A085317 -id:A085317 - cp's OEIS frontend

A123592 Primes of the form p^2 + q^2 + r^2, where p,q,r are primes.

17, 43, 59, 67, 83, 107, 139, 179, 227, 251, 307, 347, 379, 419, 467, 491, 547, 563, 587, 659, 827, 859, 971, 1019, 1091, 1259, 1427, 1499, 1667, 1699, 1811, 1867, 1907, 1931, 1979, 2027, 2243, 2267, 2339, 2531, 2579, 2699, 2819, 2843, 2939, 3347, 3371, 3499

Offset: 1

Alexander Adamchuk, Nov 14 2006

nonn

a(n) is a subset of A085317(n) = {3, 11, 17, 19, 29, 41, 43, 53, 59, 61, 67, 73, 83, ...} Primes of form x^2 + y^2 + z^2. All terms except a(1) = 17 are congruent to 3 mod 8.

If neither p, q, nor r is 3, then p^2 + q^2 + r^2 is always divisible by 3. Therefore all terms in a(n) have at least one 3^2 in their summation. - Richard R. Forberg, Aug 29 2013

			a(1) = 17 because 17 = 2^2 + 2^2 + 3^2 is prime and 2^2 + 2^2 + 2^2 = 12 is composite.

Cf. A085317 (primes of form x^2 + y^2 + z^2).

With[{nn=50},Take[Union[Select[Total/@Tuples[Prime[Range[nn/2]]^2, 3], PrimeQ]],nn]] (* Harvey P. Dale, Aug 26 2015 *)

A227994 Primes that are the sum of the squares of three integers that form an arithmetic sequence with difference 7.

Original entry on oeis.org

461, 773, 1181, 1973, 2621, 6173, 7901, 9173, 11261, 21773, 29501, 37061, 44021, 50021, 51581, 54773, 58061, 66701, 68501, 72173, 75941, 81773, 85781, 96221, 109541, 118901, 126173, 143981, 204461, 210773, 220421, 233621, 236981, 254141, 279173, 286541, 328781, 336773

Offset: 1

Will Gosnell, Aug 14 2013

nonn

Primes of the form 3k^2 + 42k + 245. - Charles R Greathouse IV, Aug 14 2013

			461 is a term since 4^2 + 11^2 + 18^2 = 461;
773 is a term since 8^2 + 15^2 + 22^2 = 773;
1181 is a term since 12^2 + 19^2 + 26^2 = 1181;
1973 is a term since 18^2 + 25^2 + 32^2 = 1973.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A085317. - Michel Marcus, Apr 01 2019

for x in range(1, 2000): b=x**2  : c= (x+7)**2: d=(x+14)**2:e=(b+c+d): print x,e

Select[Table[Total[(n+{0,7,14})^2],{n,500}],PrimeQ] (* Harvey P. Dale, Jun 10 2021 *)

for(n=1,1e3,if(isprime(t=3*(n+7)^2+98),print1(t", "))) \\ Charles R Greathouse IV, Aug 14 2013

a(14)-a(38) from Charles R Greathouse IV, Aug 14 2013

Name clarified by Jon E. Schoenfield, Apr 01 2019

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.

Original entry on oeis.org

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]

A084685 a(n) is the least x such that length of fixed-point-list when function-A085307[] was iterated and started at a(n) equals n.

Original entry on oeis.org

2, 4, 6, 14, 22, 115, 55, 105, 155, 145, 341, 501, 489, 143, 437, 301, 395, 665

Offset: 1

Labos Elemer, Jul 16 2003

			Some lists of iterated values started at a(n):
n=1: a(1)=first prime; n=2: a(2)=first true prime factor;
n=3: a(3)=first 2^j.3^i number, that is 6;
n=5: {22,112,72,32,2}
n=12: length=a(12); iv=501; fixed-point=1354674597313; list as follows
{501, 1673, 2397, 47173, 293237, 2571637, 23593109, 273116353, 522211523, 8866073119, 57914987307, 1354674597313}

Cf. A085307, A085308, A085309, A085317-A085323.

A161665 Primes that can be represented as a sum of 2 and also as a sum of 3 distinct nonzero squares, sharing a term in the sums.

Original entry on oeis.org

29, 101, 109, 149, 173, 181, 229, 233, 241, 269, 293, 389, 401, 409, 421, 433, 449, 521, 569, 641, 661, 677, 701, 757, 761, 769, 797, 821, 857, 877, 881, 941, 1021, 1069, 1097, 1109, 1117, 1181, 1229, 1237, 1277, 1289, 1301, 1373, 1381, 1429, 1433, 1481, 1549

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 15 2009

nonn

Dropping the requirement of one shared term, we would get the supersequence 17, 29, 41, 53, 61, 73, ... - R. J. Mathar, Oct 04 2009

			The prime 29 has the representations 29 = 2^2+ 5^2 = 2^2+3^2+4^2, sharing 2^2.
The prime 101 has the representations 101 = 1^2+10^2 = 1^2+6^2+8^2, sharing 1^2.
The prime 109 has the representations 109 = 3^2+10^2 = 3^2+6^2+8^2, sharing 3^2.
The prime 149 has the representations 149 = 7^2+10^2 = 6^2+7^2+8^2, sharing 7^2.

Cf. A002114, A085317.

f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[a=f[n];If[a>0,b=f[n-(f[n])^2]; If[b>0,c=(n-a^2-b^2)^(1/2);If[a!=b&&a!=c,If[PrimeQ[n],AppendTo[lst, n]]]]],{n,3,4*6!}];lst

Definition reverse-engineered from program by R. J. Mathar, Oct 04 2009

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A123592 Primes of the form p^2 + q^2 + r^2, where p,q,r are primes.

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A227994 Primes that are the sum of the squares of three integers that form an arithmetic sequence with difference 7.

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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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A084685 a(n) is the least x such that length of fixed-point-list when function-A085307[] was iterated and started at a(n) equals n.

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A161665 Primes that can be represented as a sum of 2 and also as a sum of 3 distinct nonzero squares, sharing a term in the sums.

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Mathematica

Extensions