Jimmy Zotos - cp's OEIS frontend

A227847 Number of tuples (x_1, x_2, ..., x_n) with 1 <= x_1 <= x_2 <= ... <= x_n such that Sum_{i=1..n} x_i^3 = (Sum_{i=1..n} x_i)^2 and Sum_{i=1..n-1} x_i^3 + (x_n-1)^3 + (x_n+1)^3 = (Sum_{i=1..n-1} x_i + 2x_n)^2.

0, 1, 1, 1, 2, 2, 2, 6, 10, 31, 77, 206, 568, 1704, 5037, 15554

Offset: 1

Jimmy Zotos, Aug 01 2013

An n-tuple meeting the first condition is called an n-SCESS ("sum of cubes equals square of sum").
In other words, a(n) is the number of tuples (x_1, x_2, ..., x_n) satisfying SCESS such that (x_1, x_2, ..., x_{n-1}, x_n - 1, x_n + 1) also satisfies SCESS. - Max Alekseyev, Mar 04 2025
x_1 + x_2 + ... + x_{n-1} = A152948(x_n). - Balarka Sen, Aug 01 2013

			a(3) = 1 since the only 3-SCESS is (1, 2, 3) for which the corresponding ordered tuple (1, 2, 2, 4) satisfy the SCESS property. (See Mason et al.)
a(5) = 2 since the only 5-SCESS are (1, 2, 2, 3, 5) and (3, 3, 3, 3, 6) for which the corresponding ordered tuples (1, 2, 2, 3, 4, 6) and (3, 3, 3, 3, 5, 7) satisfy the SCESS property.
a(8) = 6 since the only 8-SCESS are (1, 1, 2, 4, 5, 5, 5, 8), (1, 2, 2, 3, 4, 5, 6, 8), (2, 2, 4, 4, 6, 6, 6, 9), (2, 6, 6, 6, 6, 6, 6, 10), (3, 3, 3, 3, 5, 6, 7, 9) and (3, 5, 5, 5, 6, 7, 7, 10) for which the corresponding ordered tuples (1, 1, 2, 4, 5, 5, 5, 7, 9), (1, 2, 2, 3, 4, 5, 6, 7, 9), (2, 2, 4, 4, 6, 6, 6, 8, 10), (2, 6, 6, 6, 6, 6, 6, 9, 11), (3, 3, 3, 3, 5, 6, 7, 8, 10) and (3, 5, 5, 5, 6, 7, 7, 9, 11) satisfy the SCESS property.

Edward Barbeau and Samer Seraj, Sum of cubes is square of sum, arXiv:1306.5257 [math.NT], 2013.
John Mason, Generalising 'sums of cubes equal to squares of sums', The Mathematical Gazette 85:502 (2001), pp. 50-58.

Cf. A001055, A152948, A158649, A225819.

a(n)=my(v=vector(n, i, 1), N=n^(4/3), k); while(v[#v]N, for(i=2, N, if(v[i]Balarka Sen, Aug 01 2013 */

A001055(n) <= a(n) <= A158649(n). - Balarka Sen, Aug 01 2013

a(11)-a(15) from Balarka Sen, Aug 01 2013
a(16) from Balarka Sen, Aug 11 2013
Definition corrected by Max Alekseyev, Mar 04 2025

A225808 Values (Sum_{1<=i<=k} x_i)^2 = Sum_{1<=i<=k} x_i^3 for 1 <= x_1 <= x_2 <=...<= x_k ordered lexicographically according to (x1, x2,..., xk).

Original entry on oeis.org

1, 9, 16, 36, 81, 81, 100, 144, 256, 169, 225, 324, 361, 625, 144, 256, 324, 441, 324, 361, 441, 625, 256, 576, 729, 784, 576, 729, 900, 961, 1089, 1296, 484, 625, 784, 900, 484, 441, 576, 729, 784, 900, 1089, 1089, 1156, 1369, 625, 784, 729, 900, 1089, 1369, 1296, 1600, 900, 961, 1089

Offset: 1

Charles R Greathouse IV, Jimmy Zotos, and Balarka Sen, Jul 29 2013

a(n) <= k^4 where k is the size of the ordered tuple (x_1, x_2,..., x_k).

This sequence is closed under multiplication, that is, if m and n are in this sequence, so is m*n.

			1;
9, 16;
36, 81;
81, 100, 144, 256;
169, 225, 324, 361, 625;
144, 256, 324, 441, 324, 361, 441, 625, 256, 576, 729, 784, 576, 729, 900, 961, 1089, 1296;
484, 625, 784, 900, 484, 441, 576, 729, 784, 900, 1089, 1089, 1156, 1369, 625, 784, 729, 900, 1089, 1369, 1296, 1600, 900, 961, 1089, 1600, 1296, 1600, 2025, 2401;

Balarka Sen, Rows n = 1..10 of irregular triangle, flattened
Edward Barbeau and Samer Seraj, Sum of cubes is square of sum, arXiv:1306.5257 [math.NT], 2013.
John Mason, Generalising 'sums of cubes equal to squares of sums', The Mathematical Gazette 85:502 (2001), pp. 50-58.
Alasdair McAndrew, A cute result relating to sums of cubes
David Pagni, 82.27 An interesting number fact, The Mathematical Gazette 82:494 (1998), pp. 271-273.
Balarka Sen, Table of rows, n = 1..10
W. R. Utz, The Diophantine Equation (x_1 + x_2 + ... + x_n)^2 = x_1^3 + x_2^3 + ... + x_n^3, Fibonacci Quarterly 15:1 (1977), pp. 14, 16. Part 1, part 2.

Cf. A158649, A055012, A118881.

row[n_] := Reap[Module[{v, m}, v = Table[1, {n}]; m = n^(4/3); While[ v[[-1]] < m, v[[1]]++; If[v[[1]] > m, For[i = 2, i <= m, i++, If[v[[i]] < m, v[[i]]++; For[j = 1, j <= i - 1, j++, v[[j]] = v[[i]]]; Break[]]]]; If[Total[v^3] == Total[v]^2, Sow[Total[v]^2]]]]][[2, 1]];
Array[row, 7] // Flatten (* Jean-François Alcover, Feb 23 2019, from PARI *)

row(n)=my(v=vector(n,i,1),N=n^(4/3)); while(v[#v]N,for(i=2, N,if(v[i]

A225819 Consider the set of n-tuples such that the sum of cubes of the elements is equal to square of their sum; sequence gives largest element in all such tuples.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 42, 44, 46, 48, 51, 53, 55, 58, 60, 62, 65, 67, 70, 72, 75, 77, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 109, 112, 115, 117, 120, 123, 126, 129, 132, 134, 137, 140, 143, 146, 149, 152, 155

Offset: 1

Charles R Greathouse IV, Jimmy Zotos, and Balarka Sen, Jul 30 2013

nonn

Conjecture [Sen]: lim inf log_n a(n) >= 5/4.

			Call an n-multiset with the sum of cubes of the elements equal to square of their sum an n-SCESS.
a(6) = 7 since the only 6-SCESS with the largest element >= 7 are (2, 4, 4, 5, 5, 7), (3, 3, 3, 3, 5, 7), (3, 4, 5, 5, 6, 7), (3, 5, 5, 5, 6, 7) and (4, 5, 5, 6, 6, 7) and none have an element larger than 7.
a(7) = 9 since the only 7-SCESS with the largest element >= 9 are (4, 4, 4, 5, 5, 5, 9), (4, 5, 5, 5, 6, 6, 9) and (6, 6, 6, 6, 6, 6, 9) and none have an element larger than 9.
a(8) = 10 since the only 8-SCESS with the largest element >= 10 are (2, 5, 5, 5, 5, 5, 6, 10), (2, 6, 6, 6, 6, 6, 6, 10), (3, 4, 5, 5, 5, 6, 7, 10), (3, 4, 5, 5, 6, 6, 7, 10), (3, 5, 5, 5, 6, 7, 7, 10), (3, 6, 6, 6, 7, 7, 7, 10), (4, 4, 4, 4, 4, 4, 6, 10), (4, 4, 4, 4, 5, 5, 7, 10), (4, 5, 5, 6, 6, 7, 8, 10), (5, 5, 5, 7, 7, 7, 8, 10) and (6, 6, 6, 6, 6, 6, 9, 10) and none have an element larger than 10.

Balarka Sen, Table of n, a(n) for n = 1..500
John Mason, Generalising 'sums of cubes equal to squares of sums', The Mathematical Gazette 85:502 (2001), pp. 50-58.
W. R. Utz, The Diophantine Equation (x_1 + x_2 + ... + x_n)^2 = x_1^3 + x_2^3 + ... + x_n^3, Fibonacci Quarterly 15:1 (1977), pp. 14, 16. Part 1, part 2.

Cf. A158649, A225808.

a(n)=my(v=vector(n, i, 1), N=n^(4/3), m=n); while(v[#v]N, for(i=2, N, if(v[i]

n <= a(n) <= n^(4/3), see A158649.

A225572 Continued fraction expansion of Pi/(4e) + e/(3Pi).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 5, 3, 1, 2, 2, 1, 1, 24, 1, 2, 2, 9, 78, 1, 94, 5, 1, 27, 2, 1, 1, 2, 1, 4, 2, 2, 2, 20, 1, 4, 1, 1, 5, 23, 43, 1, 9, 1, 3, 1, 3, 1, 2, 1, 2, 4, 1, 1, 3, 8, 32, 1, 7, 1

Offset: 0

Balarka Sen, May 11 2013, based on an idea from Jimmy Zotos

The first five terms match with the continued fraction of the Euler-Mascheroni constant. See analogous comment in A225155.

Balarka Sen, Table of n, a(n) for n = 0..1000

Cf. A225155.

ContinuedFraction[Pi/(4E)+E/(3Pi),100] (* Harvey P. Dale, Oct 22 2023 *)

default(realprecision, 300); contfrac(Pi/(4*exp(1))+exp(1)/(3*Pi))

A225155 Decimal expansion of Pi/(4e) + e/(3Pi).

Original entry on oeis.org

5, 7, 7, 3, 5, 0, 4, 9, 7, 2, 5, 8, 4, 8, 5, 4, 5, 8, 5, 5, 0, 1, 1, 4, 8, 9, 2, 3, 7, 6, 8, 7, 0, 6, 1, 3, 9, 2, 3, 1, 2, 7, 6, 3, 0, 4, 2, 7, 2, 7, 5, 8, 0, 1, 6, 0, 9, 7, 0, 3, 3, 9, 6, 8, 1, 4, 5, 0, 3, 8, 3, 9, 4, 8, 3, 7, 8, 8, 2, 5, 4, 4, 0, 4, 0, 2, 1, 9, 9, 6, 0, 6, 4, 3, 1, 3, 2, 2, 8, 7, 9, 6

Offset: 0

Jimmy Zotos, May 01 2013

Approximates Euler Mascheroni constant giving 3 correct decimal places.

The constant is not known to be transcendental, or even irrational. - Balarka Sen, May 06 2013

			0.5773504972584854585501148923768706139231...

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

Cf. A000796, A001113, A061360, A086056, A001620, A225572 (continued fraction).

R:= RealField(100); (3*Pi(R)^2 + 4*Exp(2))/(12*Pi(R)*Exp(1)); // G. C. Greubel, Aug 30 2018

RealDigits[(3*Pi^2 + 4*E^2)/(12*E*Pi), 10, 100][[1]] (* G. C. Greubel, Aug 30 2018 *)

default(realprecision, 1020); x=10*Pi/exp(1)/4+10*exp(1)/Pi/3; d=0; for (n=0, 1000, d=floor(x); x=(x-d)*10; print1(d", ")) \\

More terms from Balarka Sen, May 02 2013

A211878 Decimal expansion of positive constant C such that 1 = Sum_{k>=1} 1/C^(2^k).

Original entry on oeis.org

1, 3, 2, 9, 0, 5, 9, 1, 0, 8, 7, 4, 9, 5, 5, 9, 5, 6, 4, 6, 0, 9, 9, 1, 6, 8, 2, 6, 7, 9, 2, 4, 3, 6, 2, 5, 1, 9, 4, 9, 7, 7, 6, 5, 9, 3, 8, 8, 4, 1, 8, 2, 8, 7, 8, 7, 3, 4, 2, 2, 9, 8, 5, 0, 2, 7, 3, 0, 4, 0, 8, 5, 4, 4, 9, 2, 0, 4, 4, 7, 6, 3, 4, 8, 0, 3, 8, 3, 8, 2, 7, 7, 9, 7, 8, 1, 9, 1, 2, 2, 9, 6, 8, 0, 1, 9, 3, 2, 3, 8, 6, 6

Offset: 1

Jimmy Zotos and Balarka Sen, Feb 13 2013

			C = 1.3290591087495595646...

Digits:= 120:
s:= convert(fsolve(sum(1/C^(2^k), k=1..infinity)=1, C=1)/10, string):
seq(parse(s[n+1]), n=1..112);

digits = 112; f[x_?NumericQ] := NSum[1/x^(2^k), {k, 1, Infinity}, WorkingPrecision -> digits]; x /. FindRoot[f[x] == 1, {x, 3/2, 1, 2}, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 20 2014 *)

More terms from Alois P. Heinz, Feb 13 2013

A211879 Decimal expansion of constant C such that 1 = Sum_{k>=1} 1/C^(k^3).

Original entry on oeis.org

1, 2, 3, 3, 8, 8, 8, 1, 4, 0, 3, 3, 7, 2, 7, 4, 1, 8, 8, 7, 5, 3, 5, 4, 7, 9, 2, 7, 3, 0, 8, 8, 6, 4, 1, 1, 5, 5, 0, 4, 7, 8, 2, 1, 4, 0, 1, 0, 9, 1, 2, 4, 2, 8, 9, 3, 1, 7, 6, 7, 7, 4, 8, 4, 5, 0, 7, 9, 9, 9, 0, 5, 6, 1, 9, 7, 8, 0, 3, 7, 4, 5, 5, 2, 7, 0, 9, 4, 0, 7, 6, 3, 9, 5, 5, 0, 8, 0, 1, 0, 8, 5, 8, 5, 0, 8, 6, 5, 6, 3, 7, 0, 1, 2, 5, 4

Offset: 1

Balarka Sen and Jimmy Zotos, Feb 13 2013

			C = 1.2338881403372741887535479273...

Cf. A078975

Digits:= 120:
s:= convert(fsolve(sum(1/C^(k^3), k=1..infinity)=1, C=2)/10, string):
seq(parse(s[n+1]), n=1..116);  # Alois P. Heinz, Feb 13 2013

digits = 116; f[x_?NumericQ] := NSum[1/x^(k^3), {k, 1, Infinity}, WorkingPrecision -> digits]; x /. FindRoot[f[x] == 1, {x, 3/2, 1, 2}, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 20 2014 *)

More terms from Alois P. Heinz, Feb 13 2013

A215267 Decimal expansion of 90/Pi^4.

Original entry on oeis.org

9, 2, 3, 9, 3, 8, 4, 0, 2, 9, 2, 1, 5, 9, 0, 1, 6, 7, 0, 2, 3, 7, 5, 0, 4, 9, 4, 0, 4, 0, 6, 8, 2, 4, 7, 2, 7, 6, 4, 5, 0, 2, 1, 6, 6, 8, 2, 7, 4, 4, 3, 6, 4, 4, 6, 3, 5, 1, 2, 3, 1, 9, 2, 4, 7, 7, 6, 2, 9, 6, 4, 0, 7, 9, 9, 6, 7, 2, 8, 2, 2, 4, 1, 6, 5, 1, 4, 3, 7, 3, 6, 5, 7, 6, 1, 4, 4, 1, 5

Offset: 0

Jimmy Zotos, Aug 07 2012

Decimal expansion of 1/zeta(4), the inverse of A013662. This is the probability that 4 randomly chosen natural numbers are relatively prime.
Also the asymptotic probability that a random integer is 4-free. See equivalent comments in A088453, A059956. - Balarka Sen, Aug 08 2012
The probability that the greatest common divisor of two numbers selected at random is squarefree (Christopher, 1956). - Amiram Eldar, May 23 2020

			0.92393840292159016702375049404068247276450216682744364463512319...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 231.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000
John Christopher, The Asymptotic Density of Some k-Dimensional Sets, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.
Math Forums, Probability that a number is composite, Aug 2012.
Index entries for transcendental numbers

Cf. A013662, A046100 (4-free numbers), A059956 (1/zeta(2)).

RealDigits[90/Pi^4, 10, 100][[1]] (* Bruno Berselli, Aug 07 2012 *)

90/%pi^4; /* Balarka Sen, Aug 08 2012 */

90/Pi^4 \\ Charles R Greathouse IV, Aug 07 2012

Reciprocal of A013662.

1/zeta(4) = 90/Pi^4 = Product_{k>=1} (1 - 1/prime(k)^4) = Sum_{n>=1} mu(n)/n^4, a Dirichlet series for the Möbius function mu. See the examples in Apostol, here for s = 4. - Wolfdieter Lang, Aug 07 2019

cp's OEIS Frontend

User: Jimmy Zotos

A227847 Number of tuples (x_1, x_2, ..., x_n) with 1 <= x_1 <= x_2 <= ... <= x_n such that Sum_{i=1..n} x_i^3 = (Sum_{i=1..n} x_i)^2 and Sum_{i=1..n-1} x_i^3 + (x_n-1)^3 + (x_n+1)^3 = (Sum_{i=1..n-1} x_i + 2x_n)^2.

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A225808 Values (Sum_{1<=i<=k} x_i)^2 = Sum_{1<=i<=k} x_i^3 for 1 <= x_1 <= x_2 <=...<= x_k ordered lexicographically according to (x1, x2,..., xk).

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A225819 Consider the set of n-tuples such that the sum of cubes of the elements is equal to square of their sum; sequence gives largest element in all such tuples.

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A225572 Continued fraction expansion of Pi/(4*e) + e/(3*Pi).

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A225155 Decimal expansion of Pi/(4*e) + e/(3*Pi).

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Magma

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A211878 Decimal expansion of positive constant C such that 1 = Sum_{k>=1} 1/C^(2^k).

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Maple

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A211879 Decimal expansion of constant C such that 1 = Sum_{k>=1} 1/C^(k^3).

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Examples

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Maple

Mathematica

Extensions

A215267 Decimal expansion of 90/Pi^4.

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A225572 Continued fraction expansion of Pi/(4e) + e/(3Pi).

A225155 Decimal expansion of Pi/(4e) + e/(3Pi).