A085317 -id:A085317 - cp's OEIS frontend

A094712 Primes that are not the sum of three positive squares.

2, 5, 7, 13, 23, 31, 37, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 359, 367, 383, 431, 439, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 823, 839, 863, 887, 911, 919, 967, 983, 991

Offset: 1

T. D. Noe, May 21 2004

nonn

Except for primes 2, 5, 13 and 37, this sequence consists all primes p such that p = 7 (mod 8). The density of these primes is 0.25.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A085317 (primes that are the sum of three positive squares).

lst={}; lim=32; Do[n=a^2+b^2+c^2; If[n?(AllTrue[Sqrt[#],IntegerQ]&)]==0&] (* _Harvey P. Dale, May 04 2025 *)

A094713 Number of ways that prime(n) can be represented as a^2+b^2+c^2 with c >= b >= a > 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 1, 0, 1, 2, 1, 1, 0, 1, 0, 2, 3, 1, 3, 0, 2, 1, 2, 0, 3, 2, 2, 3, 0, 1, 1, 0, 3, 3, 2, 0, 1, 2, 0, 2, 0, 3, 2, 3, 0, 3, 4, 4, 0, 5, 0, 1, 5, 2, 4, 2, 0, 2, 2, 2, 2, 3, 3, 4, 0, 0, 2, 2, 0, 5, 1, 5, 4, 5, 2, 0, 3, 0, 3, 5, 2, 7, 0, 4, 0, 0, 5, 2, 0, 7, 8, 3, 2, 2, 4, 5, 8, 3

Offset: 1

T. D. Noe, May 21 2004

nonn

			a(13) = 2 because prime(13) = 41 = 1+4+36 = 9+16+16.

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

Cf. A085317 (primes that are the sum of three positive squares), A094712 (primes that are not the sum of three positive squares), A094714 (least prime having exactly n representations as the sum of three positive squares).

lim=25; pLst=Table[0, {PrimePi[lim^2]}]; Do[n=a^2+b^2+c^2; If[n?(Min[#]>0&)],{n,110}]  (* _Harvey P. Dale, Feb 17 2011 *)

A193142 Primes which are the sum of 5 distinct positive squares.

Original entry on oeis.org

79, 103, 127, 131, 139, 151, 157, 163, 167, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 16 2011

nonn

A004434 INTERSECTION A000040. [Charles R Greathouse IV, Jul 17 2011]

			79=1^2+2^2+3^2+4^2+7^2, 103=2^2+3^2+4^2+5^2+7^2, 127=1^2+2^2+3^2+7^2+8^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Cf. A085317, A004434, A193141, A193143.

lst = {}; Do[Do[Do[Do[Do[p = a^2 + b^2 + c^2 + d^2 + e^2; If[PrimeQ[p], AppendTo[lst, p]], {e, d - 1, 1, -1}], {d, c - 1, 1, -1}], {c, b - 1, 1, -1}], {b, a - 1, 1, -1}], {a, 6, 20}]; OEISTrim[Take[Union[lst], 80]]
With[{upto=500},Select[Union[Total/@Subsets[Range[Ceiling[Sqrt[upto-30]]]^2, {5}]],PrimeQ[#]&&#<=upto&]] (* Harvey P. Dale, Jun 05 2016 *)

upto(lim)=my(v=List(),tb,tc,td,te);for(a=6,sqrt(lim),for(b=4,min(a-1,sqrt(lim-a^2)),tb=a^2+b^2;for(c=3,min(b-1,sqrt(lim-tb)),tc=tb+c^2;for(d=2,min(c-1,sqrt(lim-tc)),td=tc+d^2;forstep(e=1+td%2,d-1,2,te=td+e^2;if(te>lim,break);if(isprime(te),listput(v,te)))))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jul 17 2011

Conjecture: a(n) = prime(n+32) for n > 13. [Charles R Greathouse IV, Jul 17 2011]

A193143 Primes which are the sum of 5 distinct positive squares in more than one way.

Original entry on oeis.org

103, 127, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 229, 239, 241, 251, 263, 271, 277, 281, 283, 307, 311, 313, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 16 2011

nonn

All terms from 103 onwards in A068229 are primes which are the sum of 5 distinct positive squares in more than one way.

			103 = 1^2 + 2^2 + 3^2 + 5^2 + 8^2 = 2^2 + 3^2 + 4^2 + 5^2 + 7^2.
127 = 1^2 + 2^2 + 3^2 + 7^2 + 8^2 = 1^2 + 4^2 + 5^2 + 6^2 + 7^2 = 1^2 + 2^2 + 4^2 + 5^2 + 9^2.

Cf. A068229, A085317, A193141, A193142.

sum5sqP = {}; Do[Do[Do[Do[Do[p = a^2 + b^2 + c^2 + d^2 + e^2; If[PrimeQ[p], AppendTo[sum5sqP, p]], {e, d - 1, 1, -1}], {d, c - 1, 1, -1}], {c, b - 1, 1, -1}], {b, a - 1, 1, -1}], {a, 6, 30}]; a = Take[Sort[sum5sqP], 1000]; a = Select[Table[If[a[[n]] == a[[n - 1]] && a[[n]] != a[[n - 2]], a[[n]], ""], {n, 3, Length[a]}], IntegerQ]

A283017 Primes which are the sum of three nonzero 6th powers.

Original entry on oeis.org

3, 857, 1459, 4889, 50753, 51481, 66377, 119107, 210961, 262937, 308801, 525017, 531569, 539633, 562691, 766739, 797681, 840241, 1000793, 1046657, 1078507, 1772291, 1864873, 2303003, 2834443, 2986777, 3032641, 3107729, 3365777, 4757609, 4804201, 5135609, 5987593, 7530329, 7534361, 7743529, 8061041

Offset: 1

Ilya Gutkovskiy, Feb 26 2017

nonn

Primes of form x^6 + y^6 + z^6 where x, y, z > 0.

			3 = 1^6 + 1^6 + 1^6;
857 = 2^6 + 2^6 + 3^6;
1459 = 1^6 + 3^6 + 3^6, etc.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001014, A003359, A007490, A085317, A085318, A085319, A283018, A283019.

N:= 10^8: # to get all terms <= N
S:= [seq(i^6, i=1..floor(N^(1/6)))]:
S3:= {seq(seq(seq(S[i]+S[j]+S[k],k=1..j),j=1..i),i=1..nops(S))}:
sort(convert(select(t -> t <= N and isprime(t), S3), list)); # Robert Israel, Mar 09 2017

nn = 15; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^6)], # <= nn^6 && PrimeQ[#] &]

list(lim)=my(v=List(),a6,a6b6,t); lim\=1; for(a=1,sqrtnint(lim-2,6), a6=a^6; for(b=1,min(sqrtnint(lim-a6-1,6),a), a6b6=a6+b^6; forstep(c=if(a6b6%2,2,1),min(sqrtnint(lim-a6b6,6),b),2, if(isprime(t=a6b6+c^6), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Mar 09 2017

A283018 Primes which are the sum of three positive 7th powers.

Original entry on oeis.org

3, 257, 82499, 823799, 1119863, 2099467, 4782971, 5063033, 5608699, 6880249, 7160057, 10018571, 10078253, 10094509, 10279937, 10389481, 10823671, 19503683, 20002187, 20388839, 24782969, 31584323, 35850379, 36189869, 37931147, 50614777, 57416131, 62765029, 64845797, 68355029, 71663617, 73028453

Offset: 1

Ilya Gutkovskiy, Feb 26 2017

nonn

Primes of form x^7 + y^7 + z^7 where x, y, z > 0.

			3 = 1^7 + 1^7 + 1^7;
257 = 1^7 + 2^7 + 2^7;
82499 = 3^7 + 3^7 + 5^7, etc.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001015, A003370, A007490, A085317, A085318, A085319, A283017, A283019.

N:= 10^9: # to get all terms <= N
Res:= {}:
for x from 1 to floor(N^(1/7)) do
  for y from 1 to min(x, floor((N-x^7)^(1/7))) do
    for z from 1 to min(y, floor((N-x^7-y^7)^(1/7))) do
      p:= x^7 + y^7 + z^7;
      if isprime(p) then Res:= Res union {p} fi
od od od:
sort(convert(Res,list)); # Robert Israel, Feb 26 2017

nn = 14; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^7)], # <= nn^7 && PrimeQ[#] &]

list(lim)=my(v=List(),x7,y7,t,p); for(x=1,sqrtnint(lim\3,7), x7=x^7; for(y=x,sqrtnint((lim-x7)\2,7), y7=y^7; t=x7+y7; forstep(z=y+(x+1)%2,sqrtnint((lim-t)\1,7),2, if(isprime(p=t+z^7), listput(v,p))))); Set(v) \\ Charles R Greathouse IV, Feb 27 2017

A283019 Primes which are the sum of three nonzero 8th powers.

Original entry on oeis.org

3, 6563, 72353, 137633, 787811, 1745153, 7444673, 44726593, 49202147, 61503553, 86093443, 91858243, 100006817, 100072097, 101686177, 107444417, 143046977, 200006561, 214756067, 257412163, 300452323, 430372577, 431661313, 435812033, 447149537, 452523713, 489805633, 530372321, 744340577

Offset: 1

Ilya Gutkovskiy, Feb 26 2017

nonn

Primes of form x^8 + y^8 + z^8 where x, y, z > 0.

			3 = 1^8 + 1^8 + 1^8;
6563 = 1^8 + 1^8 + 3^8;
72353 = 2^8 + 3^8 + 4^8, etc.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001016, A003381, A006686, A007490, A085317, A085318, A085319, A283017, A283018.

nn = 13; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^8)], # <= nn^8 && PrimeQ[#] &]

list(lim)=my(v=List(),A,B,t); lim\=1; for(a=1,sqrtnint(lim-2,8), A=a^8; for(b=1,min(sqrtnint(lim-A-1,8),a), B=A+b^8; forstep(c=if(B%2,2,1),sqrtnint(lim-B,8),2, if(isprime(t=B+c^8), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017

A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

14, 29, 35, 50, 56, 66, 77, 83, 93, 107, 110, 116, 126, 140, 149, 155, 158, 165, 179, 194, 197, 200, 210, 219, 224, 242, 245, 251, 261, 264, 275, 290, 293, 302, 308, 315, 318, 332, 341, 350, 365, 371, 372, 381, 395, 398, 413, 428, 434, 435, 440, 450, 461, 462, 464, 482

Offset: 1

Antonio Roldán, Jan 29 2019

nonn

			35 = 1^2 + 3^2 + 5^2, with 3 - 1 = 5 - 3 = 2;
371 = 1^2 + 9^2 + 17^2, with 9 - 1 = 17 - 9 = 8. Also 371 = 9^2 + 11^2 + 13^2, with 11 - 9 = 13 - 11 = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000290, A000378, A000408, A085317, A120328, A292313, A306213, A306214.

N:= 1000: # for terms <= N
S:= {seq(seq(3*a^2+2*b^2, b=1..min(a-1, floor(sqrt((N-3*a^2)/2)))),a=1..floor(sqrt(N/3)))}:
sort(convert(S,list)); # Robert Israel, Jun 08 2020

for(n=3, 600, k=sqrt(n/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^2)/2; if(b==truncate(b)&&issquare(b), d=sqrt(b); if(d>=1&&d<=a-1, v=1; print1(n,", "))); a+=1))

w=List(); for(n=3, 600, k=sqrt(n/3); for(a=2, k, for(c=1, a-1, v=(a-c)^2+a^2+(a+c)^2; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))

A218797 Number of ways to write 2n - 1 as p + q + r with p <= q <= r and p, q, r, p^2 + q^2 + r^2 all prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 0, 1, 2, 2, 1, 3, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 4, 1, 3, 2, 2, 2, 2, 2, 2, 4, 3, 3, 3, 2, 4, 4, 3, 0, 2, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 3, 3, 2, 3, 4, 2, 2, 3, 2, 1, 3, 3, 1, 2, 2, 5, 1, 4, 2, 2, 1, 1, 6, 3, 1, 5, 1, 1, 5, 4, 1, 4, 1, 2, 6, 2, 4, 2, 2, 2, 1, 4, 4

Offset: 1

Zhi-Wei Sun, Nov 05 2012

nonn

Conjecture: a(n) > 0 for all n=1715,1716,....

This conjecture is stronger than the weak Goldbach conjecture. It has been verified for n up to 500,000. Those 0

			a(7)=2 since 13=3+3+7=3+5+5, and both 3^2+3^2+7^2=67 and 3^2+5^2+5^2=59 are primes.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A054860, A085317, A218754, A218585, A218654, A218656.

a[n_]:=a[n]=Sum[If[PrimeQ[n-Prime[j]-Prime[k]]==True&&PrimeQ[Prime[j]^2+Prime[k]^2+(n-Prime[j]-Prime[k])^2]==True,1,0],{j,1,PrimePi[n/3]},{k,j,PrimePi[(n-Prime[j])/2]}]
Do[Print[n," ",a[2n-1]],{n,1,10000}]

A269840 Lesser of twin primes where both are the sum of 3 nonzero squares.

Original entry on oeis.org

17, 41, 59, 107, 137, 179, 227, 281, 347, 419, 521, 569, 617, 641, 659, 809, 827, 857, 881, 1019, 1049, 1091, 1289, 1427, 1451, 1481, 1619, 1667, 1697, 1721, 1787, 1931, 2027, 2081, 2129, 2267, 2339, 2657, 2729, 2801, 2969, 3251, 3257, 3299, 3329, 3371, 3467, 3539

Offset: 1

Altug Alkan, Mar 13 2016

nonn

			17 is a term because 17 = 2^2 + 2^2 + 3^2 and 19 = 1^2 + 3^2 + 3^2.
41 is a term because 41 = 3^2 + 4^2 + 4^2 and 43 = 3^2 + 3^2 + 5^2.
59 is a term because 59 = 3^2 + 5^2 + 5^2 and 61 = 3^2 + 4^2 + 6^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000408, A001359, A085317.

isA000408(n) = my(a, b) ; a=1 ; while(a^2+10, ); p-2}
for(n=1, 1e2, if(isA000408(t(n)) && isA000408(t(n)+2), print1(t(n), ", ")));

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A094712 Primes that are not the sum of three positive squares.

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A094713 Number of ways that prime(n) can be represented as a^2+b^2+c^2 with c >= b >= a > 0.

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A193142 Primes which are the sum of 5 distinct positive squares.

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A193143 Primes which are the sum of 5 distinct positive squares in more than one way.

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A283017 Primes which are the sum of three nonzero 6th powers.

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A283018 Primes which are the sum of three positive 7th powers.

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A283019 Primes which are the sum of three nonzero 8th powers.

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A306212 Numbers that are the sum of squares of three distinct positive integers in arithmetic progression.

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