A264498 -id:A264498 - cp's OEIS frontend

A131574 Numbers n that are the product of two distinct odd primes and x^2 + y^2 = n has integer solutions.

65, 85, 145, 185, 205, 221, 265, 305, 365, 377, 445, 481, 485, 493, 505, 533, 545, 565, 629, 685, 689, 697, 745, 785, 793, 865, 901, 905, 949, 965, 985, 1037, 1073, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313, 1345, 1385, 1405, 1417, 1465, 1469

Offset: 1

Colin Barker, Aug 28 2007, corrected Aug 29 2007

nonn

The two primes are of the form 4*k + 1.

			65 is in the sequence because x^2 + y^2 = 65 = 5*13 has solutions (x,y) = (1,8), (4,7), (7,4) and (8,1).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)

Cf. A000415, A121387, A248649, A248712, A264498, A264499

dop(d, nmax) = {
  my(L=List(), v=vector(d,m,1)~, f);
  for(n=1, nmax,
    f=factorint(n);
    if(#f~==d && f[1,1]>2 && f[,2]==v && f[,1]%4==v, listput(L, n))
  );
  Vec(L)
}
dop(2, 3000) \\ Colin Barker, Nov 15 2015

A264499 Numbers n that are the product of four distinct odd primes and x^2 + y^2 = n has integer solutions.

Original entry on oeis.org

32045, 40885, 45305, 58565, 67405, 69745, 77285, 80665, 91205, 98345, 98605, 99905, 101065, 107185, 111605, 114985, 120445, 124865, 127465, 128945, 130645, 137605, 141245, 146705, 150365, 151385, 162565, 164645, 166685, 167765, 173485, 175565, 179945, 182845

Offset: 1

Colin Barker, Nov 15 2015

nonn

The four primes are of the form 4*k + 1.

			32045 is in the sequence because x^2 + y^2 = 32045 = 5*13*17*29 has solutions (x,y) = (2,179), (19,178), (46,173), (67,166), (74,163), (86,157), (109,142) and (122,131).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)

Cf. A131574, A264498.

dop(d, nmax) = {
  my(L=List(), v=vector(d,m,1)~, f);
  for(n=1, nmax,
    f=factorint(n);
    if(#f~==d && f[1,1]>2 && f[,2]==v && f[,1]%4==v, listput(L, n))
  );
  Vec(L)
}
dop(4, 200000)

A248649 Numbers n that are the product of three distinct primes such that x^2+y^2 = n has integer solutions.

Original entry on oeis.org

130, 170, 290, 370, 410, 442, 530, 610, 730, 754, 890, 962, 970, 986, 1010, 1066, 1090, 1105, 1130, 1258, 1370, 1378, 1394, 1490, 1570, 1586, 1730, 1802, 1810, 1885, 1898, 1930, 1970, 2074, 2146, 2290, 2314, 2330, 2378, 2405, 2410, 2465, 2482, 2522, 2570

Offset: 1

Colin Barker, Oct 12 2014

nonn

Union of 2*A131574 and A264498. - Ray Chandler, Dec 09 2019

			130 is in the sequence because 130 = 2*5*13, and x^2+y^2=130 has integer solutions (x,y) = (3,11) and (7,9).
1105 is in the sequence because x^2 + y^2 = 1105 = 5*13*17 has solutions (x,y) = (4,33), (9,32), (12,31) and (23,24).

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A131574, A264498, A248712.

Select[Range[3000],PrimeNu[#]==PrimeOmega[#]==3&&FindInstance[x^2+y^2==#,{x,y},Integers]!={}&] (* Harvey P. Dale, Dec 16 2023 *)

A248712 Numbers n that are the product of four distinct primes such that x^2+y^2 = n has integer solutions.

Original entry on oeis.org

2210, 3770, 4810, 4930, 5330, 6290, 6890, 6970, 7930, 9010, 9490, 10370, 10730, 11570, 11890, 12410, 12610, 12818, 13130, 14170, 14690, 15130, 15170, 15370, 16354, 16490, 17170, 17690, 17810, 18122, 18530, 19210, 19370, 19610, 20410, 21170, 21730, 22490

Offset: 1

Colin Barker, Oct 12 2014

nonn

Union of 2*A264498 and A264499. - Ray Chandler, Dec 09 2019

			2210 is in the sequence because 2210 = 2*5*13*17, and x^2+y^2=2210 has integer solutions (x,y) = (1,47), (19,43), (23,41) and (29,37).
32045 is in the sequence because x^2 + y^2 = 32045 = 5*13*17*29 has solutions (x,y) = (2,179), (19,178), (46,173), (67,166), (74,163), (86,157), (109,142) and (122,131).

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A248649, A264498, A264499.

cp's OEIS Frontend

A131574 Numbers n that are the product of two distinct odd primes and x^2 + y^2 = n has integer solutions.

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A264499 Numbers n that are the product of four distinct odd primes and x^2 + y^2 = n has integer solutions.

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PARI

A248649 Numbers n that are the product of three distinct primes such that x^2+y^2 = n has integer solutions.

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Mathematica

A248712 Numbers n that are the product of four distinct primes such that x^2+y^2 = n has integer solutions.

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