A000446 -id:A000446 - 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

A093195 Least number which is the sum of two distinct nonzero squares in exactly n ways.

Original entry on oeis.org

5, 65, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 126953125, 160225, 1221025, 3453125, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 30994415283203125, 5928325, 303460625, 53955078125, 35409725, 100140625

Offset: 1

Lekraj Beedassy, Apr 22 2004

nonn

An algorithm to compute the n-th term of this sequence: Write each of 2n and 2n+1 as products of their divisors in all possible ways and in decreasing order. For each product, equate each divisor in the product to (a1+1)(a2+1)...(ar+1), so that a1 >= a2 >= a3 >= ... >= ar, and solve for the ai. Evaluate A002144(1)^a1 * A002144(2)^a2 * ... * A002144(r)^ar for each set of values determined above, then the smaller of these products is the least integer to have precisely n partitions into a sum of two distinct positive squares. [Ant King, Dec 14 2009; May 26 2010]

Ray Chandler, Table of n, a(n) for n = 1..1438 (a(1439) exceeds 1000 digits).

Cf. A002144, A018782, A054994, A025302-A025311 (first entries). See A016032, A000446 and A124980 for other versions.

b(k)=my(c=0);for(i=1,sqrtint((k-1)\2),if(issquare(k-i^2),c+=1));c \\ A025441
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(A018782(2n), A018782(2n+1)).

More terms from Ant King, Dec 14 2009 and Feb 07 2010

A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

Original entry on oeis.org

1, 5, 25, 65, 625, 325, 15625, 1105, 4225, 8125, 9765625, 5525, 244140625, 203125, 105625, 27625, 152587890625, 71825, 3814697265625, 138125, 2640625, 126953125, 2384185791015625, 160225, 17850625, 3173828125, 1221025, 3453125

Offset: 1

N. J. A. Sloane

nonn

a(n) is least term of A054994 with exactly n divisors. - Ray Chandler, Jan 05 2012
From Jianing Song, Apr 24 2021: (Start)
a(n) is the smallest k such that A004018(k) = 4n.
Also a(n) is the smallest index of n in A002654.
a(n) is the smallest term in A004613 that has exactly n divisors.
This is a subsequence of A054994. (End)

			4225 = 5^2 * 13^2 is the smallest number all of whose prime factors are congruent to 1 modulo 4 with exactly 9 divisors, so a(9) = 4225. - _Jianing Song_, Apr 24 2021

Ray Chandler, Table of n, a(n) for n = 1..1432 (a(1433) exceeds 1000 digits).

Cf. A002144, A002654, A004018, A005179, A054994, A000446, A124980, A016032, A093195.

(* This program is not convenient to compute huge terms - A054994 is assumed to be computed with maxTerm = 10^16 *) a[n_] := Catch[ For[k = 1, k <= Length[A054994], k++, If[DivisorSigma[0, A054994[[k]]] == n, Throw[A054994[[k]]]]]]; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jan 21 2013, after Ray Chandler *)

primelist(d,r,l) = my(v=vector(l), i=0); if(l>0, forprime(p=2, oo, if(Mod(p,d)==r, i++; v[i]=p; if(i==l, break())))); v
prodR(n, maxf)=my(dfs=divisors(n), a=[], r); for(i=2, #dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a, [[n]]), r=prodR(n/dfs[i], min(dfs[i], maxf)); for(j=1, #r, a=concat(a, [concat(dfs[i], r[j])]))))); a
A018782(n)=my(pf=prodR(n, n), a=1, b, v=primelist(4, 1, bigomega(n))); for(i=1, #pf, b=prod(j=1, length(pf[i]), v[j]^(pf[i][j]-1)); if(bJianing Song, Apr 25 2021, following R. J. Mathar's program for A005179.

A000446(n) = min(a(2n-1), a(2n)) for n > 1.
A124980(n) = min(a(2n-1), a(2n)).
A016032(n) = min(2*a(2n-1), a(2n), a(2n+1)).
A093195(n) = min(a(2n), a(2n+1)).
From Jianing Song, Apr 24 2021: (Start)
If the factorization of n into primes is n = Product_{i=1..r} p_i with p_1 >= p_2 >= ... >= p_r, then a(n) <= (q_1)^((p_1)-1) * (q_2)^((p_2)-1) * ... * (q_r)^((p_r)-1), where q_1 < q_2 < ... < q_r are the first r primes congruent to 1 modulo 4. The smallest n such that the equality does not hold is n = 16.
a(n) <= 5^(n-1) for all n, where the equality holds if and only if n = 1 or n is a prime.
a(p*q) = 5^(p-1) * 13^(q-1) for primes p >= q. (End)

A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

Original entry on oeis.org

1, 25, 325, 1105, 4225, 5525, 203125, 27625, 71825, 138125, 2640625, 160225, 17850625, 1221025, 1795625, 801125, 1650390625, 2082925, 49591064453125, 4005625, 44890625, 2158203125, 30525625, 5928325, 303460625, 53955078125, 35409725, 100140625

Offset: 1

Artur Jasinski, Nov 15 2006

nonn

The number must be strictly positive, but one of the squares may be zero, as we see from a(1) = 1 = 1^2 + 0^2 and a(2) = 25 = 3^2 + 4^2 = 5^0 + 0^2. - M. F. Hasler, Jul 07 2024

			a(3) = 325 is decomposable in 3 ways: 15^2 + 10^2 = 17^2 + 6^2 = 18^2 + 1^2.

Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).

Cf. A002144, A018782, A054994, A118882.

See A016032, A000446 and A093195 for other versions.

A124980(n)={for(a=1, oo, A000161(a)==n && return(a))} \\ R. J. Mathar, Nov 29 2006, edited by M. F. Hasler, Jul 07 2024

PD(n, L=n, D=Vecrev(divisors(n)[^1])) = { if(n>1, concat(vector(#D, i, if(D[i] > L, [], D[i] < n, [concat(D[i], P) | P <- PD(n/D[i], D[i])], [[n]]))), [[]])}
apply( {A124980(n)=vecmin([prod(i=1, #a, A002144(i)^(a[i]-1)) | a<-concat([PD(n*2,n), PD(n*2-1)])])}, [1..44]) \\ M. F. Hasler, Jul 07 2024

from sympy import divisors, isprime, prod
def PD(n, L=None): return [[]] if n==1 else [
    [d]+P for d in divisors(n)[:0:-1] if d <= (L or n) for P in PD(n//d, d)]
A2144=lambda upto=999: filter(isprime, range(5, upto, 4))
def A124980(n):
    return min(prod(a**(f-1) for a,f in zip(A2144(), P))
               for P in PD(n*2, n)+PD(n*2-1)) # M. F. Hasler, Jul 07 2024

a(n) = A000446(n), n > 1. - R. J. Mathar, Jun 15 2008
a(n) = min(A018782(2n-1), A018782(2n)).
a(n) = min { k > 0 | A000161(k) = n }. - M. F. Hasler, Jul 07 2024

More terms from R. J. Mathar, Nov 29 2006

Edited and extended by Ray Chandler, Jan 07 2012

A000448 Smallest number that is the sum of 2 squares in at least n ways.

Original entry on oeis.org

0, 25, 325, 1105, 4225, 5525, 27625, 27625, 71825, 138125, 160225, 160225, 801125, 801125, 801125, 801125, 2082925, 2082925, 4005625, 4005625, 5928325, 5928325, 5928325, 5928325, 29641625, 29641625, 29641625, 29641625, 29641625, 29641625, 29641625, 29641625

Offset: 1

N. J. A. Sloane

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A000446 (smallest number that is the sum of 2 squares in exactly n ways).

a(n) = min { A000446(k) ; k >= n }. - M. F. Hasler, Jul 06 2024

Extended by Ray Chandler, Jan 13 2012

A085625 Numbers that are the sum of 2 squares in exactly 2 ways.

Original entry on oeis.org

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 338, 340, 365, 370, 377, 400, 410, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 629, 676, 680, 685, 689, 697, 730

Offset: 1

Hugo Pfoertner, Jul 09 2003

nonn

Wells erroneously writes that this sequence begins as 50, 65, 85, 145, ... . - Stefano Spezia, Sep 07 2024

			a(3) = 65 because 65 = 8^2 + 1^2 = 7^2 + 4^2;
a(4) = 85 because 85 = 9^2 + 2^2 = 7^2 + 6^2.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.

Robert Price, Table of n, a(n) for n = 1..16794

Cf. A000161, A000446, A224443, A294282, A295153, A295489, A295748.

Select[Range[730], Length[PowersRepresentations[#,2,2]]==2 &] (* Stefano Spezia, Sep 07 2024 *)

n such that A000161(n) = 2.

A255018 Smallest number that is the sum of 3 nonnegative cubes in exactly n ways.

Original entry on oeis.org

4, 0, 216, 5104, 13896, 161568, 1259712, 2016496, 2562624, 14926248, 58995000, 34012224, 150547032, 471960000, 119095488, 1259712000, 952763904, 5159780352, 3974344704, 2176782336, 10077696000, 2985984000, 36330467328, 30723115968, 23887872000, 17414258688, 72825163776, 75686967000

Offset: 0

Alex Ratushnyak, Feb 25 2015

nonn

			a(0) = 4 because the smallest number that cannot be represented as a sum of 3 nonnegative cubes is 4.
a(1) = 0 is the sum of three 0's.
a(2) = 216 = 3^3 + 4^3 + 5^3 = 6^3 + 0 + 0.
a(3) = 5104 = 1 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A000578, A000446, A011541, A025418, A025419.

TOP = 6000000
a = [0]*TOP
for b in range(TOP):
  b3 = b**3
  if b3*3>=TOP: break
  for c in range(b,TOP):
    c3 = b3 + c**3
    if c3>=TOP: break
    for d in range(c,TOP):
      res = c3 + d**3
      if res>=TOP: break
      a[res] += 1
m = max(a)
r = [-1] * (m+1)
for i in range(TOP):
    if r[a[i]]==-1:  r[a[i]]=i
print(r)

More terms from Rémy Sigrist, Jul 14 2020

A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.

Original entry on oeis.org

2, 0, 121, 2299, 6061, 43681, 66671, 33659659, 187891, 1266749, 8067191, 639533521, 2066801

Offset: 0

Seiichi Manyama, Jul 02 2024

a(n) is the smallest nonnegative k such that A374276(k) = n.
a(14) = 36735721.
a(15) = 153276629.
a(16) = 7703531.
a(18) = 39269219.
a(20) = 250082921.
a(24) = 84738841.
a(32) = 454508329.

Cf. A000446, A198799.

Cf. A031363, A374091, A374095, A374276.

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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A093195 Least number which is the sum of two distinct nonzero squares in exactly n ways.

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A018782 Smallest k such that circle x^2 + y^2 = k passes through exactly 4n integer points.

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A124980 Smallest strictly positive number decomposable in n different ways as a sum of two squares.

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A000448 Smallest number that is the sum of 2 squares in at least n ways.

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A085625 Numbers that are the sum of 2 squares in exactly 2 ways.

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A255018 Smallest number that is the sum of 3 nonnegative cubes in exactly n ways.

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A374275 Smallest k such that k can be written in exactly n ways as x^2 + 3xy + y^2 with 0 <= x <= y.