A093195 -id:A093195 - 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

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)

A000446 Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

Original entry on oeis.org

0, 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

Offset: 1

N. J. A. Sloane

nonn

Could start with a(0) = 3: the smallest nonnegative integer that can be written as sum of two squares in 0 ways. - M. F. Hasler, Jul 05 2024

			a(1) = 0 because 0 is the smallest integer which is uniquely a unique sum of two squares, namely 0^2 + 0^2.
a(2) = 25 from 25 = 5^2 + 0^2 = 3^2 + 4^2.
a(3) = 325 from 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(4) = 1105 from 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.

Ray Chandler, Table of n, a(n) for n = 1..1458 (a(1459) exceeds 1000 digits).
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A000448 (similar, but "in at least n ways").
Cf. A002144, A018782, A054994.
See A016032, A093195 and A124980 for other versions.

A000446(n)=if(n>1, A124980(n), 0) \\ M. F. Hasler, Jul 06 2024

A000446=lambda n: A124980(n) if n>1 else 0 # M. F. Hasler, Jul 07 2024

An algorithm to compute the n-th term of this sequence for n>1: Write each of 2n and 2n-1 as products of their divisors, in decreasing order and in all possible ways. 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 x A002144(2)^a2 x ... x 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 squares. [Ant King, Oct 07 2010]
a(n) = min(A018782(2n-1), A018782(2n)) for n > 1.
a(n) = A124980(n) for n > 1. - M. F. Hasler, Jul 07 2024

Better description and more terms from David W. Wilson, Aug 15 1996

Definition improved by several correspondents, Nov 12 2007

A328151 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + y^2 = k.

Original entry on oeis.org

0, 2, 5, 50, 65, 1250, 325, 31250, 1105, 8450, 8125, 19531250, 5525, 488281250, 105625, 211250, 27625, 305175781250, 71825, 7629394531250, 138125, 5281250, 126953125, 4768371582031250, 160225, 35701250, 1221025, 2442050, 3453125

Offset: 0

Felix Fröhlich, Oct 05 2019

a(n) is the smallest nonnegative i such that A063725(i) = n.

If a(n) exists, then a(n) is of the form 2*m^2 if and only if n is odd. - Chai Wah Wu, Jun 28 2024

			For n = 3: The sums of the two members of each of the pairs (1, 49), (25, 25) and (49, 1) is 50 and 50 is the smallest nonnegative integer where exactly 3 such pairs exist, so a(3) = 50.

Chai Wah Wu, terms a(n) for n odd and a(n) <= 3433037534088061250.

Cf. A063725, A093195.

a063725(n) = if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ after Charles R Greathouse IV in A063725
a(n) = for(x=0, oo, if(a063725(x)==n, return(x)))

# uses Python code from A063725
from itertools import count
def A328151(n): return next(m for m in ((k**2<<1) if n&1 else k for k in count(0)) if A063725(m)==n) # Chai Wah Wu, Jun 28 2024

Conjecture: a(2k) = A093195(k) for k >= 1, a(2k+1) = 2*A006339(k)^2 for k >= 0. - Jon E. Schoenfield, Jan 23 2022

a(13)-a(22) from Bert Dobbelaere, Oct 20 2019

a(23)-a(28) from Chai Wah Wu, Jun 28 2024

A332989 a(n) is the smallest number writable in n different ways as the sum of two distinct nonzero pentagonal numbers.

Original entry on oeis.org

6, 211, 2999, 13352, 205427, 250927, 1134927, 2177527, 5002427, 6422352, 17349697, 30135652, 45997927, 55075502, 168570052, 130917177, 101275552, 249483677, 441561407, 433742427, 771789552, 1546505052, 1316582177, 1701923302, 2288827477, 1073520852, 3110207127

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

I conjecture this sequence is infinite.

			211 can be written 35 + 176 and 1 + 210;
2999 can be written 852 + 2147, 247 + 2752, 117 + 2882;
13352 = P(52) + P(79) = P(29) + P(90) = P(17) + (93) = P(10) + P(94).

Giovanni Resta, Table of n, a(n) for n = 1..40

Cf. A000326, A332988.

Cf. A093195 (analog sequence for perfect squares).

a(n) = for(k=1, oo, if(sum(i=1, sqrt(1+12*k)\6, sqrt(1+24*k+12*i-36*i*i)%6==5)==n, return(k))); \\ Jinyuan Wang, Mar 06 2020

Name clarified by Jinyuan Wang, Mar 06 2020

Terms a(12) and beyond from Giovanni Resta, Mar 08 2020

A350241 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

1, 65, 101, 142, 175, 255, 316, 380, 501, 625, 794, 995, 1155, 1456, 1696, 2012, 2373, 2709, 3118, 3566, 4158, 4608, 5211, 5852, 6500, 7221, 8065, 8906, 9766, 11089, 11855, 12868, 14020, 15337, 16601, 17854, 19255, 20840, 22364, 23964, 25813, 27665, 29650, 31635

Offset: 1

Ilya Gutkovskiy, Dec 21 2021

nonn

			For n = 2: 65 = 1^2 + 8^2 = 4^2 + 7^2.
For n = 3: 101 = 1^2 + 6^2 + 8^2 = 2^2 + 4^2 + 9^2 = 4^2 + 6^2 + 7^2.

Index entries for sequences related to sums of squares

Cf. A000290, A025341, A025379, A025415, A025417, A052261, A093195, A097563, A215908, A341040.

More terms from Jinyuan Wang, Dec 21 2021

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

A374141 a(n) is the smallest number which can be represented as the sum of two distinct nonzero hexagonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

7, 384, 4995, 51106, 204805, 483031, 2443431, 4674256, 10476781, 17272531, 25600656, 60765331, 90406956, 206602126, 332808531, 481676406, 303826656, 435211156, 789949306, 1406495106, 2260173906, 2704798281, 3220562556, 4435869181, 5165053156, 5309576106, 9818788281

Offset: 1

Ilya Gutkovskiy, Jun 28 2024

nonn

			a(2) = 384 = 6 + 378 = 153 + 231.

Michael S. Branicky, Table of n, a(n) for n = 1..39
Michael S. Branicky, Python program for A374141, A374142, and A374143
Eric Weisstein's World of Mathematics, Hexagonal Number

Cf. A000384, A093195, A332989, A342326, A374142, A374143.

Python
```
# see linked program
```

a(9)-a(27) from Michael S. Branicky, Jun 29 2024

A374094 a(n) is the smallest nonnegative integer k where there are exactly n solutions to x^2 + x*y + y^2 = k with 0 < x < y.

Original entry on oeis.org

0, 7, 91, 637, 1729, 31213, 12103, 405769, 53599, 157339, 593047

Offset: 0

Seiichi Manyama, Jun 28 2024

a(n) is the smallest nonnegative k such that A374092(k) = n.

a(11) > 10^8. - Robert Israel, Jun 28 2024

Cf. A093195, A374095.

Cf. A374090, A374092.

N:= 10^6:
V:= Array(0..N):
for x from 1 to floor(sqrt(N/3)) do
  for y from x+1 do
     v:= x^2 + x*y + y^2;
     if v > N then break fi;
     V[v]:= V[v]+1;
od od:
W:= Array(0..10);
for i from 1 to N while count < 11 do
  v:= V[i];
  if W[v] = 0 then W[v]:= i; count:= count+1 fi
od:
0, seq(W[i],i=1..10); # Robert Israel, Jun 28 2024

from itertools import count
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A374094(n): return next(m for m in count(0) if sum(1 for d in diop_quadratic(x*(x+y)+y**2-m) if 0Chai Wah Wu, Jun 28 2024

a(n) <= 13 * 7^(n-1).

A374142 a(n) is the smallest number which can be represented as the sum of two distinct nonzero heptagonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

8, 617, 8726, 255575, 1339801, 2419165, 9402323, 25764500, 35486953, 144568133, 385495261, 735503569, 638279039, 1183071664, 1571634527, 4449717748, 3584182298, 3871587494, 5693954599, 27084640649, 24205505111, 32489035067, 31973745058, 38935021406, 47570693867, 44749048300, 53075499329

Offset: 1

Ilya Gutkovskiy, Jun 28 2024

nonn

			a(2) = 617 = 1 + 616 = 148 + 469.

Michael S. Branicky, Python program for A374141, A374142, and A374143
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A000566, A093195, A332989, A342326, A374141, A374143.

Python
```
# see linked program
```

a(8)-a(27) from Michael S. Branicky, Jun 29 2024

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A016032 Least positive integer that is the sum of two squares of positive integers 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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A000446 Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.

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A328151 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + y^2 = k.

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A332989 a(n) is the smallest number writable in n different ways as the sum of two distinct nonzero pentagonal numbers.

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A350241 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly n ways, or 0 if no such number exists.

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

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