A025285 -id:A025285 - cp's OEIS frontend

A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways.

50, 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 325, 338, 340, 365, 370, 377, 410, 425, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 650, 680, 685, 689, 697, 725, 730, 740, 745, 754, 765

Offset: 1

N. J. A. Sloane

A025426(a(n)) > 1. - Reinhard Zumkeller, Aug 16 2011
For the question that is in the link AskNRICH Archive: It is easy to show that (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2 = (a*d + b*c)^2 + (a*c - b*d)^2. So terms of this sequence can be generated easily. 5 is the least number of the form a^2 + b^2 where a and b distinct positive integers and this is a list sequence. This is the why we observe that there are many terms which are divisible by 5. - Altug Alkan, May 16 2016
Square roots of square terms: {25, 50, 65, 75, 85, 100, 125, 130, 145, 150, 169, 170, 175, 185, 195, 200, 205, 221, 225, 250, 255, 260, 265, 275, 289, 290, 300, 305, ...}. They are also listed by A009177. - Michael De Vlieger, May 16 2016

			50 is a term since 1^2 + 7^2 and 5^2 + 5^2 equal 50.
25 is not a term since though 3^2 + 4^2 = 25, 25 is square, i.e., 0^2 + 5^2 = 25, leaving it with only one possible sum of 2 nonzero squares.
625 is a term since it is the sum of squares of {0,25}, {7,24}, and {15,20}.

Ming-Sun Li, Kathryn Robertson, Thomas J. Osler, Abdul Hassen, Christopher S. Simons and Marcus Wright, "On numbers equal to the sum of two squares in more than one way", Mathematics and Computer Education, 43 (2009), 102 - 108.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 125.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
AskNRICH Archive, Numbers expressible as the sum of 2 squares in more than one way
D. J. C. Mackay and S. Mahajan, Numbers that are Sums of Squares in Several Ways
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Subsequence of A001481. A subsequence is A025285 (2 ways).

Cf. A004431, A118882, A000404, A018825, A025284 (one way).

import Data.List (findIndices)
a007692 n = a007692_list !! (n-1)
a007692_list = findIndices (> 1) a025426_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range@ 800, Length@ Select[PowersRepresentations[#, 2, 2], First@ # != 0 &] > 1 &] (* Michael De Vlieger, May 16 2016 *)

isA007692(n)=nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb >= 2; \\ Altug Alkan, May 16 2016

is(n)=my(t); if(n<9, return(0)); for(k=sqrtint(n\2-1)+1,sqrtint(n-1), if(issquare(n-k^2)&&t++>1, return(1))); 0 \\ Charles R Greathouse IV, Jun 08 2016

A025294 Numbers that are the sum of 2 nonzero squares in 3 or more ways.

Original entry on oeis.org

325, 425, 650, 725, 845, 850, 925, 1025, 1105, 1250, 1300, 1325, 1445, 1450, 1525, 1625, 1690, 1700, 1825, 1850, 1885, 2050, 2125, 2210, 2225, 2405, 2425, 2465, 2525, 2600, 2650, 2665, 2725, 2825, 2873, 2890, 2900, 2925, 3050, 3125, 3145, 3250, 3380

Offset: 1

David W. Wilson

nonn

A025426(a(n)) > 2. - Reinhard Zumkeller, Feb 26 2015

Robert Israel and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 2508 terms from Robert Israel)
Index entries for sequences related to sums of squares

Complement of A025285 relative to A007692. - Washington Bomfim, Oct 24 2010

Cf. A025426.

a025294 n = a025294_list !! (n-1)
a025294_list = filter ((> 2) . a025426) [1..]
-- Reinhard Zumkeller, Feb 26 2015

  N:= 100000: # generate all entries <=N
SSQ:= {}: SSQ2:= {}: SSQ3:= {}:
for a from 1 to floor(sqrt(N)) do
  for b from a to floor(sqrt(N-a^2)) do
    n:= a^2 + b^2;
    if member(n,SSQ2) then SSQ3:= SSQ3 union {n}
    elif member(n,SSQ) then SSQ2:= SSQ2 union {n}
    else SSQ:= SSQ union {n}
    end if
end do end do:
SSQ3;  # Robert Israel, Jan 20 2013

okQ[n_] := Length[Select[PowersRepresentations[n, 2, 2][[All, 1]], Positive] ] > 2; Select[Range[5000], okQ] (* Jean-François Alcover, Mar 04 2019 *)

A343708 Numbers that are the sum of two positive cubes in exactly two ways.

Original entry on oeis.org

1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597, 439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, 684019, 704977, 805688, 842751, 885248, 886464

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A001235 at term 455 because 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3 = A011541(3). Thus, this term is not in this sequence but is in A001235.

			13832 is in this sequence because 13832 = 2^3 + 24^3 = 18^3 + 20^3.

David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Cf. A001235, A025285, A025396, A338667, A344804.

Select[Range@70000,Length@Select[PowersRepresentations[#,2,3],FreeQ[#,0]&]==2&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,1000)]#n
for pos in cwr(power_terms,2):#m
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v == 2])#s
for x in range(len(rets)):
    print(rets[x])

A273318 Numbers n such that n+k-1 is the sum of two nonzero squares in exactly k ways for all k = 1, 2, 3.

Original entry on oeis.org

58472, 79208, 104616, 150048, 160848, 205648, 224648, 234448, 252808, 259648, 259920, 294048, 297448, 387648, 421648, 433448, 462976, 488448, 506248, 563048, 621448, 683648, 770976, 790848, 799648, 837448, 1008648, 1040848, 1084904, 1186632, 1195648, 1205648, 1212064

Offset: 1

Altug Alkan, May 19 2016

nonn

Numbers n such that n+k-1 is the sum of two nonzero squares in exactly 4-k ways for all k = 1, 2, 3 are 22984, 65600, 80800, 85544, ...

			58472 is a term because;
58472 = 86^2 + 226^2.
58473 = 48^2 + 237^2 = 147^2 + 192^2.
58474 = 57^2 + 235^2 = 125^2 + 207^2 = 143^2 + 195^2.

Robert Israel, Table of n, a(n) for n = 1..1095

Cf. A000404, A025284, A025285, A025286.

N:= 10^6: # get all terms <= N-2
R:= Vector(N):
for x from 1 to floor(sqrt(N)) do
  for y from 1 to min(x,floor(sqrt(N-x^2))) do
    R[x^2+y^2]:= R[x^2+y^2]+1
od od:
count:= 0:
for n from 1 to N-2 do
  if [R[n],R[n+1],R[n+2]] = [1,2,3] then
  count:= count+1; A[count]:= n;
fi
od:
seq(A[i],i=1..count); # Robert Israel, May 19 2016

is(n,k) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == k; }
isok(n) = is(n,1) && is(n+1,2) && is(n+2,3);

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

A273529 Primes p of the form x^2 + y^2 such that p+1 is the sum of the two nonzero squares in exactly 2 ways.

Original entry on oeis.org

337, 409, 449, 577, 1009, 1129, 1489, 1801, 2377, 2521, 2609, 2689, 2833, 3041, 3169, 3329, 3361, 3433, 3529, 3889, 4049, 4513, 4657, 5209, 5569, 5689, 5857, 5881, 5953, 6529, 6553, 6569, 7177, 7297, 8009, 8089, 8209, 8329, 8641, 8737, 8761, 9433, 9697, 9769, 10169, 10321

Offset: 1

Altug Alkan, May 24 2016

nonn

Number of prime divisors (counted with multiplicity) of p+1 is 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 5, 3, 5, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 3, 3, 3, 3, 5, 3, 7, 3, 3, 5, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, ...

In this sequence, 20249 is the first p such that p+1 has even number of prime divisors (counted with multiplicity).

			The prime 409 is a term because 409 = 3^2 + 20^2 and 410 = 7^2 + 19^2 = 11^2 + 17^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002313, A025285.

is(n, k)=nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == k;
isok(n) = isprime(n) && is(n, 1) && is(n+1, 2);

is(n)=if(n%8!=1 || !isprime(n), return(0)); my(f=factor((n+1)/2), t=1); for(i=1, #f~, if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); t==3 || t==4 \\ Charles R Greathouse IV, May 24 2016

a(n) mod 8 = 1.

A076520 n appears once if n is the sum of 2 nonzero squares in 1 way, twice if n is the sum of 2 squares in 2 ways, 3 times if n is the sum of 2 squares 3 ways etc.

Original entry on oeis.org

2, 5, 5, 8, 10, 10, 13, 13, 17, 17, 18, 20, 20, 25, 25, 26, 26, 29, 29, 32, 34, 34, 37, 37, 40, 40, 41, 41, 45, 45, 50, 50, 50, 52, 52, 53, 53, 58, 58, 61, 61, 65, 65, 65, 65, 68, 68, 72, 73, 73, 74, 74, 80, 80, 82, 82, 85, 85, 85, 85, 89, 89, 90, 90, 97, 97, 98, 100, 100

Offset: 1

Benoit Cloitre, Nov 09 2002

nonn

Index entries for sequences related to sums of squares

Cf. A000404, A001105, A025285, A025302.

lista(nn) = for (n=1, nn, for (i=1, n-1, if (issquare(i) && issquare(n-i), print1(n, ", ")))) \\ Michel Marcus, Nov 30 2013

list(lim)=my(v=List());for(n=2,lim,for(k=1,sqrtint(n-1),if(issquare(n-k^2),listput(v,n))));Vec(v) \\ Charles R Greathouse IV, Dec 01 2013

Does limit n -> infinity a(n)/n exist?

Answer: Yes, and it is equal to 4/Pi. - Charles R Greathouse IV, Dec 01 2013

Typo in definition corrected by Michel Marcus, Dec 01 2013

A273236 Primes p such that p + k is the sum of two nonzero squares in exactly k ways for all k = 1, 2, 3.

Original entry on oeis.org

563047, 1186631, 1205647, 1421647, 1871503, 2058047, 2615047, 2739103, 2795047, 3703463, 3743647, 4106447, 4723847, 4748047, 4758847, 5744447, 6991847, 8376847, 9951047, 10014847, 12214303, 12773447, 14161183, 14402447, 15232031, 15630847

Offset: 1

Altug Alkan, May 26 2016

nonn

All terms of this sequence are the sum of 4 but no fewer nonzero squares.

			The prime 563047 is a term because 563048 = 218^2 + 718^2, 563049 = 165^2 + 732^2 = 357^2 + 660^2 and 563050 = 71^2 + 747^2 = 141^2 + 737^2 = 505^2 + 555^2.

Cf. A025284, A025285, A025286, A273318.

is(n, k) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == k; }
isok(n) = isprime(n) && is(n+1, 1) && is(n+2, 2) && is(n+3, 3);

A273293 Numbers k such that k and k^2 are the sums of two nonzero squares in exactly two ways.

Original entry on oeis.org

50, 200, 338, 450, 578, 800, 1352, 1682, 1800, 2312, 2450, 2738, 3042, 3200, 3362, 4050, 5202, 5408, 5618, 6050, 6728, 7200, 7442, 9248, 9800, 10658, 10952, 12168, 12800, 13448, 15138, 15842, 16200, 16562, 18050, 18818, 20402, 20808, 21632, 22050, 22472, 23762, 24200, 24642, 25538

Offset: 1

Altug Alkan, May 19 2016

nonn

If k is the sum of 2 nonzero squares in exactly 2 ways, then k = a^2 + b^2 = c^2 + d^2 where (a, b), (c, d) are distinct and a, b, c, d are nonzero. For k^2,
   k^2 = (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2,
   k^2 = (a^2 + b^2)*(c^2 + d^2) = (a*d + b*c)^2 + (a*c - b*d)^2,
   k^2 = (a^2 + b^2)*(a^2 + b^2) = (a^2 - b^2)^2 + (2*a*b)^2,
   k^2 = (c^2 + d^2)*(c^2 + d^2) = (c^2 - d^2)^2 + (2*c*d)^2.
Note that if k is of the form 2*m^2 where m is a nonzero integer, then the first two representations will be the same and one of the last two identities will not be the sum of two nonzero squares and we will have two distinct representations for k^2. This is the case that gives motivation for this sequence.
a(n) is the sum of the areas of the squares on the sides of the integer-sided triangle with hypotenuse A084645(n). - Wesley Ivan Hurt, Jan 05 2022

			50 is a term because 50 = 1^2 + 7^2 = 5^2 + 5^2 and 2500 = 14^2 + 48^2 = 30^2 + 40^2.

Cf. A025285, A084645.

isA273293(n) = {nb = 0; lim = sqrtint(n); for (x=1, lim, if ((n-x^2 >= x^2) && issquare(n-x^2), nb++); ); nb == 2; }
lista(nn) = for(n=1, nn, if(isA273293(n) && isA273293(n^2), print1(n, ", ")));

cp's OEIS Frontend

A007692 Numbers that are the sum of 2 nonzero squares in 2 or more ways.

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A025294 Numbers that are the sum of 2 nonzero squares in 3 or more ways.

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A343708 Numbers that are the sum of two positive cubes in exactly two ways.

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A273318 Numbers n such that n+k-1 is the sum of two nonzero squares in exactly k ways for all k = 1, 2, 3.

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

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A273529 Primes p of the form x^2 + y^2 such that p+1 is the sum of the two nonzero squares in exactly 2 ways.

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A076520 n appears once if n is the sum of 2 nonzero squares in 1 way, twice if n is the sum of 2 squares in 2 ways, 3 times if n is the sum of 2 squares 3 ways etc.

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