A342902 -id:A342902 - cp's OEIS frontend

A342903 a(n) is the smallest number that is the sum of n positive squares in two ways.

50, 27, 28, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 2

N. J. A. Sloane, Apr 03 2021

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

			a(2) = 50 = 1+49 = 25+25.
a(3) = 27 = 1+1+25 = 9+9+9.
a(5) = 20 = 1+1+1+1+16 = 4+4+4+4+4.

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 Eq. (15), page 188.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A230563, A342902.

a(n) = n+15 for n >= 5.

A343077 a(n) is the smallest number that is the sum of n positive 4th powers in two ways.

Original entry on oeis.org

635318657, 2673, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292

Offset: 2

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,4,2) in Alter's notation.

			a(2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.
a(3) = 2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4.

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 3, page 190.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003824, A046881, A342902, A343078, A343079, A343082, A343086.

a(n) = n + 240 for n >= 16.

A343081 a(n) is the smallest number that is the sum of n positive cubes in three or more ways.

Original entry on oeis.org

5104, 1225, 766, 221, 222, 223, 224, 197, 163, 164, 165, 166, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177

Offset: 3

Sean A. Irvine, Apr 04 2021

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

			a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
a(4) = 1225 = 1^3 + 2^3 + 6^3 + 10^3 = 3^3 + 7^3 + 7^3 + 8^3 = 4^3 + 6^3 + 6^3 + 9^3.
a(9) = 224 = 6^3 + 8*1^3 = 3*4^3 + 3^3 + 5*1^3 = 5^3 + 4^3 + 4*2^3 + 3*1^3.

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 7, page 192.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A342902, A343080, A343082, A343083, A343085.

a(n) = n + 124 for n >= 15.

Corrected by Robert Israel, Apr 05 2021

a(9) reverted by Sean A. Irvine, Apr 18 2021

A343085 a(n) is the smallest number that is the sum of n positive cubes in four ways.

Original entry on oeis.org

13896, 1979, 1252, 626, 470, 256, 224, 225, 226, 227, 221, 222, 223, 203, 204, 205, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,3,4) in Alter's notation.

			a(3) = 13896 = 1^3 + 12^3 + 23^3 = 2^3 + 4^3 + 24^3 = 4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3.
a(4) = 1979 = 1^3 + 5^3 + 5^3 + 12^3 = 2^3 + 3^3 + 6^3 + 12^3 = 5^3 + 5^3 + 9^3 + 10^3 = 6^3 + 6^3 + 6^3 + 11^3.

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 11, page 194.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A342902, A343081, A343084, A343086.

LinearRecurrence[{2,-1},{13896,1979,1252,626,470,256,224,225,226,227,221,222,223,203,204,205,171,172},60] (* Harvey P. Dale, Aug 06 2022 *)

a(n) = n + 152 for n >= 19.

A343078 a(n) is the smallest number that is the sum of n positive 5th powers in two ways.

Original entry on oeis.org

1375298099, 51445, 4097, 4098, 4099, 4100, 4101, 4102, 4103, 4104, 4105, 4106, 4107, 4108, 4109, 4110, 4111, 4112, 4113, 4114, 4115, 4116, 4117, 4118, 4119, 4120, 4121, 4122, 4123, 4124, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,5,2) in Alter's notation.

			a(3) = 1375298099 = 3^5 + 54^5 + 62^5 = 24^5 + 28^5 + 67^5.
a(4) = 51445 = 4^5 + 7^5 + 7^5 + 7^5 = 5^5 + 6^5 + 6^5 + 8^5.

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 4, page 190.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A230563, A342902, A343077, A343079, A343083.

a(n) = n + 1023 for n >= 33.

A343079 a(n) is the smallest number that is the sum of n positive 6th powers in two ways.

Original entry on oeis.org

160426514, 1063010, 1063011, 570947, 570948, 63232, 63233, 52489, 52490, 52491, 16393, 16394, 16395, 16396, 16397, 13122, 13123, 13124, 13125, 13126, 13127, 13128, 13129, 13130, 13131, 13132, 13133, 13134, 13135, 13136, 13137, 13138, 8225, 8226, 8227, 8228, 8229, 6592, 6593, 6594, 6595, 6596, 6597, 6598, 6599, 6600, 6601, 6602, 6603, 6604, 6605, 6606, 6607, 6608, 6609, 6610, 6611, 6612, 6613, 6614, 6615, 6616, 4160

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,6,2) in Alter's notation.

Alter paper has a typographical error a(3)=106426514.

			a(3) = 160426514 = 3^6 + 19^6 + 22^6 = 10^6 + 15^6 + 23^6.
a(4) = 1063010 = 2^6 + 2^6 + 9^6 + 9^6 = 3^6 + 5^6 + 6^6 + 10^6.

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. See Table 5, page 191.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A116643, A230563, A342902, A343077, A343078.

a(n) = n + 4095 for n >= 65.

A343080 a(n) is the smallest number that is the sum of n positive squares in three ways.

Original entry on oeis.org

325, 54, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 2

Sean A. Irvine, Apr 04 2021

nonn

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

			a(2) = 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(3) = 54 = 1^2 + 2^2 + 7^2 = 2^2 + 5^2 + 5^2 = 3^2 + 3^2 + 6^2.

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 6, page 191.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A342902, A343083, A343081, A343082, A343083, A343084.

a(n) = n + 24 for n >= 4.

A343887 a(1) = 1. Thereafter if a(n) is a novel term, a(n+1) = number of prior terms > a(n). If a(n) has been seen already, a(n+1) = a(n) + smallest prior term (which, once used, cannot be used again).

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 2, 3, 5, 0, 2, 2, 4, 1, 3, 4, 6, 0, 3, 3, 6, 9, 0, 3, 3, 6, 9, 12, 0, 4, 4, 8, 3, 7, 4, 7, 11, 1, 5, 6, 11, 16, 0, 6, 6, 12, 18, 0, 6, 6, 12, 18, 24, 0, 6, 6, 12, 18, 25, 0, 7, 7, 14, 6, 13, 7, 13, 20, 2, 10, 16, 18, 27, 0

Offset: 1

David James Sycamore, May 02 2021

The sequence is nontrivial if and only if a(1) > 0. a(n) <= n for n <= 10000, but it is not known if this holds for all n. a(n) + a(n+1) <= n is usually but not always true (first exception is at n=509; a(509) + a(510) = 248 + 311 = 559).
For n > 1, a(n) = 0 if and only if a(n-1) is a record novel term, whereas every non-record novel term is followed by a nonzero term. Let S(n) be the set of unused terms prior to a(n), then step function |S(n)| increments +1 at a(k+1), where a(k) is a novel term. S(n) typically contains multiple copies of each unused number, providing a continuously incremented supply of least prior terms to add to repeat leading terms as the sequence extends. This suggests that there is always a next record, and hence that zero occurs infinitely many times. Indices of records: 1, 5, 10, 16, 24, 29, 35, 49, 54, 60, 66, 80, 86, 114, 136, 166, 176, 192, 198, 231, ...
If a(k) is a record term, we see a(k), 0, m, m, ... where m is the least member of S(k). Between any consecutive pair of zeros we see either no novel terms, in which case the trajectory climbs quickly to the next record term, or there are novel terms, each of which disturbs and extends the trajectory to the next record (see plots).

			a(2)=0 since a(1)=1 is a novel term and there are zero terms prior to a(1) which are greater than 1. a(3)=1 since a(2)=0 is a novel term and there is one prior term (a(1)=1) which is > 0. a(4)=1+0=1 because a(3) is a repeat term and the smallest unused prior term is 0.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n) for n = 1..2^18.
Michael De Vlieger, Labeled scatterplot of a(n) for n = 1..2^9 showing records in red, zeros in blue, repeated terms in black, terms instigated by a new previous term in gold, and green otherwise.

Cf. A181391, A341846, A342902.

Block[{a = {1}, s = {}}, Do[If[FreeQ[#2, #1], AppendTo[a, Count[#2, ?(# > a[[-1]] &)] ], AppendTo[a, a[[-1]] + First[s] ]; Set[s, Rest@ s]] & @@ {First[#1], #2} & @@ TakeDrop[a, -1]; Set[s, Insert[s, a[[-2]], LengthWhile[s, # < a[[-2]] &] + 1]], 80]; a] (* _Michael De Vlieger, May 03 2021 *)

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A342903 a(n) is the smallest number that is the sum of n positive squares in two ways.

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A343077 a(n) is the smallest number that is the sum of n positive 4th powers in two ways.

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A343081 a(n) is the smallest number that is the sum of n positive cubes in three or more ways.

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A343085 a(n) is the smallest number that is the sum of n positive cubes in four ways.

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A343078 a(n) is the smallest number that is the sum of n positive 5th powers in two ways.

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A343079 a(n) is the smallest number that is the sum of n positive 6th powers in two ways.

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A343080 a(n) is the smallest number that is the sum of n positive squares in three ways.

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A343887 a(1) = 1. Thereafter if a(n) is a novel term, a(n+1) = number of prior terms > a(n). If a(n) has been seen already, a(n+1) = a(n) + smallest prior term (which, once used, cannot be used again).

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