A016032 -id:A016032 - cp's OEIS frontend

A287165 Smallest number with exactly n representations as a sum of 6 nonzero squares or 0 if no such number exists.

6, 21, 30, 36, 63, 54, 60, 87, 78, 81, 84, 111, 102, 117, 108, 116, 126, 129, 134, 137, 132, 150, 172, 165, 161, 156, 177, 164, 195, 191, 182, 213, 180, 188, 198, 0, 204, 206, 215, 222, 243, 212, 251, 262, 233, 230

Offset: 1

Ilya Gutkovskiy, May 20 2017

nonn

			a(1) = 6 because 6 is the smallest number with exactly 1 representation as a sum of 6 nonzero squares: 6 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2;
a(2) = 21 because 21 is the smallest number with exactly 2 representations as a sum of 6 nonzero squares: 21 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2, etc.

Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A016032, A025414, A025416, A025430, A080654, A287166, A287167.

A025430(a(n)) = n for a(n) > 0.

A287167 Smallest number with exactly n representations as a sum of 8 nonzero squares or 0 if no such number exists.

Original entry on oeis.org

8, 23, 35, 32, 46, 58, 72, 56, 62, 70, 71, 79, 80, 83, 88, 89, 91, 86, 103, 94, 109, 104, 107, 112, 113, 110, 122, 119, 126, 121, 118, 144, 0, 128, 131, 136, 137, 153, 143, 139, 149, 134, 0, 0, 142, 152, 164, 154

Offset: 1

Ilya Gutkovskiy, May 20 2017

nonn

			a(1) = 8 because 8 is the smallest number with exactly 1 representation as a sum of 8 nonzero squares: 8 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2;
a(2) = 23 because 23 is the smallest number with exactly 2 representations as a sum of 8 nonzero squares: 23 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2, etc.

Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A016032, A025414, A025416, A025432, A080654, A287165, A287166.

A025432(a(n)) = n for a(n) > 0.

A374168 a(n) is the smallest number which can be represented as the sum of two nonzero square pyramidal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

2, 60, 9692375

Offset: 1

Ilya Gutkovskiy, Jun 30 2024

There are no further positive terms <= 10^15. - Michael S. Branicky, Jul 01 2024

			a(2) = 60 = 5 + 55 = 30 + 30.

Eric Weisstein's World of Mathematics, Square Pyramidal Number.

Cf. A000330, A011541, A016032, A334013, A374193, A374194.

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

A273279 Least perfect power that is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

8, 125, 3125, 4225, 1953125, 48828125, 105625, 274625, 762939453125, 2640625, 476837158203125, 17850625, 1221025, 34328125, 186264514923095703125, 1650390625, 446265625, 1160290625, 41259765625, 4291015625, 45474735088646411895751953125, 30525625

Offset: 1

Altug Alkan, May 19 2016

nonn

Least m^k that is the sum of two nonzero squares in exactly n ways where m > 0 and k >= 2.
Terms of this sequence are 2^3, 5^3, 5^5, 65^2, 5^10, 5^11, 325^2, 65^3, ...
Prime powers that are listed in this sequence are 2^3, 5^3, 5^5, 5^10, 5^11, ...

			8 is a term because 8 = 2^3 = 2^2 + 2^2.
125 is a term because 125 = 5^3 = 2^2 + 11^2 = 5^2 + 10^2.
3125 is a term because 3125 = 5^5 = 10^2 + 55^2 = 25^2 + 50^2 = 38^2 + 41^2.

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A001597, A006339, A016032, A025426, A266927, A273238.

p = Select[Prime@ Range@ 90, Mod[#, 4] == 1 &]; f[w_] := Times @@ (Take[p, Length@w]^Reverse[w]); c[w_] := Floor[(1/2) Times @@ (w+1)];r[w_] := Block[{v, k = If[Length@w == 1, 1,2]}, While[(v = cn[k w]) < trg, k++]; If[v == trg, b = Min[b, f[k*w]]]; If[cn[w] < trg, r[Append[w, 1]]; v=w; v[[-1]]++; r[v]]]; a[1]=8; a[n_] := (b=Infinity; trg = n; r[{2}]; r[{1, 1}]; b); Array[a, 50] (* Giovanni Resta, May 19 2016 *)

a(9)-a(22) from Giovanni Resta, May 19 2016

A274548 Least number k such that k and k+1 are the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

17, 985, 33524, 19720, 116643124, 2263924, 411769906249, 7342945, 2027986649, 1348336249

Offset: 1

Altug Alkan, Jun 27 2016

			a(2) = 985 because 985 = 12^2 + 29^2 = 16^2 + 27^2 and 986 = 5^2 + 31^2 = 19^2 + 25^2.

Cf. A000404, A016032.

a(5)-a(10) from Giovanni Resta, Jun 28 2016

A274686 Least number k such that k-th triangular number is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

4, 40, 25, 145, 625, 169, 31249, 985, 2600, 2500, 87890625, 3649, 384199200, 15625, 33124, 6409

Offset: 1

Altug Alkan, Jul 02 2016

From Robert Israel, Jul 04 2016: (Start)
Least k such that A025426(A000217(k)) = n.
A025426(A000217(18463134765625))=17, but I don't know if this is minimal. (End)
a(18) = 24649, a(20) = 40000, a(21) = 250000. 25*10^6, 25*10^8, 25*10^12 are not terms. Are there other terms of the form 25*10^(2k)? - Chai Wah Wu, Jul 23 2020

			a(2) = 40 because 40*41 / 2 = 820 = 6^2 + 28^2 = 12^2 + 26^2.

Cf. A000217, A000404, A006339, A016032, A025426, A273238.

a(11)-a(16) from Giovanni Resta, Jul 04 2016

A273354 Least number that is the sum of 2 positive cubes in exactly n ways and also the sum of 2 positive squares in exactly n ways.

Original entry on oeis.org

2, 4624776

Offset: 1

Altug Alkan, May 20 2016

4624776 is the first term of A272701.
A011541(k) is not the sum of two nonzero squares for 2 <= k <= 6.
If it exists, what is the a(3)?

			a(1) = 2 because 2 = 1^3 + 1^3 = 1^2 + 1^2.
a(2) = 4624776 because 4624776 = 51^3 + 165^3 = 72^3 + 162^3 = 1026^2 + 1890^2 = 1350^2 + 1674^2.

Cf. A001235, A011541, A016032, A272701.

A273545 Least number k such that k*n is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

2, 25, 975, 1105, 1625, 16575, 739375, 27625, 71825, 27625, 58093750, 480675, 93925, 8547175, 1077375, 801125, 76765625, 2082925, 783935546875, 801125, 942703125, 23740234375, 1404178750, 17784975, 12138425, 8300781250, 106229175, 700984375, 221252441406250

Offset: 1

Altug Alkan, May 26 2016

nonn

			a(2) = 25 because 25*2 = 50 is the least even number that is the sum of two nonzero squares in exactly 2 ways; 50 = 1^2 + 7^2 = 5^2 + 5^2.

Cf. A006339, A016032, A025426, A273238.

nR[n_] := (SquaresR[2, n] + Plus @@ Pick[{-4, 4}, IntegerQ /@ Sqrt[{n, n/2}]])/8; a[n_] := Block[{k=1}, While[nR[n * k] != n, k++]; k]; Array[a, 10] (* Giovanni Resta, May 27 2016 *)

a(6)-a(29) from Giovanni Resta, May 26 2016

A273787 Least number k such that A001844(k) (sums of two consecutive squares) is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

1, 6, 21, 23, 221, 78, 7278, 153, 703, 1653, 6695846, 496, 670758346, 8346, 1471, 1081

Offset: 1

Altug Alkan, May 30 2016

a(18) = 1978, a(20) = 4596, a(21) = 304153, a(22) = 137903, a(24) = 2628. - Chai Wah Wu, Feb 13 2018

			a(2) = 6 from 6^2 + 7^2 = 2^2 + 9^2.
a(3) = 21 from 21^2 + 22^2 = 5^2 + 30^2 = 14^2 + 27^2.
a(4) = 23 form 23^2 + 24^2 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2.

Cf. A001844, A006339, A016032, A025426, A273238.

A025426(n)=my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f~, if(f[i, 1]%4>1, if(f[i, 2]%2, return(0)), t*=f[i, 2]+1)); if(t%2, t-(-1)^v, t)/2
a(n)=my(k=1); while(A025426(2*k*(k+1)+1)!=n, k++); k \\ Charles R Greathouse IV, Jun 03 2016

a(10)-a(14) from Giovanni Resta, Jun 03 2016

a(15)-a(16) from Chai Wah Wu, Feb 13 2018

cp's OEIS Frontend

A287165 Smallest number with exactly n representations as a sum of 6 nonzero squares or 0 if no such number exists.

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A287167 Smallest number with exactly n representations as a sum of 8 nonzero squares or 0 if no such number exists.

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A374168 a(n) is the smallest number which can be represented as the sum of two nonzero square pyramidal numbers in exactly n ways, or -1 if no such number exists.

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

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A273279 Least perfect power that is the sum of two nonzero squares in exactly n ways.

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A274548 Least number k such that k and k+1 are the sum of two nonzero squares in exactly n ways.

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A274686 Least number k such that k-th triangular number is the sum of two nonzero squares in exactly n ways.

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A273354 Least number that is the sum of 2 positive cubes in exactly n ways and also the sum of 2 positive squares in exactly n ways.

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A273545 Least number k such that k*n is the sum of two nonzero squares in exactly n ways.

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A273787 Least number k such that A001844(k) (sums of two consecutive squares) is the sum of two nonzero squares in exactly n ways.

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