A078134 -id:A078134 - cp's OEIS frontend

A298642 Number of partitions of n^2 into distinct squares > 1.

1, 0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 5, 2, 10, 4, 12, 12, 11, 19, 23, 43, 50, 55, 78, 120, 126, 234, 207, 407, 385, 701, 712, 1090, 1231, 1850, 2102, 3054, 3385, 4988, 5584, 7985, 9746, 12205, 15737, 18968, 25157, 30927, 39043, 47708, 61915, 74592, 99554

Offset: 0

Ilya Gutkovskiy, Jan 24 2018

nonn

			a(5) = 2 because we have [25] and [16, 9].

Cf. A000290, A001156, A030273, A033461, A037444, A078134, A092362, A093115, A093116, A280129, A298640.

a(n) = [x^(n^2)] Product_{k>=2} (1 + x^(k^2)).

a(n) = A280129(A000290(n)).

A334542 Numbers m such that m^2 = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 58, 85, 375, 666, 1968, 1998, 3578, 3665, 3891, 4658, 4995, 6675, 7735, 18434, 27475, 28784, 46692, 56763, 58896, 59577, 59949, 76965, 186633, 186673, 795848, 949968, 965667, 1339575, 1587616, 1929798, 2765388, 2989584, 3674195, 4763568, 5762784, 36741656, 58988961, 134369685, 188959392

Offset: 1

Scott R. Shannon, May 05 2020

			58 is a term as p = 5*8 = 40 and 58^2 = 3364 = 40^2 + 42^2.
3891 is a term as p = 3*8*9*1 = 216 and 3891^2 = 15139881 = 216^2 + 3885^2.

Giovanni Resta, Table of n, a(n) for n = 1..140 (terms < 2*10^13)

Cf. A007954, A000404, A078134, A334557, A334558.

Subsequence of A052382 (zeroless numbers).

isok(m) = my(p=vecprod(digits(m))); p && issquare(m^2 - p^2); \\ Michel Marcus, May 06 2020

A334557 Numbers m such that m = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

Original entry on oeis.org

1, 13, 41, 61, 125, 212, 281, 613, 1156, 1424, 2225, 3232, 3316, 4113, 11125, 11281, 11525, 12816, 14913, 16317, 16441, 19125, 21284, 21625, 24128, 25216, 27521, 31525, 53125, 56116, 61321, 65161, 71325, 82116, 82217, 83521, 84313, 111812, 113125, 113625, 115336, 115681, 117125, 118372

Offset: 1

Scott R. Shannon, May 06 2020

			13 is a term as p = 1*3 = 3 and 13 = 3^2 + 2^2.
281 is a term as p = 2*8*1 = 16 and 281 = 16^2 + 5^2.
118372 is a term as p = 1*1*8*3*7*2 = 336 and 118372 = 336^2 + 74^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007954, A000404, A078134, A334542, A334558.

isok(m) = my(p=vecprod(digits(m))); p && issquare(m - p^2); \\ Michel Marcus, May 06 2020

A334558 Numbers m such that m^2 + p^2 = k^2, with p > 0, where p = A007954(m) = the product of digits of m.

Original entry on oeis.org

429, 437, 598, 1938, 3584, 3875, 5576, 6864, 16758, 36828, 43778, 47775, 47859, 56637, 56672, 82928, 91798, 129584, 156782, 165688, 165838, 178857, 215985, 379488, 655578, 798847, 1881576, 2893337, 3918768, 4816872, 5439798, 5829795, 7558299, 9675288, 11943887

Offset: 1

Scott R. Shannon, May 06 2020

			429 is a term as p = 4*2*9 = 72 and 429^2 + 72^2 = 189225 = 435^2.
16758 is a term as p = 1*6*7*5*8 = 1680 and 16758^2 + 1680^2 = 283652964 = 16842^2.

Giovanni Resta, Table of n, a(n) for n = 1..147 (terms < 2*10^13)

Cf. A007954, A000404, A078134, A334542, A334557.

isok(m) = my(p=vecprod(digits(m))); p && issquare(m^2 + p^2); \\ Michel Marcus, May 06 2020

A281155 Expansion of (Sum_{k>=2} x^(k^2))^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 3, 0, 0, 1, 0, 6, 0, 0, 0, 3, 3, 0, 3, 0, 6, 0, 0, 3, 0, 3, 3, 6, 0, 0, 1, 6, 6, 0, 0, 0, 6, 0, 6, 6, 0, 3, 0, 6, 6, 0, 0, 6, 3, 3, 3, 6, 6, 0, 3, 0, 6, 1, 3, 12, 6, 0, 0, 6, 3, 6, 6, 0, 3, 0, 3, 15, 6, 0, 0, 6, 12, 0, 3, 3, 6, 6, 0, 12, 3, 0, 6, 6

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Number of ways to write n as an ordered sum of 3 squares > 1.

			G.f. = x^12 + 3*x^17 + 3*x^22 + 3*x^24 + x^27 + 6*x^29 + 3*x^33 + 3*x^34 + 3*x^36 + ...
a(17) = 3 because we have [9, 4, 4], [4, 9, 4] and [4, 4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A005875, A002102, A006456, A063691, A078134, A280542.

nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^3, {x, 0, nmax}], x]
CoefficientList[Series[(-1 - 2 x + EllipticTheta[3, 0, x])^3/8, {x, 0, 105}], x]

G.f.: (Sum_{k>=2} x^(k^2))^3.

G.f.: (1/8)*(-1 - 2*x + theta_3(0,x))^3, where theta_3 is the 3rd Jacobi theta function.

A303906 Expansion of Product_{k>=2} 1/(1 - x^(k*(k+1)/2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 3, 1, 0, 4, 2, 0, 5, 2, 1, 7, 3, 1, 8, 4, 2, 10, 6, 2, 13, 8, 3, 15, 10, 4, 20, 12, 6, 22, 16, 8, 28, 19, 10, 33, 25, 12, 40, 29, 16, 48, 36, 19, 55, 44, 26, 65, 53, 30, 76, 64, 38, 88, 75, 46, 106, 88, 56, 119, 105, 68, 141, 122, 80, 160

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

First differences of A007294.

Number of partitions of n into triangular numbers > 1.

Index entries for sequences related to partitions

Cf. A000217, A002865, A007294, A078134, A302835.

nmax = 75; CoefficientList[Series[Product[1/(1 - x^(k (k + 1)/2)), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[1 + Sum[x^(j (j + 1)/2)/Product[(1 - x^(k (k + 1)/2)), {k, 2, j}], {j, 2, nmax}], {x, 0, nmax}], x]

G.f.: 1 + Sum_{j>=2} x^(j*(j+1)/2)/Product_{k=2..j} (1 - x^(k*(k+1)/2)).

a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2)^(5/3) / (2^(9/2) * sqrt(3) * Pi^(2/3) * n^(13/6)). - Vaclav Kotesovec, May 04 2018

A367116 a(0) = 1; for n >= 1, a(n) is the largest number m = Product_{j=1..k} (b(j)-1) where {b(1), ..., b(k)} is a vector of positive integers such that Sum_{i=1..k} b(i)^2 = n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 3, 2, 4, 0, 3, 2, 4, 0, 3, 6, 4, 8, 3, 6, 4, 8, 9, 6, 12, 8, 16, 6, 12, 8, 16, 18, 12, 24, 16, 32, 12, 24, 27, 32, 36, 24, 48, 32, 64, 24, 48, 54, 64, 72, 48, 96, 64, 128, 81, 96, 108, 128, 144, 96, 192, 128, 256

Offset: 0

Yifan Xie, Dec 16 2023

nonn

a(n) = 0 if and only if n cannot be expressed as the sum of positive squares other than 1. The largest such n is 23. See A078134.
All terms can be expressed in the form of 2^x*3^y, with y <= 4.
Proof: (Start)
If prime p >= 5 is a factor of a(n), where n = Sum_{i=1..k} b(i)^2 and a(n) = Product_{i=1..k} (b(k)-1), there must be a number 1 <= j <= k where b(j) = q*p + 1, where q is a positive integer.
If q is odd, b(j)^2 = (q*p + 1)^2 = 4*((q*p + 1)/2)^2 results in q*p, a factor of a(n), being replaced by (q*p - 1)^4/16. Since q*p >= 5, q*p < (q*p - 1)^4/16.
If q is even, b(j)^2 = (2*(q/2)*p + 1)^2 = 4*(q*p/2)^2 + (q*p/2 - 2)*2^2 + 3^2 results in q*p being replaced by (q*p - 2)^4/8. Since q*p >=5, q*p < (q*p - 2)^4/8.
In conclusion, if prime p >= 5 is a factor of a(n), the value of a(n) can be improved, so the a(n) is invalid.
If y >= 5, since 5*4^2 = 80 = 8^3^2 + 2*2^2 results in 3^5 = 243 being replaced by 2^8 = 256 > 243, so the a(n) is invalid. (End)

			For n = 23, it's impossible to write 23 as the sum of positive squares other than 1, so a(23) = 0;
For n = 69, a(69) = max{0, a(65), 2*a(60), 3*a(53), 4*a(44), 5*a(33), 6*a(20), 7*a(5)} = 2*a(60) = 256.

Yifan Xie, Table of n, a(n) for n = 0..1000

Cf. A000290, A000792, A078134

a[nn_] := Module[{v},v = {1};For[n = 2, n <= nn, n++,Module [{i=1,t=0},While[i^2James C. McMahon, Dec 17 2023 From PARI *)

lista(nn) = {my(v = vector(nn+1)); v[1] = 1; for(n=2, nn+1, my(i=1, t=0); for(i=1, sqrtint(n-1), t=max(t, (i-1)*v[n-i*i])); v[n] = t); v;}

a(n) = Max_{i^2 <= n} (i-1)*a(n^2-i).

Let n = 4*k + r, 0 <= r <= 3. a(n) = max{2^(4*ceiling((k - 2*r)/9) + r)*3^(ceiling((k - 2*r - 9*ceiling((k - 2*r)/9))/4)), 2^(4*(ceiling((k - 2*r)/9) - 1) + r)*3^(ceiling((k - 2*r - 9*ceiling((k - 2*r)/9 - 1))/4))}

A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Number of ways to write n as an ordered sum of 2 squares > 1.

			G.f. = x^8 + 2*x^13 + x^18 + 2*x^20 + 2*x^25 + 2*x^29 + x^32 + 2*x^34 + 2*x^40 + ...
a(13) = 2 because we have [9, 4] and [4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A000925, A004018, A006456, A063725, A078134, A085989, A280542.

nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^2, {x, 0, nmax}], x]
CoefficientList[Series[(1 + 2 x - EllipticTheta[3, 0, x])^2/4, {x, 0, 105}], x]

G.f.: (Sum_{k>=2} x^(k^2))^2.

G.f.: (1/4)*(1 + 2*x - theta_3(0,x))^2, where theta_3 is the 3rd Jacobi theta function.

A298600 Expansion of Product_{k>=2} 1/(1 + x^(k^2)).

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 0, -2, 2, -1, 1, 2, -2, 1, -1, -2, 3, -2, 1, 2, -3, 2, -1, -1, 2, -3, 1, 1, -2, 3, 0, 0, 2, -3, 1, -1, -2, 3, -1, -2, 2

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

sign

The difference between the number of partitions of n into an even number of squares > 1 and the number of partitions of n into an odd number of squares > 1.

Cf. A001156, A033461, A078134, A276516, A280129, A292520, A298601.

nmax = 82; CoefficientList[Series[Product[1/(1 + x^k^2), {k, 2, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} 1/(1 + x^(k^2)).

A298601 Expansion of Product_{k>=2} (1 - x^(k^2)).

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, -2, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 0, 1, 1, 0, 0, -2, 0, 0, 1, 0, 0, -1, -1, 2, 1, 1, 0, -1, -1

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

sign

The difference between the number of partitions of n into an even number of distinct squares > 1 and the number of partitions of n into an odd number of distinct squares > 1.

Partial sums of A276516.

Cf. A001156, A033461, A078134, A276516, A280129, A292520, A298600.

nmax = 92; CoefficientList[Series[Product[1 - x^k^2, {k, 2, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 - x^(k^2)).

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A298642 Number of partitions of n^2 into distinct squares > 1.

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A334542 Numbers m such that m^2 = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

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A334557 Numbers m such that m = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

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A334558 Numbers m such that m^2 + p^2 = k^2, with p > 0, where p = A007954(m) = the product of digits of m.

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PARI

A281155 Expansion of (Sum_{k>=2} x^(k^2))^3.

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A303906 Expansion of Product_{k>=2} 1/(1 - x^(k*(k+1)/2)).

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A367116 a(0) = 1; for n >= 1, a(n) is the largest number m = Product_{j=1..k} (b(j)-1) where {b(1), ..., b(k)} is a vector of positive integers such that Sum_{i=1..k} b(i)^2 = n.

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A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

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