A045636 -id:A045636 - cp's OEIS frontend

A086119 Numbers of the form p^3 + q^3, p, q primes.

16, 35, 54, 133, 152, 250, 351, 370, 468, 686, 1339, 1358, 1456, 1674, 2205, 2224, 2322, 2540, 2662, 3528, 4394, 4921, 4940, 5038, 5256, 6244, 6867, 6886, 6984, 7110, 7202, 8190, 9056, 9826, 11772, 12175, 12194, 12292, 12510, 13498, 13718, 14364

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

			133 belongs to the sequence because it can be written as 2^3 + 5^3.

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

Cf. A001235, A003325, A045636, A045699, A086120, A086121.

sumList[x_List, y_List] := Module[{t = {}}, Do[t = Union[t, x[[i]] + y], {i, Length[x]}];  t]; nn = 10; Select[sumList[Prime[Range[nn]]^3, Prime[Range[nn]]^3], # < Prime[nn]^3 &]

More terms from Alexander Adamchuk, Nov 10 2006

A214511 Least number having n orderless representations as p^2 + q^2, where p and q are primes.

Original entry on oeis.org

8, 338, 2210, 10370, 202130, 229970, 197210, 81770, 18423410, 16046810, 12625730, 21899930, 9549410, 370247930, 416392730, 579994610, 338609570, 2155919090, 601741010, 254885930, 10083683090, 4690939370, 29207671610, 30431277890, 22264417370, 23231920010

Offset: 1

T. D. Noe, Jul 26 2012

nonn

A045698(a(n)) = n and A045698(m) < n for m < a(n). - Reinhard Zumkeller, Jul 29 2012

a(53) = 3374376505370. a(52) and terms following a(53) are greater than 4*10^13. - Giovanni Resta, Jul 02 2018

			a(2) = 338 because 338 = 7^2 + 17^2 = 13^2 + 13^2 and 338 is the least number with this property.

Giovanni Resta, Table of n, a(n) for n = 1..51
Index entries for sequences related to sums of squares

Cf. A088919, A045636, A214723.
Cf. A016032 (p and q integers).
Cf. A214512, A214513.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a214511 = (+ 1) . fromJust . (`elemIndex` a045698_list)
-- Reinhard Zumkeller, Jul 29 2012

nn = 10^6; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2, {i, Length[ps]}, {j, i, Length[ps]}]]; t = Select[t, # <= nn &]; t2 = Sort[Tally[t]]; u = Union[Transpose[t2][[2]]]; d = Complement[Range[u[[-1]]], u]; If[d == {}, nLim = u[[-1]], nLim = d[[1]]-1]; t3 = Table[Select[t2, #[[2]] == n &, 1][[1]], {n, nLim}]; Transpose[t3][[1]]

a(21)-a(26) from Donovan Johnson, Jul 29 2012

A214723 Numbers of the form p^2 + q^2, with p and q prime, in exactly one way.

Original entry on oeis.org

8, 13, 18, 29, 34, 50, 53, 58, 74, 98, 125, 130, 146, 170, 173, 178, 194, 218, 242, 290, 293, 298, 314, 365, 370, 386, 458, 482, 530, 533, 538, 554, 698, 722, 818, 845, 850, 866, 962, 965, 970, 986, 1058, 1082, 1202, 1250, 1322, 1370, 1373, 1378, 1394, 1418

Offset: 1

J. Stauduhar, Jul 26 2012

nonn

A045698(a(n)) = 1. - Reinhard Zumkeller, Jul 29 2012

			a(1) = 8 = 2^2 + 2^2, and no other p^2 + q^2 sums to 8.
Both 7^2 + 17^2 and 13^2 + 13^2 sum to 338, so 338 is not in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares

Subsequence of A045636 (numbers of the form p^2 + q^2).

import Data.List (elemIndices)
a214723 n = a214723_list !! (n-1)
a214723_list = elemIndices 1 a045698_list
-- Reinhard Zumkeller, Jul 29 2012

nn = 2000; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2, {i, Length[ps]}, {j, i, Length[ps]}]]; t = Select[t, # <= nn &]; Sort[Transpose[Select[Tally[t], #[[2]] == 1 &]][[1]]] (* T. D. Noe, Jul 26 2012 *)

A045698 Number of ways n can be written as the sum of two squares of primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Felice Russo

a(A214879(n)) = 0; a(A045636(n)) > 0; a(A214723(n)) = 1; a(A214511(n)) = n and a(m) < n for m < A214511(n). - Reinhard Zumkeller, Jul 29 2012

The smallest value of n such that a(n) = 2 is 338. (This helps distinguish it from the characteristic function of A045636.) - Wesley Ivan Hurt, Jun 13 2013

			For example, a(29) = 1 because 29 = 2^2 + 5^2. a(3) = 0 because there is no way to write 3 as sum of two squares of primes.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to sums of squares

Cf. A001248, A010051, A037213.

a045698 n = length $ filter (\x -> x > 0 && a010051' x == 1) $
map (a037213 . (n -)) $
takeWhile (<= div n 2) a001248_list
-- Reinhard Zumkeller, Jul 29 2012

a(n)=my(s=0,q);forprime(p=2,sqrtint(n\2),if(issquare(n-p^2,&q)&&isprime(q),s++));s \\ Charles R Greathouse IV, Jun 04 2014

More terms from Erich Friedman

A124865 Numbers of the form p^2-q^2 with p > q prime.

Original entry on oeis.org

5, 16, 21, 24, 40, 45, 48, 72, 96, 112, 117, 120, 144, 160, 165, 168, 192, 240, 264, 280, 285, 288, 312, 336, 352, 357, 360, 408, 432, 480, 504, 520, 525, 528, 552, 600, 648, 672, 720, 768, 792, 816, 832, 837, 840, 888, 912, 936, 952, 957, 960, 1008, 1032, 1080

Offset: 1

Alexander Adamchuk, Nov 10 2006

nonn

The only prime term is a(1) = 5.
All odd terms are of the form p^2-4.
All even terms are divisible by 8.
Numbers of the form (p^2-q^2)/8 (p, q odd primes, p>q) are listed in A124866.
Elliott & Richner call these "ans numbers". - Charles R Greathouse IV, Feb 17 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
N. E. Elliott and D. Richner, An investigation of the set of ans numbers, Missouri J. of Math. Sci. 15 (2003), pp. 189-199.
Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.

Apart from a(1), a subsequence of A177713.
Cf. A045636 (numbers of the form p^2+q^2, p, q primes).
Cf. A124866 (numbers of the form (p^2-q^2)/8, p, q odd primes, p>q).

With[{nn=60},Take[Union[#[[2]]^2-#[[1]]^2&/@Subsets[Prime[Range[nn]],{2}]],nn]] (* Harvey P. Dale, Aug 21 2015 *)

is(n)=if(n%24, isprimepower(n+4)==2 || isprimepower(n+9)==2, fordiv(n/4,d, if(isprime(n/d/4+d) && isprime(n/d/4-d), return(1))); 0) \\ Charles R Greathouse IV, Feb 17 2014

a(n) >> n log n, see Luca. - Charles R Greathouse IV, Feb 17 2014

A143850 Numbers of the form (p^2 + q^2)/2, for odd primes p and q.

Original entry on oeis.org

9, 17, 25, 29, 37, 49, 65, 73, 85, 89, 97, 109, 121, 145, 149, 157, 169, 185, 193, 205, 229, 241, 265, 269, 277, 289, 325, 349, 361, 409, 425, 433, 445, 481, 485, 493, 505, 529, 541, 565, 601, 625, 661, 685, 689, 697, 709, 745, 769, 829, 841, 845, 853, 865

Offset: 1

T. D. Noe, Sep 03 2008

nonn

The primes in this sequence are listed in A103739.
a(n) mod 4 = 1. See A227697 for related sequence. - Richard R. Forberg, Sep 22 2013
The squares of primes in this sequence form the subsequence A001248 \ {4}. - Bernard Schott, Jul 09 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001248, A045636, A103739, A227697.

Cf. A075892 (a subsequence).

Take[Union[Total[#]/2&/@(Tuples[Prime[Range[2,20]],2]^2)],60] (* Harvey P. Dale, Dec 28 2014 *)

list(lim)=my(v=List(), p2); lim\=1; if(lim<9, lim=8); forprime(p=3, sqrtint(2*lim-9), p2=p^2; forprime(q=3, min(sqrtint(2*lim-p2), p), listput(v, (p2+q^2)/2))); Set(v) \\ Charles R Greathouse IV, Feb 14 2017

A086120 Natural numbers of the form p^3 - q^3, where p and q are primes.

Original entry on oeis.org

19, 98, 117, 218, 316, 335, 866, 988, 1206, 1304, 1323, 1854, 1946, 2072, 2170, 2189, 2716, 3582, 4570, 4662, 4788, 4886, 4905, 5308, 5402, 5528, 6516, 6734, 6832, 6851, 7254, 9970, 10586, 10836, 11824, 12042, 12140, 12159, 12222, 17530, 17624, 18268

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

To find all differences p^3 - q^3 less than N, it is required that all primes p and q up to sqrt(N/6) be tested.

			117 belongs to the sequence because it can be written as 5^3 - 2^3.

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

Cf. A086119, A086121. Also see A045636, A045699.

sumList[x_List, y_List] := (punchline = {}; Do[punchline = Union[punchline, x[[i]] + y], {i, Length[x]}]; punchline); posPart[x_List] := (punchline = {}; Do[If[x[[i]] > 0, punchline = Union[punchline, {x[[i]]}]], {i, Length[x]}]; punchline); posPart[sumList[Prime[Range[10]]^3, - Prime[Range[10]]^3]]
nn=10^5; Union[Reap[Do[n=Prime[i]^3-Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[Sqrt[nn/6]]}, {j,i-1}]][[2,1]]] (* T. D. Noe, Oct 04 2010 *)
With[{upto=20000},Select[Abs[#[[1]]-#[[2]]]&/@Subsets[Prime[ Range[ Sqrt[ upto/6]]]^3,{2}]//Union,#<=upto&]] (* Harvey P. Dale, Dec 10 2017 *)

Corrected by T. D. Noe, Oct 04 2010

A214515 Numbers of the form p^2 + q^2 + r^2 + s^2, where p, q, r, and s are primes.

Original entry on oeis.org

16, 21, 26, 31, 36, 37, 42, 47, 52, 58, 61, 63, 66, 68, 71, 76, 79, 82, 84, 87, 92, 100, 103, 106, 108, 111, 116, 124, 127, 132, 133, 138, 143, 148, 151, 154, 156, 159, 164, 172, 175, 178, 180, 181, 183, 186, 188, 191, 196, 199, 202, 204, 207, 212, 220, 223

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), A214514 (three primes).

nn = 10^3; ps = Prime[Range[PrimePi[Sqrt[nn]]]]; t = Flatten[Table[ps[[i]]^2 + ps[[j]]^2 + ps[[k]]^2 + ps[[l]]^2, {i, Length[ps]}, {j, i, Length[ps]}, {k, j, Length[ps]}, {l, k, Length[ps]}]]; t = Select[t, # <= nn &]; Union[t]

A214879 Numbers that cannot be written as sum of the squares of two primes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76

Offset: 1

Reinhard Zumkeller, Jul 29 2012

nonn

A045698(a(n)) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A045636 (complement).

import Data.List (elemIndices)
a214879 n = a214879_list !! (n-1)
a214879_list = elemIndices 0 a045698_list
-- Reinhard Zumkeller, Jul 29 2012

is(n)=forprime(p=2,sqrtint(n), if(isprimepower(n-p^2)==2, return(0))); 1 \\ Charles R Greathouse IV, Sep 01 2015

from sympy import primerange
def aupto(limit):
    primes = list(primerange(2, int((limit-4)**.5)+2))
    nums = [p*p + q*q for i, p in enumerate(primes) for q in primes[i:]]
    return sorted(set(range(limit+1)) - set(k for k in nums if k <= limit))
print(aupto(76)) # Michael S. Branicky, Aug 13 2021

a(n) ~ n. - Charles R Greathouse IV, Sep 01 2015

A086121 Positive sums or differences of two cubes of primes.

Original entry on oeis.org

16, 19, 35, 54, 98, 117, 133, 152, 218, 250, 316, 335, 351, 370, 468, 686, 866, 988, 1206, 1304, 1323, 1339, 1358, 1456, 1674, 1854, 1946, 2072, 2170, 2189, 2205, 2224, 2322, 2540, 2662, 2716, 3528, 3582, 4394, 4570, 4662, 4788, 4886, 4905, 4921, 4940, 5038

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

			117 and 133 each belong to the (set) sequence because can be written as 117 = 5^3 - 2^3 and 133 = 5^3 + 2^3.

Hans Havermann and T. D. Noe, Table of n, a(n) for n = 1..20000.

Cf. A086119, A086120. Also see A045636, A045699.

nn=10^6; td=Reap[Do[n=Prime[i]^3-Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[Sqrt[nn/6]]}, {j,i-1}]][[2,1]]; ts=Reap[Do[n=Prime[i]^3+Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[nn^(1/3)]}, {j,i}]][[2,1]]; Union[td,ts] (* T. D. Noe, Oct 04 2010 *)
n = 100; Select[Sort@Flatten@ Table[Prime[i]^3 + (-1)^k Prime[j]^3, {i, n}, {j, i}, {k, 2}], 0 < # < (Prime[n] + 2)^3 - Prime[n]^3 &] (* Ray Chandler, Oct 05 2010 *)

Edited by N. J. A. Sloane, Oct 05 2010 to remove a discrepancy between the terms of the sequence and the b-file. The old Mma program and b-file were wrong.

cp's OEIS Frontend

A086119 Numbers of the form p^3 + q^3, p, q primes.

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A214511 Least number having n orderless representations as p^2 + q^2, where p and q are primes.

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A214723 Numbers of the form p^2 + q^2, with p and q prime, in exactly one way.

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A045698 Number of ways n can be written as the sum of two squares of primes.

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A124865 Numbers of the form p^2-q^2 with p > q prime.

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A143850 Numbers of the form (p^2 + q^2)/2, for odd primes p and q.

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A086120 Natural numbers of the form p^3 - q^3, where p and q are primes.

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A214515 Numbers of the form p^2 + q^2 + r^2 + s^2, where p, q, r, and s are primes.

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