Balarka Sen - cp's OEIS frontend

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).

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.

A225567 Primes with nonzero digits such that sum of cubes of digits equal to square of sums.

Original entry on oeis.org

1423, 2143, 2341, 4231, 12253, 21523, 22153, 22531, 23251, 25321, 32251, 35221, 36343, 36433, 43633, 52321, 64333, 114451, 144511, 224461, 244261, 246241, 365557, 415141, 424261, 426421, 446221, 446461, 451411, 462421, 466441, 541141, 555637, 556537, 556573

Offset: 1

Balarka Sen, Jul 26 2013

Largest term of this sequence is the 20-digit prime 99151111111111111111.
The Pagni article mentioned below has no bearing on this problem because it deals with the well-known identity sum_{i=1..n} i^3 = (sum_{i=1..n} i)^2. However, the article is interesting. - T. D. Noe, Jul 26 2013
This sequence has exactly 14068465 provable primes. This result required about one hour of Mathematica on fairly fast computer having 16 GB of memory. - T. D. Noe, Jul 30 2013

			a(5) = 12253 since 1^3 + 2^3 + 2^3 + 5^3 + 3^3 = (1 + 2 + 2 + 5 + 3)^2.

T. D. Noe, Table of n, a(n) for n = 1..1201 (terms < 10^7)
David Pagni, 82.27 An interesting number fact, The Mathematical Gazette 82:494 (1998), pp. 271-273.
C. Rivera, PP&P Puzzle 158: Sum of Cubes equal to Square of Sum

Cf. A055012 (sum of cubes of digits), A118881 (square of sum of the digits).

Cf. A227072, A227073.

(* let tz[[i]] be numbers computed in A227073 *) Select[tz, PrimeQ] (* T. D. Noe, Jul 30 2013 *)
pQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&Total[idn^3] == Total[ idn]^2]; Select[Prime[Range[50000]],pQ] (* Harvey P. Dale, Sep 17 2013 *)

forprime(n=1, 10^7, v=digits(n); if(sum(i=1, length(v), v[i]^3)==sum(i=1, length(v), v[i])^2 & setsearch(Set(v),0)!=1, print1(n", ")))

Corrected by T. D. Noe, Jul 26 2013

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))

A225037 Decimal expansion of the number Sum_{n>=1} ksexp(n,3/2)^(-1).

Original entry on oeis.org

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

Offset: 1

Balarka Sen, Apr 25 2013

The function 'ksexp' is the n-base Kneser tetration, see references below.

			1.687635215119112465518894...

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung phi(phi(x))=e^x und verwandter Funktionalgleichungen. J. Reine Angew. Math., 187 (1949), 56-67.
H. Trappmann & D. Kouznetsov, Uniqueness of Holomorphic Superlogarithms (2009)

Sheldon Levenstein, Fast accurate Kneser superexponential algorithm

\\ Download the algorithm for ksexp (see the link)
\r kneserquiet.gp; \\ Load the algorithm
b(i)=init(i); sexp(3/2)
return(1+sumalt(i=1,1/b(i)));

A221648 Floor(integral_{n=0..infinity} (log(x) - log(1 + x^2))^n/(1 + x^2)^2 dx).

Original entry on oeis.org

0, -2, 2, -7, 24, -121, 720, -5042, 40324, -362893, 3628841, -39916952, 479002203, -6227023411, 87178303368, -1307674428809, 20922790212198, -355687429932747, 6402373716747122, -121645100478614633, 2432902008641837022, -51090942174965733088, 1124000727801486784372

Offset: 0

Balarka Sen, May 04 2013

Balarka Sen, Table of n, a(n) for n = 0..100
G. Boros, V. H. Moll, An Integral with three parameters, SIAM Rev. 40 (1998), no. 4, 972-980.

default(realprecision,100);for(n=0,50,print1(floor(intnum(x=0,oo,log(x/(1+x^2))^n/(1+x^2)^2))", "))

A225154 Floor(Sum_{i=1..n} (Sum_{j=1..i} sqrt(1/j))).

Original entry on oeis.org

1, 2, 4, 7, 11, 14, 18, 23, 27, 32, 38, 43, 49, 55, 62, 68, 75, 82, 90, 97, 105, 113, 121, 130, 138, 147, 156, 166, 175, 185, 194, 204, 214, 225, 235, 246, 257, 267, 279, 290, 301, 313, 325, 336, 349, 361, 373, 385, 398

Offset: 1

Balarka Sen, Apr 30 2013

The fact that a(n)/n diverges (it is greater than sqrt(n)) implies sum_{k>=1} 1/sqrt(k) is not Cesaro summable.

Balarka Sen, Table of n, a(n) for n = 1..500

Cf. A022819, A025224.

for(n=1,100,print1(floor(sum(i=1,n,sum(j=1,i,1/sqrt(j))))","))

a(n)=sum(j=1,n,(n+1-j)/sqrt(j))\1 \\ Charles R Greathouse IV, May 02 2013

a(n) ~ 2*Sum_{k=1..n} sqrt(k) ~ (4/3) n^(3/2).

A225088 Floor(ksexp(n, 13/10)) where ksexp(n, z) = n^ksexp(n, z-1) is Kneser's superexponential.

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 29, 38, 47, 59, 72, 86, 102, 121, 141, 163, 187, 214, 243, 274, 308, 345, 385, 427, 473, 521, 573, 628, 687, 749, 815, 885, 959, 1037, 1119, 1206, 1297, 1393, 1493, 1598, 1708, 1824, 1944, 2070, 2202, 2339, 2482, 2631, 2785, 2947, 3114

Offset: 1

Balarka Sen, Apr 27 2013

nonn

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung ϕ(ϕ(x)) = e^x und verwandter Funktionalgleichungen, J. Reine Angew. Math. 187 (1949), 56-67.

Balarka Sen, Table of n, a(n) for n = 1..100

Cf. A225087, A225086.

A225087 Floor(ksexp(n,12/10)) where ksexp(n, z) = n^ksexp(n, z-1) is Kneser's superexponential.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 14, 17, 21, 25, 29, 34, 39, 45, 51, 57, 63, 70, 77, 85, 92, 101, 109, 118, 127, 137, 147, 157, 168, 179, 190, 202, 214, 226, 239, 252, 266, 280, 294, 309, 324, 339, 355, 371, 387, 404, 422, 439, 457, 476, 494, 514, 533, 553, 574, 594, 615

Offset: 1

Balarka Sen, Apr 27 2013

nonn

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung ϕ(ϕ(x)) = e^x und verwandter Funktionalgleichungen, J. Reine Angew. Math. 187 (1949), 56-67.

Balarka Sen, Table of n, a(n) for n = 1..100

Cf. A225086, A225088.

A225086 Floor(ksexp(n,11/10)) where ksexp(n, z) = n^ksexp(n, z-1) is Kneser's superexponential.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 28, 30, 32, 35, 37, 40, 43, 45, 48, 51, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 87, 90, 93, 96, 100, 103, 106, 110, 113, 117, 120, 124, 128, 131, 135, 139, 142, 146, 150, 154, 157, 161, 165, 169, 173, 177

Offset: 1

Balarka Sen, Apr 27 2013

nonn

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung ϕ(ϕ(x)) = e^x und verwandter Funktionalgleichungen, J. Reine Angew. Math. 187 (1949), 56-67.

Balarka Sen, Table of n, a(n) for n = 1..100

Cf. A225088, A225087.

cp's OEIS Frontend

User: Balarka Sen

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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A225567 Primes with nonzero digits such that sum of cubes of digits equal to square of sums.

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

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A225037 Decimal expansion of the number Sum_{n>=1} ksexp(n,3/2)^(-1).

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A221648 Floor(integral_{n=0..infinity} (log(x) - log(1 + x^2))^n/(1 + x^2)^2 dx).

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A225154 Floor(Sum_{i=1..n} (Sum_{j=1..i} sqrt(1/j))).

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A225088 Floor(ksexp(n, 13/10)) where ksexp(n, z) = n^ksexp(n, z-1) is Kneser's superexponential.

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