A113505 -id:A113505 - cp's OEIS frontend

A135693 Not the sum of three positive squares or cubes.

1, 2, 4, 5, 7, 8, 15, 23, 31, 87, 111, 119, 148, 167, 263, 311, 335, 391, 407, 455, 559, 599, 839, 951, 1159, 1231, 1287, 1303, 1391, 1455, 1463, 1607, 1660, 1679, 1751, 1863, 1991, 2351, 2615, 2799, 3247, 3983, 4327, 4367, 5199, 5655, 6047, 6159, 6351

Offset: 1

Zak Seidov and Giovanni Resta, Feb 24 2008

nonn

No other terms < 10^9. Presumably the sequence is finite.

Sum can include a mix of squares and cubes. - James C. McMahon, Apr 19 2025

Chai Wah Wu, Table of n, a(n) for n = 1..82

Cf. A113505, A135367.

Cf. A004214, A057904.

lim=40000; s=Range[Sqrt[lim]]^2; c=Range[Surd[lim, 3]]^3; A135693=Complement[Range[lim], Select[Total/@Tuples[Union[s, c], {3}], #<=lim&]] (* James C. McMahon, Apr 19 2025 *)

Edited by N. J. A. Sloane, Mar 01 2008

A135367 Not the sum of three perfect powers (assuming 1, but not 0, is a perfect power).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 15, 23, 31, 87, 111, 119, 167, 335, 1391, 1455, 1607, 1679, 1991, 25887, 26375

Offset: 1

Zak Seidov and Giovanni Resta, Feb 24 2008

nonn

No other terms < 10^9. Presumably the sequence is finite.

Cf. A113505, A135693, A135393, A135402.

Complement of A074499.

A056828 Numbers that are not the sum of at most three powerful (1) numbers.

Original entry on oeis.org

7, 15, 23, 87, 111, 119

Offset: 1

Henry Bottomley, Aug 30 2000

Mollin and Walsh conjectured that there are no further terms.
Heath-Brown proved that the sequence is finite.
No other terms less than 40000000. - Paul.Jobling(AT)WhiteCross.com, May 14 2001

			Smallest powerful numbers are 1, 4, 8, 9, 16, 25,... so 7, 15 and 23 are not the sum of one, two or three of them.

D. R. Heath-Brown, "Ternary Quadratic Forms and Sums of Three Square-Full Numbers." In Séminaire de Théorie des Nombres, Paris 1986-87 (Ed. C. Goldstein). Boston, MA: Birkhauser, pp. 137-163, 1988.

R. A. Mollin and P. G. Walsh, On Powerful Numbers, Intern. J. Math. and Math. Sci, 9:801-806, 1986.
Eric Weisstein's World of Mathematics, Powerful Number.

Cf. A001694, A076871, A113505.

A135402 Not the sum of three squares or cubes greater than 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 18, 19, 23, 30, 31, 46, 55, 64, 87, 91, 111, 119, 128, 130, 148, 151, 167, 263, 311, 335, 391, 407, 455, 487, 540, 559, 599, 839, 951, 967, 1159, 1231, 1287, 1303, 1391, 1455, 1463, 1607, 1660, 1679, 1751

Offset: 1

Giovanni Resta, Feb 24 2008

nonn

Giovanni Resta, Table of n, a(n) for n = 1..104

Cf. A113505, A135693, A135367, A135393.

A274459 Least number of perfect powers that add up to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 1, 2, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 3, 2, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 2, 1, 2, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 1, 2, 3, 3, 2

Offset: 1

Sergio Pimentel, Jun 23 2016

nonn

Least number of perfect powers (A001597) needed to add up to n.
This sequence is close to but not exactly equal to A063274.
a(n) is at most 4 since any number can be written as a sum of 4 squares (Lagrange's theorem), but it is possible that for a sufficiently large n, a(n) < 4.
a(n) <= a(i) + a(n-i) for 1 <= i <= n-1. (for computational ease, the maximum value for i can be chosen as floor(n/2)). a(1991) = 4. for 1992 <= k <= 20000, there is no k such that a(k) = 4. - David A. Corneth, Jun 24 2016 [Next such k is 25887, see A113505. - Vaclav Kotesovec, Jun 25 2016]

			a(31) = 2 since 31 can be written as the sum of two (31 = 3^3 + 2^2 = 27 + 4) but no fewer than two perfect powers.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001597, A002376, A002377, A002828, A002804, A079611, A063274, A188462, A225926.

Cf. A063275 (indices for which a(n)=3), A113505 (indices for which a(n)=4).

nn = 72; t = Select[Range@ nn, # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &]; Table[Min@ Map[Length, Select[IntegerPartitions@ n, AllTrue[#, MemberQ[t, #] &] &]], {n, nn}] (* Michael De Vlieger, Jun 23 2016, after Ant King at A001597 *)

lista(n) = {my(v = vector(n)); for(i = 2,sqrtint(n), for(j = 2, logint(n, i), v[i^j] = 1)); v[1]=1; v[2]=2; for(i=3, #v, if(v[i]==0, v[i] = vecmin(vector( i\2, k,v[k] + v[i-k]))));v} \\ David A. Corneth, Jun 24 2016; corrected by Peter Schorn, Jun 09 2022

More terms from Michael De Vlieger, Jun 23 2016

Terms from a(74) from David A. Corneth, Jun 24 2016

A135393 Not the sum of three perfect powers greater than 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 18, 19, 23, 30, 31, 46, 55, 87, 111, 119, 151, 167, 335, 1391, 1455, 1607, 1679, 1991, 25887, 26375

Offset: 1

Giovanni Resta, Feb 24 2008

nonn

Probably there are no further terms.

Cf. A113505, A135693, A135367, A135402.

A282499 Expansion of (Sum_{k = i^j, i>=1, j>=2, excluding duplicates} x^k)^3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 0, 0, 3, 3, 3, 1, 6, 6, 0, 3, 6, 9, 3, 3, 12, 3, 0, 4, 9, 9, 4, 6, 9, 6, 0, 9, 12, 12, 9, 9, 18, 9, 6, 12, 18, 18, 6, 21, 21, 6, 6, 10, 24, 9, 12, 15, 18, 12, 3, 18, 12, 18, 12, 18, 24, 15, 9, 9, 18, 24, 15, 18, 24, 9, 6, 18, 24, 12, 13, 15, 27, 6, 9, 15, 19, 18, 9, 24, 12, 18, 0, 15, 24, 27, 9, 12, 24, 12, 12

Offset: 0

Ilya Gutkovskiy, Feb 16 2017

nonn

Number of ways to write n as an ordered sum of 3 perfect powers (A001597).

			a(14) = 6 because we have  [9, 4, 1], [9, 1, 4], [4, 9, 1], [4, 1, 9], [1, 9, 4] and [1, 4, 9].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Perfect Powers

Cf. A001597, A078635, A113505.

nmax = 95; CoefficientList[Series[(x + Sum[Boole[GCD @@ FactorInteger[k][[All, 2]] > 1] x^k, {k, 2, nmax}])^3, {x, 0, nmax}], x]

G.f.: (Sum_{k = i^j, i>=1, j>=2, excluding duplicates} x^k)^3.

A135578 Not the sum of three distinct perfect powers (assuming 1, but not 0, is a perfect power).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 19, 20, 22, 23, 24, 27, 31, 43, 55, 87, 96, 102, 103, 111, 119, 167, 223, 335, 1391, 1455, 1607, 1679, 1775, 1991, 25887, 26375

Offset: 1

Zak Seidov, Feb 24 2008

nonn

Presumably the sequence is finite.

Cf. A113505.

A135579 Not the sum of three distinct perfect powers > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 30, 31, 32, 34, 35, 36, 41, 43, 46, 54, 55, 58, 59, 64, 86, 87, 91, 96, 102, 103, 111, 118, 119, 128, 130, 147, 151, 167, 223, 335, 1391, 1455, 1607, 1679, 1775, 1991, 25887

Offset: 1

Zak Seidov, Feb 24 2008

nonn

Presumably the sequence is finite.

Cf. A113505.

cp's OEIS Frontend

A135693 Not the sum of three positive squares or cubes.

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A135367 Not the sum of three perfect powers (assuming 1, but not 0, is a perfect power).

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A056828 Numbers that are not the sum of at most three powerful (1) numbers.

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A135402 Not the sum of three squares or cubes greater than 1.

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A274459 Least number of perfect powers that add up to n.

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A135393 Not the sum of three perfect powers greater than 1.

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A282499 Expansion of (Sum_{k = i^j, i>=1, j>=2, excluding duplicates} x^k)^3.

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A135578 Not the sum of three distinct perfect powers (assuming 1, but not 0, is a perfect power).

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A135579 Not the sum of three distinct perfect powers > 1.

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