin primes), A037073, A069496.
Cf. A045636 (sum of two squared primes is a superset).
Cf. A214511 (least number having n representations).
Cf. A225104 (numbers having at least three representations is a superset).
Cf. A226539, A226562 (sums decomposed in exactly two and three ways).

Prime2PairsSum := s -> select(x ->`if`(andmap(isprime, x), true, false),
   numtheory:-sum2sqr(s)):
for n from 2 to 10^6 do
  if nops(Prime2PairsSum(n)) = 4 then print(n, Prime2PairsSum(n)) fi;
od;

(* Assuming mod(a(n),24) = 2 *) Reap[ For[ k = 2, k <= 2 + 240000, k = k + 24, pr = Select[ PowersRepresentations[k, 2, 2], PrimeQ[#[[1]]] && PrimeQ[#[[2]]] &]; If[Length[pr] == 4 , Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 14 2013 *)

a(n) = p^2 + q^2; p, q are (not necessarily different) primes

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A045636 Numbers of the form p^2 + q^2, with p and q primes.

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A045698 Number of ways n can be written as the sum of two squares of primes.

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A214512 Least number having n orderless representations as p^2 + q^2 + r^2, where p, q, and r are primes.

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A214513 Least number having n orderless representations as p^2 + q^2 + r^2 + s^2, where p, q, r, and s are primes.

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A226539 Numbers which are the sum of two squared primes in exactly two ways (ignoring order).

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A226562 Numbers which are the sum of two squared primes in exactly three ways (ignoring order).

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A242230 Primes p of the form p^2 + q + 1 where p < q are consecutive primes.

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A242231 Primes p of the form p^2 + q - 1 where p < q are consecutive primes.

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