A182479 -id:A182479 - cp's OEIS frontend

A263723 Number of representations of the prime P = A182479(n) as P = p^2 + q^2 + r^2, where p < q < r are also primes.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 4, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

nonn

According to Sierpinski and Schinzel (1988), it is easy to prove that the smallest of p, q, r is always p = 3, and under Schinzel's hypothesis H the sequence is infinite.

			A182479(1) = 83 = 3^2 + 5^2 + 7^2 and A182479(2) = 179 = 3^2 + 7^2 + 11^2 are the only ways to write 83 and 179 as sums of squares of 3 distinct primes, so a(1) = 1 and a(2) = 1.
A182479(5) = 419 = 3^2 + 7^2 + 19^2 = 3^2 + 11^2 + 17^2 are the only such representations of 419, so a(5) = 2.

W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988; see pp. 220-221.

Wikipedia, Schinzel's Hypothesis H

Cf. A085317, A125516, A182479.

lst = {}; r = 7; While[r < 132, q = 5; While[q < r, P = 9 + q^2 + r^2; If[PrimeQ@P, AppendTo[lst, P]];
  q = NextPrime@q]; r = NextPrime@r]; Take[Transpose[Tally@Sort@lst][[2]], 105]

A137364 Prime numbers n such that n = p1^2 + p2^2 + p3^2, a sum of squares of 3 distinct prime numbers.

Original entry on oeis.org

83, 179, 227, 347, 419, 419, 467, 491, 563, 587, 659, 659, 827, 971, 1019, 1019, 1091, 1259, 1427, 1499, 1499, 1667, 1811, 1811, 1907, 1907, 1979, 1979, 2027, 2243, 2267, 2339, 2339, 2531, 2579, 2699, 2819, 2843, 2939, 3347, 3539, 3539, 3659, 3659, 3779

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 09 2008

Multiple solutions with different sets {p1,p2,p3} are indicated by repeating the entry for each solution. - R. J. Mathar, Apr 12 2008

All terms are congruent to 5 modulo 6. The smallest of the primes {p1,p2,p3} is always 3. - Zak Seidov, Jun 03 2014

			83 = 3^2 + 5^2 + 7^2;
179 = 3^2 + 7^2 + 11^2;
227 = 3^2 + 7^2 + 13^2.

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

Cf. A182479, A243342. - Zak Seidov, Jun 03 2014

Array[r, 99]; Array[y, 99]; For[i = 0, i < 10^2, r[i] = y[i] = 0; i++ ]; z = 4^2; n = 0; For[i1 = 1, i1 < z, a = Prime[i1]; a2 = a^2; For[i2 = i1 + 1, i2 < z, b = Prime[i2]; b2 = b^2; For[i3 = i2 + 1, i3 < z, c = Prime[i3]; c2 = c^2; p = a2 + b2 + c2; If[PrimeQ[p], Print[a2, " + ", b2, " + ", c2, " = ", p]; n++; r[n] = p]; i3++ ]; i2++ ]; i1++ ]; Sort[Array[r, 39]]
lst= {}; Do[p = Prime[q]^2 + Prime[r]^2 + Prime[s]^2; If[PrimeQ@p, AppendTo[lst, p]], {q, 26}, {r, q-1}, {s, r-1}]; Take[Sort@lst,72] (* Vincenzo Librandi, Jun 15 2013 *)

More terms from R. J. Mathar, Apr 12 2008

A243342 Least prime p that is expressible as the sum of three distinct primes squared in exactly n ways.

Original entry on oeis.org

83, 419, 3779, 10739, 240899, 229979, 1180019, 369419, 36964859, 33670379, 13235699, 21899939, 412547339, 370247939, 467152019, 579994619

Offset: 1

Zak Seidov, Jun 03 2014

All terms are congruent to 5 modulo 6 since the first square must be 9.

			p = 83, {a,b,c} = {3,5,7}, 1 way
p = 419, {a,b,c} = {3,7,19}, {3,11,17}, 2 ways
p = 3779, {a,b,c} ={3,7,61}, {3,17,59}, {3,31,53}, 3 ways
p = 10739, {a,b,c} = {3,11,103}, {3,23,101}, {3,53,89}, {3,67,79}, 4 ways.

Cf. A137364, A182475, A182479.

p = a^2 + b^2 + c^2 with a < b < c primes, note that a = 3 in all cases.

cp's OEIS Frontend

A263723 Number of representations of the prime P = A182479(n) as P = p^2 + q^2 + r^2, where p < q < r are also primes.

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A137364 Prime numbers n such that n = p1^2 + p2^2 + p3^2, a sum of squares of 3 distinct prime numbers.

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A243342 Least prime p that is expressible as the sum of three distinct primes squared in exactly n ways.

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