A342903 -id:A342903 - cp's OEIS frontend

A342893 Smallest number that is the sum of three positive n-th powers in three ways.

54, 5104, 811538

Offset: 2

N. J. A. Sloane, Apr 03 2021

			a(2) = 54 = 1+4+49 = 4+25+25 = 9+9+36.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342894-A342903.

A342902 a(n) is the smallest number that is the sum of n positive cubes in two ways.

Original entry on oeis.org

1729, 251, 219, 157, 158, 131, 132, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126

Offset: 2

N. J. A. Sloane, Apr 03 2021

nonn

This is r(n,3,2) in Alter's notation.

			a(2) = 1729 = 12^3 + 1^3 = 10^3 + 9^3 (the famous Hardy-Ramanujan number).
a(3) = 251 = 5^3 + 5^3 + 1^3 = 6^3 + 3^3 + 2^3.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 2, page 190.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001235, A011541, A230563, A342903, A343077, A343078, A343079, A343081, A343085.

a(n) = n+63 for n >= 9.

A342895 Smallest number that is the sum of five positive n-th powers in three ways.

Original entry on oeis.org

29, 766, 4225, 13124675, 224688658

Offset: 2

N. J. A. Sloane, Apr 04 2021

a(7) > 2*10^11. - Jon E. Schoenfield, Apr 11 2021

			From _Jon E. Schoenfield_, Apr 11 2021: (Start)
a(2) = 29 = 1^2 + 1^2 + 1^2 + 1^2 + 5^2
          = 1^2 + 1^2 + 3^2 + 3^2 + 3^2
          = 1^2 + 2^2 + 2^2 + 2^2 + 4^2.
.
a(3) = 766 = 1^3 + 1^3 + 2^3 + 3^3 + 9^3
           = 1^3 + 4^3 + 4^3 + 5^3 + 8^3
           = 2^3 + 2^3 + 4^3 + 7^3 + 7^3.
.
a(4) = 4225 = 2^4 + 2^4 + 2^4 + 3^4 + 8^4
            = 2^4 + 4^4 + 4^4 + 6^4 + 7^4
            = 3^4 + 4^4 + 6^4 + 6^4 + 6^4.
.
a(5) = 13124675 =  1^5 +  9^5 + 10^5 + 20^5 + 25^5
                =  2^5 +  5^5 + 12^5 + 23^5 + 23^5
                = 16^5 + 19^5 + 20^5 + 20^5 + 20^5.
.
a(6) = 224688658 = 3^6 +  8^6 + 19^6 + 20^6 + 22^6
                 = 5^6 +  5^6 + 16^6 + 16^6 + 24^6
                 = 8^6 + 10^6 + 15^6 + 20^6 + 23^6. (End)

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

a(6) from Jinyuan Wang, Apr 11 2021

A342894 Smallest number that is the sum of four positive n-th powers in three ways.

Original entry on oeis.org

28, 1225, 16578, 1479604544, 1885800643779

Offset: 2

N. J. A. Sloane, Apr 04 2021

			a(2) = 28 = 1+1+1+25 = 4+4+4+16 = 1+9+9+9.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

a(5)-a(6) from Jinyuan Wang, Apr 10 2021

A342896 Smallest number that is the sum of six positive n-th powers in three ways.

Original entry on oeis.org

30, 221, 2676, 696467, 17886882

Offset: 2

N. J. A. Sloane, Apr 04 2021

			From _Ilya Gutkovskiy_, Aug 12 2024: (Start)
a(6) = 17886882 = 1^6 + 8^6 + 8^6 + 12^6 + 12^6 + 15^6
                = 2^6 + 2^6 + 6^6 +  9^6 +  9^6 + 16^6
                = 3^6 + 5^6 + 6^6 +  6^6 + 10^6 + 16^6. (End)

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

a(6) from Jinyuan Wang, Apr 11 2021

A342897 Smallest number that is the sum of three positive n-th powers in four ways.

Original entry on oeis.org

129, 13896, 5978882

Offset: 2

N. J. A. Sloane, Apr 04 2021

			a(2) = 129 = 1+64+64 = 4+4+121 = 16+49+64 = 4+25+100.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

A342898 Smallest number that is the sum of four positive n-th powers in four ways.

Original entry on oeis.org

52, 1979, 236674

Offset: 2

N. J. A. Sloane, Apr 04 2021

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

A342899 Smallest number that is the sum of five positive n-th powers in four ways.

Original entry on oeis.org

53, 1252, 20995, 287618651, 351060139210

Offset: 2

N. J. A. Sloane, Apr 04 2021

			53 = 1^2 + 1^2 + 1^2 + 1^2 + 7^2
   = 1^2 + 1^2 + 1^2 + 5^2 + 5^2
   = 1^2 + 2^2 + 4^2 + 4^2 + 4^2
   = 1^2 + 3^2 + 3^2 + 3^2 + 5^2
   = 2^2 + 2^2 + 2^2 + 4^2 + 5^2.
1252 = 1^3 + 1^3 + 5^3 + 5^3 + 10^3
     = 1^3 + 2^3 + 3^3 + 6^3 + 10^3
     = 3^3 + 3^3 + 7^3 + 7^3 +  8^3
     = 3^3 + 4^3 + 6^3 + 6^3 +  9^3.
20995 = 1^4 + 1^4 + 1^4 + 4^4 + 12^4
      = 2^4 + 3^4 + 3^4 + 3^4 + 12^4
      = 2^4 + 6^4 + 9^4 + 9^4 +  9^4
      = 4^4 + 6^4 + 7^4 + 7^4 + 11^4.
287618651 =  8^5 + 21^5 + 27^5 + 27^5 + 48^5
          =  9^5 + 13^5 + 26^5 + 37^5 + 46^5
          = 11^5 + 12^5 + 23^5 + 41^5 + 44^5
          = 11^5 + 20^5 + 22^5 + 30^5 + 48^5.
351060139210 =  8^6 + 50^6 + 53^6 + 56^6 + 81^6
             = 18^6 + 43^6 + 49^6 + 64^6 + 80^6
             = 27^6 + 31^6 + 56^6 + 62^6 + 80^6
             = 35^6 + 36^6 + 38^6 + 73^6 + 76^6.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

a(5) from Robert Israel, Apr 07 2021

a(6) from Michael S. Branicky, May 08 2021

A342900 Smallest number that is the sum of six positive n-th powers in four ways.

Original entry on oeis.org

36, 626, 6626, 12047994, 353563195

Offset: 2

N. J. A. Sloane, Apr 04 2021

			From _Ilya Gutkovskiy_, Aug 12 2024: (Start)
a(6) = 353563195 = 3^6 +  8^6 + 11^6 + 19^6 + 22^6 + 24^6
                 = 4^6 +  5^6 + 15^6 + 18^6 + 20^6 + 25^6
                 = 8^6 + 10^6 + 11^6 + 15^6 + 23^6 + 24^6
                 = 8^6 + 13^6 + 15^6 + 15^6 + 16^6 + 26^6. (End)

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

a(5)-a(6) from Jinyuan Wang, Apr 11 2021

A342901 Smallest number that is the sum of seven positive n-th powers in four ways.

Original entry on oeis.org

37, 470, 2932, 893604, 82718691

Offset: 2

N. J. A. Sloane, Apr 04 2021

			From _Ilya Gutkovskiy_, Aug 12 2024: (Start)
a(6) = 82718691 = 1^6 + 1^6 +  8^6 + 14^6 + 16^6 + 17^6 + 18^6
                = 3^6 + 7^6 + 10^6 + 10^6 + 16^6 + 16^6 + 19^6
                = 4^6 + 5^6 + 10^6 + 12^6 + 14^6 + 17^6 + 19^6
                = 4^6 + 9^6 + 10^6 + 13^6 + 13^6 + 14^6 + 20^6. (End)

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.

Cf. A230563, A342893-A342903.

a(5)-a(6) from Jinyuan Wang, Apr 11 2021

cp's OEIS Frontend

A342893 Smallest number that is the sum of three positive n-th powers in three ways.

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A342902 a(n) is the smallest number that is the sum of n positive cubes in two ways.

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A342895 Smallest number that is the sum of five positive n-th powers in three ways.

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A342894 Smallest number that is the sum of four positive n-th powers in three ways.

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A342896 Smallest number that is the sum of six positive n-th powers in three ways.

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A342897 Smallest number that is the sum of three positive n-th powers in four ways.

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A342898 Smallest number that is the sum of four positive n-th powers in four ways.

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A342899 Smallest number that is the sum of five positive n-th powers in four ways.

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A342900 Smallest number that is the sum of six positive n-th powers in four ways.

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A342901 Smallest number that is the sum of seven positive n-th powers in four ways.

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