A025293 -id:A025293 - cp's OEIS frontend

A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.

2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, 160225, 1221025, 2442050, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 61051250, 5928325, 303460625, 53955078125, 35409725, 100140625, 1289367675781250

Offset: 1

Robert G. Wilson v

			a(0) = 1 as 1 is the least positive integer not expressible as the sum of two squared positives.
a(1) = 2 from 2 = 1^2 + 1^2.
a(2) = 50 from 50 = 1^2 + 7^2 = 5^2 + 5^2.

A. Beiler, Recreations in the Theory of Numbers, Dover, pp. 140-141.

T. D. Noe and Ray Chandler, Table of n, a(n) for n = 1..2178 (a(2179) exceeds 1000 digits).
C. Rivera, Puzzle 62
Eric Weisstein's World of Mathematics, Square Number
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A018825, A048610, A025284-A025293 (first entries).

See A000446, A124980 and A093195 for other versions.

Array[Block[{k = 1}, While[Length@ DeleteCases[PowersRepresentations[k, 2, 2], ?(! FreeQ[#, 0] &)] != #, k++]; k] &, 6] (* _Michael De Vlieger, Mar 31 2019 *)

b(k)=my(c=0);for(i=1,sqrtint(k\2),if(issquare(k-i^2),c+=1));c \\ A025426
for(n=1,10,k=1;while(k,if(b(k)==n,print1(k,", ");break);k+=1)) \\ Derek Orr, Mar 20 2019

a(n) = min(2*A018782(2n-1), A018782(2n), A018782(2n+1)).

Corrected and extended by Jud McCranie

Definition improved by several correspondents, Nov 12 2007

A025285 Numbers that are the sum of 2 nonzero squares in exactly 2 ways.

Original entry on oeis.org

50, 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 338, 340, 365, 370, 377, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 680, 685, 689, 697, 730, 740, 745, 754, 765, 785, 793, 800, 820

Offset: 1

David W. Wilson

nonn

Order and signs don't count. E.g. 50 = 5^2+5^2 = 7^2+1^2 (= (-5)^2+5^2, but that doesn't count as different).
A131574 is a subsequence. - Zak Seidov, Jan 31 2014
A025426(a(n)) = 2. - Reinhard Zumkeller, Feb 26 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Square Number.
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A025284, A025286, A025287, A025288, A025289, A025290, A025291, A025292, A025293, A025426.

Cf. A007692.

a025285 n = a025285_list !! (n-1)
a025285_list = filter ((== 2) . a025426) [1..]
-- Reinhard Zumkeller, Feb 26 2015

selQ[n_] := Length[ Select[ PowersRepresentations[n, 2, 2], Times @@ # != 0 &]] == 2; Select[Range[1000], selQ] (* Jean-François Alcover, Oct 03 2013 *)

is(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n-1), issquare(n-k^2))==2 \\ Charles R Greathouse IV, May 24 2016

is(n)=my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)==4 \\ Charles R Greathouse IV, May 24 2016

a(n) >= A007692(n) with equality only for n <= 16. - Alois P. Heinz, Mar 23 2023

A025301 Numbers that are the sum of 2 nonzero squares in 10 or more ways.

Original entry on oeis.org

138125, 160225, 204425, 226525, 235625, 276250, 292825, 300625, 308125, 320450, 333125, 337025, 348725, 359125, 386425, 393125, 403325, 408850, 416585, 430625, 435625, 453050, 456025, 469625, 471250, 491725, 493025, 495625, 499525, 505325

Offset: 1

David W. Wilson

nonn

Sequences A025320 and A025301 are different. 2*5^18 = 7629394531250 = 182125^2 + 2756125^2 = 390625^2 + 2734375^2 = 596875^2 + 2696875^2 = 799687^2 + 2643841^2 = 946555^2 + 2594885^2 = 1140625^2 + 2515625^2 = 1328125^2 + 2421875^2 = 1507975^2 + 2314175^2 = 1799375^2 + 2095625^2 = 1953125^2 + 1953125^2 (not distinct squares) is not in A025320. - Vaclav Kotesovec, Feb 27 2016

Numbers in A025301 but not in A025320 are exactly those numbers of the form 2*p_1^(2*a_1)*p_2^(2*a_2)*...*p_m^(2*a_m)*q^18 where p_i are primes of the form 4k+3 and q is a prime of the form 4k+1. Thus 2*5^18 is the smallest term in A025301 that is not in A025320. - Chai Wah Wu, Feb 27 2016

Cf. A025293, A025300, A025338.

nn = 505325; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i}]; Flatten[Position[t, ?(# >= 10 &)]] (* _T. D. Noe, Apr 07 2011 *)

A025311 Numbers that are the sum of 2 distinct nonzero squares in exactly 10 ways.

Original entry on oeis.org

138125, 235625, 276250, 300625, 308125, 333125, 393125, 430625, 435625, 471250, 495625, 552500, 563125, 593125, 601250, 616250, 648125, 666250, 670625, 723125, 743125, 775625, 786250, 788125, 820625, 861250, 871250, 885625, 918125, 942500

Offset: 1

David W. Wilson

nonn

Where does this first differ from A025293? - R. J. Mathar, Jun 24 2025

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

Cf. A025293.

nn = 942500; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i - 1}]; Flatten[Position[t, 10]] (* T. D. Noe, Apr 07 2011 *)

A236711 Numbers that are the sum of 2 nonzero squares in exactly 11 ways.

Original entry on oeis.org

5281250, 9031250, 21125000, 26281250, 36125000, 42781250, 47531250, 52531250, 81281250, 84500000, 87781250, 105125000, 116281250, 126953125, 144500000, 166015625, 166531250, 171125000, 190125000, 210125000, 236531250, 241340450, 247531250, 253906250, 258781250

Offset: 1

Zak Seidov, Jan 30 2014

nonn

Are all terms multiples of 5?
The answer is "no"; 2789895602 = 2 * 13^6 * 17^2 is a term that is not a multiple of 5. Is it the first such term? - Zak Seidov, Jul 05 2015
a(152) = 2789895602 is the first term that is not divisible by 5. In the first 1000 terms, the only powers to which 5 appears as a factor are 0 (for 10 terms, beginning with a(152), after which the next does not occur until a(331)), 2 (for only 14 terms, the smallest of which is a(22) = 241340450 = 2 * 5^2 * 13^6), 6 (for 360 terms), and 10 (for the remaining 616 terms). - Jon E. Schoenfield, Jul 07 2015

			5281250 = x^2 + y^2 with {x,y} = {71,2297}, {245,2285}, {325,2275}, {575,2225}, {875,2125}, {949,2093}, {1105,2015}, {1175,1975}, {1435,1795}, {1567,1681}, {1625,1625}.

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
C. Rivera, Puzzle 62

Cf. A000404, A016032, A025285-A025293.

More terms from Jon E. Schoenfield, Jul 05 2015

A115592 Number of distinct representations of n as the sum of two nonzero squares nontrivially divides the number of distinct representations of n as the sum of two primes.

Original entry on oeis.org

50, 200, 260, 290, 370, 530, 578, 610, 650, 740, 884, 962, 1060, 1170, 1300, 1370, 1460, 1508, 1530, 1690

Offset: 1

Jonathan Vos Post, Apr 02 2006

"Nontrivially" meaning the number of distinct representations of n as the sum of two nonzero squares is at least 2.

			a(1) = 50 because 50 = 1^2 + 49^2 = 5^2 + 5^2 (2 distinct ways as sum of nonzero squares) and 50 = 3 + 47 = 7 + 43 = 13 + 37 = 19 + 31 (4 distinct ways as sum of two primes) and 2 | 4.
a(2) = 200 because 200 = 2^2 + 14^2 = 10^2 + 10^2 (2 distinct ways as sum of nonzero squares) and 200 = 3 + 197 = 7 + 193 = 19 + 181 = 37 + 163 = 43 + 157 = 61 + 139 = 73 + 127 = 97 + 103, (8 distinct ways as sum of two primes) and 2 | 8.
a(3) = 260 because (2 distinct ways as sum of nonzero squares) divides (10 distinct ways as sum of two primes).
a(4) = 290 because (2 distinct ways as sum of nonzero squares) divides (10 distinct ways as sum of two primes).
a(5) = 370 because (2 distinct ways as sum of nonzero squares) divides (14 distinct ways as sum of two primes).
a(6) = 530 because (2 distinct ways as sum of nonzero squares) divides (14 distinct ways as sum of two primes).
a(7) = 578 because (2 distinct ways as sum of nonzero squares) divides (12 distinct ways as sum of two primes).
a(8) = 610 because (2 distinct ways as sum of nonzero squares) divides (20 distinct ways as sum of two primes).
a(9) = 650 because (3 distinct ways as sum of nonzero squares) divides (21 distinct ways as sum of two primes).
a(10) = 740 because (2 distinct ways as sum of nonzero squares) divides (18 distinct ways as sum of two primes).
1300 is in the sequence because (3 distinct ways as sum of nonzero squares) divides (33 distinct ways as sum of two primes).

Cf. A000040, A025284-A025293.

Numbers n such that #{a^2 + b^2 = n and a>0 and b>0 and a>= b} > 1 and #{a^2 + b^2 = n and a>0 and b>0 and a>= b} | #{p(i) + p(j) = n and i >= j where p(k) = A000040(k)}.

More terms from Nate Falkenstein (njf127(AT)psu.edu), Apr 25 2006

cp's OEIS Frontend

A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.

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A025285 Numbers that are the sum of 2 nonzero squares in exactly 2 ways.

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A025301 Numbers that are the sum of 2 nonzero squares in 10 or more ways.

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A025311 Numbers that are the sum of 2 distinct nonzero squares in exactly 10 ways.

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A236711 Numbers that are the sum of 2 nonzero squares in exactly 11 ways.

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A115592 Number of distinct representations of n as the sum of two nonzero squares nontrivially divides the number of distinct representations of n as the sum of two primes.

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