A045636 -id:A045636 - cp's OEIS frontend

A214514 Numbers of the form p^2 + q^2 + r^2, where p, q, and r are primes.

12, 17, 22, 27, 33, 38, 43, 54, 57, 59, 62, 67, 75, 78, 83, 99, 102, 107, 123, 129, 134, 139, 147, 150, 155, 171, 174, 177, 179, 182, 187, 195, 198, 203, 219, 222, 227, 243, 246, 251, 267, 291, 294, 297, 299, 302, 307, 315, 318, 323, 339, 342, 347, 363, 369

Offset: 1

T. D. Noe, Jul 29 2012

nonn

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

Cf. A045636 (two primes), A214515 (four primes).

nn = 10^3; 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 &]; Union[t]

from sympy import primerange as primes
from itertools import takewhile, combinations_with_replacement as mc
def aupto(N):
    psqs = list(takewhile(lambda x: x<=N, (p**2 for p in primes(1, N+1))))
    sum3 = set(sum(c) for c in mc(psqs, 3) if sum(c) <= N)
    return sorted(sum3)
print(aupto(369)) # Michael S. Branicky, Dec 17 2021

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

A124866 Numbers of the form (p^2-q^2)/8, p, q odd primes, p>q.

Original entry on oeis.org

2, 3, 5, 6, 9, 12, 14, 15, 18, 20, 21, 24, 30, 33, 35, 36, 39, 42, 44, 45, 51, 54, 60, 63, 65, 66, 69, 75, 81, 84, 90, 96, 99, 102, 104, 105, 111, 114, 117, 119, 120, 126, 129, 135, 141, 144, 150, 156, 159, 165, 168, 170, 171, 174, 180, 186, 189, 195, 201, 204, 207

Offset: 1

Alexander Adamchuk, Nov 10 2006

nonn

Primes in a(n) are {2, 3, 5}.

Cf. A124865 Numbers of the form p^2-q^2, p, q primes, p>q. Cf. A045636 Numbers of the form p^2+q^2, p, q primes.

Take[Union[Flatten[Table[(Prime[p]^2 - Prime[q]^2)/8, {p, 2, 100}, {q, 2, p - 1}]]], 60] (* Alonso del Arte, Jul 14 2011 *)

A212292 Odd numbers not of the form p^2 + q^2 + r with p, q, and r prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 17, 33, 43, 83, 179, 623, 713, 1019

Offset: 1

Charles R Greathouse IV, Jun 21 2012

The corresponding sequence with the restriction to primes removed is empty.
Wang shows that all but x^{9/20+e} members of this sequence up to x are congruent to 2 mod 3, for any e > 0.
There are no more terms < 10^7. - Donovan Johnson, Jun 27 2012
There are no more terms < 4*10^9. - Jud McCranie, Jun 09 2013
There are no more terms < 10^11. - Giovanni Resta, Jun 09 2013

Wang Mingqiang, On sums of a prime, and a square of prime, and a k-power of prime, Northeastern Mathematical Journal 18:4 (2002), pp. 283-286.

Cf. A045636, A006285, A118955, A156695, A226484.

list(lim)=my(p1=vector(primepi(sqrt(lim-5.5)),i,prime(i)^2), p2=List(), v=List(), u=List([1,3,5,7,9]), t); for(i=1,#p1, for(j=i,#p1,t=p1[i]+p1[j]; if(t>lim, break, listput(p2,t)))); p2=vecsort(Vec(p2),,8); for(i=1,#p2,forprime(p=2,lim-p2[i],listput(v,p2[i]+p))); v=select(n->n%2, vecsort(Vec(v),,8)); for(i=2,#v,forstep(j=v[i-1]+2,v[i]-2,2,listput(u,j))); Vec(u)

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 *)

A167276 Primes p such that p^2=x^2+y^2-1 with x and y also prime.

Original entry on oeis.org

7, 13, 17, 23, 31, 37, 41, 43, 47, 53, 67, 73, 83, 89, 103, 107, 109, 137, 149, 151, 157, 163, 173, 191, 193, 227, 229, 233, 241, 263, 269, 293, 307, 311, 313, 317, 331, 337, 353, 359, 383, 389, 397, 401, 421, 431, 439, 443, 457, 463, 467, 487, 499, 523, 557, 577, 593, 599, 613, 619, 643, 683, 701, 727, 733, 757, 773, 829, 839, 853, 857, 863, 887, 947, 967, 977, 983, 997

Offset: 1

Juri-Stepan Gerasimov, Nov 01 2009

nonn

Appears to be infinite.

Since (5*x+13)^2 + 1 = (3*x+7)^2 + (4*x+11)^2, it appears that there are infinitely many members of this sequence of the form 5*x+13 where x is an even number, that is the form of A030431(n). See the solution 78 at page 49 in the given reference (250 Problems in Elementary Number Theory) for the related conjecture. - Altug Alkan, Mar 30 2016

			a(1)=7 (x=5, y=5); a(2)=13 (x=7, y=11); a(3)=17 (x=11, y=17); a(4)=23 (x=13, y=19); a(5)=31 (x=11, y=31);...; a(21)=463 (x=461, y=43)

W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, Problem 78 page 7.

Cf. A000040.

isA045636 := proc(n) local p,q ; p := 2 ; while p^2+4 <= n do q := p ; while p^2+q^2 <= n do if q^2+p^2 = n then return true; end if ; q := nextprime(q) ; end do ; p := nextprime(p) ; end do ; return false ; end proc: A066872 := proc(n) ithprime(n)^2+1 ; end: for n from 1 to 200 do if isA045636(A066872(n)) then printf("%d,",ithprime(n)) ; end if ; end do ; # R. J. Mathar, Nov 09 2009

Select[Prime@ Range@ 168, Resolve[Exists[{x, y}, Reduce[#^2 == x^2 + y^2 - 1, {x, y}, Primes]]] &] (* Michael De Vlieger, Mar 30 2016 *)

{ A000040(i): A066872(i) in A045636}. [R. J. Mathar, Nov 09 2009]

Edited and extended by Daniel Platt, Nov 02 2009

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

A242243 Semiprimes sp of the form p^2 + q + 1 where p and q are consecutive primes.

Original entry on oeis.org

15, 33, 187, 309, 559, 1411, 1897, 2263, 2869, 3543, 6979, 10717, 11559, 11995, 22353, 32953, 39009, 54529, 57363, 58333, 66313, 77011, 80383, 113917, 120759, 124969, 147079, 158011, 167701, 175983, 177673, 237661, 241581, 253519, 299767, 310813, 376387, 381309

Offset: 1

K. D. Bajpai, May 09 2014

nonn

			a(1) = 15 = 3^2 + 5 + 1 = 3 * 5 is semiprime, 3 and 5 are consecutive primes.
a(2) = 33 = 5^2 + 7 + 1 = 3 * 11 is semiprime, 5 and 7 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..5970

Cf. A000040, A001358, A045636, A145292, A228183.

with(numtheory): A242243:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)+1);if  bigomega(k)=2 then RETURN (k); fi;end: seq(A242243 (),x=1..500);

Select[Table[Prime[n]^2 + Prime[n + 1] + 1, {n, 500}], PrimeOmega[#] == 2 &]

cp's OEIS Frontend

A214514 Numbers of the form p^2 + q^2 + r^2, where p, q, and r 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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A124866 Numbers of the form (p^2-q^2)/8, p, q odd primes, p>q.

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A212292 Odd numbers not of the form p^2 + q^2 + r with p, q, and r prime.

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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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A167276 Primes p such that p^2=x^2+y^2-1 with x and y also prime.

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