A307597 -id:A307597 - cp's OEIS frontend

A307598 Number of partitions of n into 3 distinct positive triangular numbers.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 0, 4, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 2, 2, 2, 1, 3, 2, 0, 4, 1, 1, 5, 1, 3, 2, 2, 3, 2, 2, 1, 4, 1, 2, 4, 2, 2, 3, 2, 1, 3, 2, 4, 3, 3, 2, 2, 3, 1, 6

Offset: 0

Ilya Gutkovskiy, Apr 17 2019

nonn

The greedy inverse starts 0, 10, 19, 37, 52, 82, 109, 136, 241, 226, 217, 247, 364, 427, 457, 541, 532, 577, 637, 961, 721, 787, 1066, 1102, 1381, 1267, 1564, 1192, 1396, 1816, 1501, 1612, 1927, 1942, 2242, 1792, 2842, 2587, 2557, 2422, ... - R. J. Mathar, Apr 28 2020

			a(19) = 2 because we have [15, 3, 1] and [10, 6, 3].

Cf. A000217, A002244, A008443, A024940, A025442, A053604, A063993, A224326, A307597.

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^(k*(k+1)/2)).

A260647 Numbers that are the sum of two distinct nonzero triangular numbers.

Original entry on oeis.org

4, 7, 9, 11, 13, 16, 18, 21, 22, 24, 25, 27, 29, 31, 34, 36, 37, 38, 39, 42, 43, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 91, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108, 111, 112, 114, 115, 119, 120

Offset: 1

Arkadiusz Wesolowski, Dec 02 2015

nonn

The sequence contains every square greater than 1.

Conjecture: the sequence contains infinitely many primes.

			24 = 3 + 21, so 24 is in the sequence.

Cf. A000217, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).

Subsequence of A051533.

r = 120; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, n_ /; n > 0]
With[{nn=20},Select[Union[Total/@Subsets[Accumulate[Range[nn]],{2}]],#<= (nn(nn+1))/2+1&]] (* Harvey P. Dale, Jul 26 2020 *)

{k: A307597(k) > 0 }. - R. J. Mathar, Apr 28 2020

A341021 Number of partitions of n into 4 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 1, 1, 0, 2, 1, 3, 0, 1, 2, 0, 2, 2, 1, 3, 2, 0, 2, 2, 2, 1, 3, 0, 4, 3, 1, 3, 2, 2, 3, 2, 1, 5, 3, 3, 2, 4, 1, 2, 5, 1, 5, 3, 2, 5, 3, 3, 4, 4, 3, 4, 6, 0, 6, 4, 2, 7, 4, 3, 5, 4, 3, 5, 5, 5, 4, 5, 5, 5, 8, 2, 6, 5, 1, 10, 5, 4, 7, 7, 4

Offset: 20

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A008438, A010054, A307597, A307598, A319814, A340949, A341022, A341023, A341024, A341025, A341026, A341027.

A342326 a(n) is the smallest nonnegative integer that can be written as a sum of two distinct nonzero triangular numbers in exactly n ways or -1 if no such integer exists.

Original entry on oeis.org

0, 4, 16, 81, 471, 2031, 1381, 11781, 6906, 17956, 34531, 123256, 40056, 305256, 863281, 448906, 200281, 1957231, 520731, 10563906, 1001406, 11222656, 7631406, 3454506, 1482081, 75865156, 7172606106, 8852431, 25035156, 334020781, 13018281, 38531031, 7410406, 7014160156

Offset: 0

Robert G. Root, Mar 08 2021

nonn

Conjecture: This sequence has a positive a(n) for every positive integer n, and each sequence in the infinite indexed family, of which this sequence offers the initial terms, is infinite, as well.
From David A. Corneth, Mar 08 2021: (Start)
a(40) = 37052031, a(45) = 221310781, a(48) = 60765331, a(39) <= 2782318906, a(42) <= 325457031, a(47) <= 927577056, a(50) <= 2200089531, a(54) <= 327539956, a(56) <= 926300781, a(60) <= 481676406, a(63) <= 4598740656, a(64) <= 303826656, a(71) <= 4579579956, a(72) <= 789949306, a(80) <= 1519133281, a(96) <= 3220562556. Terms for n <= 96 not listed here and terms for which only upper bounds are known are >= 3*10^8.
Is a(n) == 6 (mod 25) for n >= 5? It holds for all terms known to date.
The triangular numbers mod 25 are periodic with period 25. Constructing all 25*25 = 625 sums of two distinct triangular numbers mod 25 gives 65 cases for 6 (mod 25). The second largest occurs 40 times. (End)
a(47) = 550240551, a(59) = 7629645156, a(67) = 6418012656, a(81) = 9498658731, a(90) = 8188498906. All upper bounds listed in the above comments for n other than 47 are the exact values of a(n). For all n for which no value is listed here or above, a(n) > 10^10 (or a(n) = -1). - Jon E. Schoenfield, Mar 09 2021
From Martin Ehrenstein, Mar 09 2021: (Start)
a(44) = 15646972656. For n<=51, all terms not mentioned here or above, a(n) >= 6.5*10^10 (or a(n) = -1).
a(47) == 1 (mod 25) and a(95) = 47652012541 == 16 (mod 25). Thus the answer to Corneth's question is 'No'. (End)

			a(1) =  4 = 1 +  3;
a(2) = 16 = 1 + 15 =  6 + 10;
a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.

Martin Ehrenstein, Table of n, a(n) for n = 0..37

Cf. A000217, A260647, A051533, A262749, A265140, A265134, A265136, A265138, A307597.

r = 125000; (* generates the first 12 terms of the sequence *)
lst = Table[0, {r}];
lim = Floor[Sqrt[2r]];
Do[ num = (i^2 + i)/2 + (j^2 + j)/2;
If[num <= r, lst[[num]]++], {i, lim}, {j,  i - 1}];
First /@ (Flatten@Position[lst, #] & /@ Range[Max[lst]])

upto(n) = {my(v = vector(n)); res = vector(10); for(i = 1, (sqrtint(8*n + 1)-1)\2, bi = binomial(i + 1, 2); for(j = i+1, (sqrtint(8*(n - bi))-1)\2, v[bi + binomial(j+1, 2)]++ ) ); for(i = 1, #v, if(v[i] > 0, if(v[i] > #res, res = concat(res, vector(v[i] - #res)); ); if(res[v[i]] == 0, res[v[i]] = i ) ) ); concat(0, res) } \\ David A. Corneth, Mar 08 2021

a(n) = min { m >= 0 : A307597(m) = n }. - Alois P. Heinz, Mar 08 2021

a(13)-a(18) from Alois P. Heinz, Mar 08 2021
a(19)-a(25) from David A. Corneth, Mar 08 2021
a(26)-a(33) from Jon E. Schoenfield, Mar 09 2021 (some terms first found by David A. Corneth)

A341027 Number of partitions of n into 10 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 1, 0, 4, 0, 2, 1, 0, 2, 2, 1, 2, 1, 0, 2, 2, 2, 2, 2, 1, 2, 3, 0, 2, 4, 1, 3, 2, 4, 1, 3, 2, 3, 4, 0, 4, 4, 1, 5, 5, 3

Offset: 220

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A010054, A226254, A307597, A307598, A319820, A340955, A341021, A341022, A341023, A341024, A341025, A341026.

A341022 Number of partitions of n into 5 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 3, 0, 2, 2, 0, 1, 3, 2, 3, 2, 1, 2, 3, 2, 0, 5, 1, 3, 3, 2, 3, 2, 4, 2, 5, 2, 2, 6, 1, 4, 6, 1, 5, 6, 3, 4, 4, 4, 4, 5, 3, 5, 7, 6, 4, 8, 2, 5, 7, 3, 7, 7, 7, 5, 7, 6, 4, 12, 5, 4, 10, 3, 11, 9, 5, 9, 8, 5

Offset: 35

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A008439, A010054, A307597, A307598, A319815, A340950, A341021, A341023, A341024, A341025, A341026, A341027.

A341023 Number of partitions of n into 6 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 2, 0, 3, 1, 1, 3, 2, 1, 3, 2, 1, 3, 3, 1, 2, 3, 3, 5, 2, 2, 5, 1, 2, 5, 3, 4, 3, 5, 2, 4, 7, 1, 6, 4, 2, 8, 4, 5, 7, 4, 4, 6, 6, 4, 9, 6, 4, 8, 5, 6, 11, 8, 4, 9, 5, 8, 8, 9, 8, 9, 11

Offset: 56

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A008440, A010054, A307597, A307598, A319816, A340951, A341021, A341022, A341024, A341025, A341026, A341027.

A341024 Number of partitions of n into 7 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 3, 0, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 4, 0, 2, 4, 0, 3, 4, 3, 4, 3, 1, 4, 3, 3, 4, 4, 4, 2, 6, 3, 6, 6, 1, 6, 3, 3, 5, 9, 4, 4, 8, 2, 6, 9, 3, 9, 7, 4, 9, 6, 6, 10, 8, 5

Offset: 84

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A010054, A226252, A307597, A307598, A319817, A340952, A341021, A341022, A341023, A341025, A341026, A341027.

A341025 Number of partitions of n into 8 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 3, 1, 1, 3, 1, 1, 2, 1, 2, 4, 2, 2, 2, 0, 2, 5, 2, 1, 5, 2, 2, 4, 3, 4, 2, 3, 4, 3, 5, 1, 8, 4, 3, 8, 1, 4, 7, 2, 5, 7, 5, 6, 6, 4, 4, 10, 6

Offset: 120

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A007331, A010054, A307597, A307598, A319818, A340953, A341021, A341022, A341023, A341024, A341026, A341027.

A341026 Number of partitions of n into 9 distinct nonzero triangular numbers.

Original entry on oeis.org

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

Offset: 165

Ilya Gutkovskiy, Feb 02 2021

nonn

Cf. A000217, A010054, A226253, A307597, A307598, A319819, A340954, A341021, A341022, A341023, A341024, A341025, A341027.

cp's OEIS Frontend

A307598 Number of partitions of n into 3 distinct positive triangular numbers.

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A260647 Numbers that are the sum of two distinct nonzero triangular numbers.

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A341021 Number of partitions of n into 4 distinct nonzero triangular numbers.

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A342326 a(n) is the smallest nonnegative integer that can be written as a sum of two distinct nonzero triangular numbers in exactly n ways or -1 if no such integer exists.

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A341027 Number of partitions of n into 10 distinct nonzero triangular numbers.

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A341022 Number of partitions of n into 5 distinct nonzero triangular numbers.

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A341023 Number of partitions of n into 6 distinct nonzero triangular numbers.

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A341024 Number of partitions of n into 7 distinct nonzero triangular numbers.

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A341025 Number of partitions of n into 8 distinct nonzero triangular numbers.

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A341026 Number of partitions of n into 9 distinct nonzero triangular numbers.

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