A003106 -id:A003106 - cp's OEIS frontend

A039899 Number of partitions satisfying 0 < cn(0,5) + cn(2,5) + cn(3,5).

0, 0, 1, 2, 3, 5, 8, 12, 18, 25, 36, 49, 68, 91, 123, 162, 214, 278, 362, 464, 596, 757, 961, 1209, 1521, 1897, 2366, 2931, 3627, 4463, 5487, 6711, 8200, 9976, 12121, 14672, 17738, 21371, 25716, 30852, 36964, 44168, 52709, 62746, 74600, 88497

Offset: 0

Olivier Gérard

nonn

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: o < 0 + 2 + 3 (OMZBBp).
Number of partitions of n such that (greatest part) > (multiplicity of greatest part), for n >= 1.  For example, a(6) counts these 8 partitions:  6, 51, 42, 411, 33, 321, 3111, 21111.  See the Mathematica program at A240057 for the sequence as a count of partitions defined in this manner, and related sequences.  - Clark Kimberling, Apr 02 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A003106, A003114, A039900, A237976.

b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i<1, 0, b(n, i-1, t)+ `if`(i>n, 0, b(n-i, i,
      `if`(irem(i, 5) in {1, 4}, t, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 03 2014

Table[Count[IntegerPartitions[n], p_ /; Min[p] < Length[p]], {n, 24}] (* Clark Kimberling, Feb 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<1, 0, b[n, i-1, t] + If[i > n, 0, b[n-i, i, If[MemberQ[{1, 4}, Mod[i, 5]], t, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=0, N, x^k*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

G.f.: Sum_{k>=0} x^k * (1-x^(k*(k-1))) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

A333155 Decimal expansion of a constant related to the asymptotics of A268188 and A333153.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 09 2020

			0.5932422150033691271841376173302559541109959549627957429060245786...

Cf. A003114, A268188, A333141, A333151, A333152.

Cf. A003106, A333153, A333154.

evalf(sqrt(15) * log((sqrt(5) + 1)/2) / Pi, 120);

RealDigits[Sqrt[15]*Log[GoldenRatio]/Pi, 10, 105][[1]]

Equals sqrt(15) * log(phi) / Pi, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
If m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k^2) / Product_{j=1..k} (1 - x^j)), then a(n) ~ A333155^m * phi^(1/2) * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2)).
If m >= 0 and g.f. is Sum_{k>=1} (k^m * x^(k*(k+1)) / Product_{j=1..k} (1 - x^j)), then a(n) ~ A333155^m * exp(2*Pi*sqrt(n/15)) * n^((2*m-3)/4) / (2 * 3^(1/4) * 5^(1/2) * phi^(1/2)).

A259361 n occurs 2n+2 times.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Jun 24 2015

Define the oblong root obrt(x) to be the (larger) solution of y * (y+1) = x; i.e., obrt(x) = sqrt(x+1/4) - 1/2. So obrt(x) is an integer iff x is an oblong number (A002378). Then a(n) = floor(obrt(n)).

a(n) gives (from the preceding comment) also the maximal number of parts of partitions of n with no part 1 and difference of parts at least two. See A003106, with the combinatorial interpretation of the sum of the Rogers-Ramanujan identity. - Wolfdieter Lang, Oct 29 2016

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000194, A002378.

Cf. A003056.

a259361 = floor . subtract (1 / 2) . sqrt . (+ 1 / 4) . fromIntegral
a259361_list = concat xss where
   xss = iterate (\(xs@(x:_)) -> map (+ 1) (x : x : xs)) [0, 0]
-- Reinhard Zumkeller, Jul 09 2015

[Floor((-1+Sqrt(1+4*n))/2): n in [0..85]]; // Vincenzo Librandi, Oct 30 2016

Flatten[Table[PadLeft[{}, 2n + 2, n], {n, 0, 8}]] (* Alonso del Arte, Jun 30 2015 *)
Table[Floor[(-1 + Sqrt[1 + 4 n])/2], {n, 0, 120}] (* Michael De Vlieger, Oct 31 2016 *)

from math import isqrt
def A259361(n): return (m:=isqrt(n-1)-1)+(n-1>m*(m+3)) if n else 0 # Chai Wah Wu, Nov 07 2024

a(n) = A000194(n+1)-1.

a(n) = floor((-1 + sqrt(1+4*n))/2). See the first comment above. - Wolfdieter Lang, Oct 29 2016

A284152 a(n) = Sum_{d|n, d == 2 or 3 mod 5} d.

Original entry on oeis.org

0, 2, 3, 2, 0, 5, 7, 10, 3, 2, 0, 17, 13, 9, 3, 10, 17, 23, 0, 2, 10, 24, 23, 25, 0, 15, 30, 37, 0, 5, 0, 42, 36, 19, 7, 35, 37, 40, 16, 10, 0, 54, 43, 24, 3, 25, 47, 73, 7, 2, 20, 67, 53, 50, 0, 45, 60, 60, 0, 17, 0, 64, 73, 42, 13, 60, 67, 87, 26, 9, 0, 115, 73

Offset: 1

Seiichi Manyama, Mar 21 2017

nonn

			Divisors of 12 are 1 2 3 4 6 12.
And 2 == 12 mod 5.
We get a(12) = 2 + 3 + 12 = 17.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A284280, A284281.

Cf. A003106, A284150 (Sum_{d|n, d==1 or 4 mod 5} d).

a(n) = A000203(n) -5*A000203(n/5) -A284150(n), where A000203(.) =0 for non-integer arguments. - R. J. Mathar, Mar 21 2017

a(n) = A284280(n) + A284281(n). - Seiichi Manyama, Mar 24 2017

A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138

Offset: 0

Vaclav Kotesovec, Mar 17 2020

nonn

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.

Cf. A192918, A376841.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024

A035406 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 1 (mod 5).

Original entry on oeis.org

5, 10, 46, 58, 63, 70, 86, 89, 273, 287, 296, 299, 354, 361, 376, 397, 400, 404, 412, 485, 490, 501, 1213, 1227, 1236, 1239, 1310, 1312, 1337, 1344, 1352, 1383, 1388, 1561, 1564, 1573, 1588, 1600, 1621, 1624, 1628, 1636, 1736, 1741, 1772, 1777, 1790, 1834

Offset: 1

Olivier Gérard

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Sean A. Irvine, Table of n, a(n) for n = 1..1000
Sean A. Irvine, Java program (github)

Cf. A003106, A035407, A035408, A035409, A035410.

Revised by Geoffrey Caveney and Sean A. Irvine, Sep 03 2025

A035407 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 2 (mod 5).

Original entry on oeis.org

1, 10, 15, 70, 86, 89, 100, 102, 127, 134, 376, 397, 400, 404, 412, 485, 490, 501, 511, 513, 539, 547, 565, 570, 658, 661, 670, 1600, 1621, 1624, 1628, 1636, 1736, 1741, 1772, 1777, 1790, 1834, 1837, 1846, 2049, 2056, 2069, 2090, 2092, 2118, 2126, 2133, 2145

Offset: 1

Olivier Gérard

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A003106, A035406, A035408, A035409, A035410.

Revised by Geoffrey Caveney and Sean A. Irvine, Sep 03 2025

A035408 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 3 (mod 5).

Original entry on oeis.org

1, 15, 24, 27, 100, 102, 127, 134, 144, 149, 180, 185, 511, 513, 539, 547, 565, 570, 658, 661, 670, 689, 694, 730, 742, 747, 752, 768, 771, 887, 894, 907, 2090, 2092, 2118, 2126, 2133, 2145, 2150, 2283, 2288, 2332, 2335, 2344, 2355, 2358, 2413, 2420, 2671

Offset: 1

Olivier Gérard

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A003106, A035406, A035407, A035409, A035410.

Revised by Geoffrey Caveney and Sean A. Irvine, Sep 03 2025

A035409 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 4 (mod 5).

Original entry on oeis.org

3, 24, 27, 31, 39, 144, 149, 180, 185, 196, 205, 210, 254, 257, 266, 689, 694, 730, 742, 747, 752, 768, 771, 887, 894, 907, 916, 925, 930, 984, 1000, 1003, 1012, 1037, 1044, 1180, 1185, 1196, 2719, 2724, 2760, 2772, 2777, 2784, 2800, 2803, 2984, 2993, 2996

Offset: 1

Olivier Gérard

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A003106, A035406, A035407, A035408, A035410.

Revised by Geoffrey Caveney and Sean A. Irvine, Sep 03 2025

A035410 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 0 (mod 5).

Original entry on oeis.org

3, 5, 31, 39, 46, 58, 63, 196, 205, 210, 254, 257, 266, 273, 287, 296, 299, 354, 361, 916, 925, 930, 984, 1000, 1003, 1012, 1037, 1044, 1180, 1185, 1196, 1213, 1227, 1236, 1239, 1310, 1312, 1337, 1344, 1352, 1383, 1388, 1561, 1564, 1573, 1588, 3507, 3516, 3521

Offset: 1

Olivier Gérard

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A003106, A035406, A035407, A035408, A035409.

Revised by Geoffrey Caveney and Sean A. Irvine, Sep 03 2025

cp's OEIS Frontend

A039899 Number of partitions satisfying 0 < cn(0,5) + cn(2,5) + cn(3,5).

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Maple

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A333155 Decimal expansion of a constant related to the asymptotics of A268188 and A333153.

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A259361 n occurs 2n+2 times.

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A284152 a(n) = Sum_{d|n, d == 2 or 3 mod 5} d.

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A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

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A035406 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 1 (mod 5).

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A035407 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 2 (mod 5).

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A035408 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 3 (mod 5).

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A035409 Limit of the position of the n-th partition into parts 5k+2 or 5k+3 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 4 (mod 5).

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A333374 G.f.: Sum_{k>=1} (x^(k(k+1)) Product_{j=1..k} (1 + x^j)/(1 - x^j)).