A090858 -id:A090858 - cp's OEIS frontend

A265251 Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.

0, 0, 0, 1, 0, 1, 2, 2, 2, 4, 6, 6, 9, 10, 14, 19, 22, 26, 35, 40, 50, 63, 74, 88, 107, 127, 150, 181, 213, 249, 296, 345, 401, 473, 546, 636, 741, 853, 983, 1138, 1306, 1498, 1722, 1967, 2247, 2574, 2925, 3327, 3788, 4294, 4866, 5516, 6233, 7036, 7947, 8953

Offset: 0

Emeric Deutsch, Dec 28 2015

nonn

Conjecture: a(n) is also the difference between the number of parts in the distinct partitions of n and the number of distinct parts in the odd partitions of n (offset 0). For example, if n = 5, there are 5 parts in the distinct partitions of 5 (5, 41, 32) and 4 distinct parts in the odd partitions of 5 (namely, 5,3,1,1 in 5,311,11111) with difference 1. - George Beck, Apr 22 2017

George E. Andrews has kindly informed me that he has proved this conjecture and the result will be included in his article "Euler's Partition Identity and Two Problems of George Beck" which will appear in The Mathematics Student, 86, Nos. 1-2, January - June (2017). - George Beck, Apr 23 2017

			a(9) = 4 because we have [2,2,2,3], [3,3,3], [1,1,1,2,4], and [1,1,1,6].

Vaclav Kotesovec, Table of n, a(n) for n = 0..15866 (terms 0..10000 from Alois P. Heinz)
George E. Andrews, Euler's Partition Identity and Two Problems of George Beck, The Mathematics Student, 86, 1-2:115-119 (2017); Preprint.
Cristina Ballantine and Richard Bielak, Combinatorial proofs of two Euler type identities due to Andrews, arXiv:1803.06394 [math.CO], 2018.
Cristina Ballantine and Amanda Welch, Beck-type identities for Euler pairs of order $r$, arXiv:2006.02335 [math.NT], 2020.
Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, Hook length biases and general linear partition inequalities, arXiv:2303.16512 [math.CO], 2023.
Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a theorem for parts congruent to t mod r, arXiv:2011.08220 [math.CO], 2020.
Cristina Ballantine and Amanda Welch, Beck-type companion identities for Franklin's identity, arXiv:2101.06260 [math.CO], 2021.
Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a modular refinement, Ramanujan J. (2021).
Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, Hook length bias in odd versus distinct partitions, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #39.
Alexander Berkovich and Aritram Dhar, Finite analogs of partition bias related to hook length two and a variant of Sylvester's map, arXiv:2503.17571 [math.CO], 2025. See p. 19.
Shishuo Fu and Dazhao Tang, Generalizing a partition theorem of Andrews, arXiv:1705.05046 [math.CO], 2017.
Runqiao Li and Andrew Y. Z. Wang, The dual form of Beck type identities, Ramanujan J. (2021).
Aritro Pathak, On certain partition bijections related to Euler's partition problem, arXiv:2004.03596 [math.CO], 2020. Also Discrete Mathematics 345.2 (2022): 112673.

Column k=3 of A266477.

Cf. A015723, A090858, A090867.

g := add(x^(3*k)/(1+x^k), k = 1 .. 100)*mul(1+x^i, i = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, m), m = 0 .. 75);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+5-4*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 3*i>n, 0, b(n-3*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 5 - 4*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t] + If[t == 1 || 3*i > n, 0, b[n - 3*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 11 2016, after Alois P. Heinz *)
Take[ CoefficientList[ Expand[ Sum[x^(3k)/(1 + x^k), {k, 60}] Product[1 + x^i, {i, 60}]], x], 60] (* slower than above *) (* Robert G. Wilson v, Apr 24 2017 *)

x='x + O('x^54); concat([0, 0, 0],Vec(sum(k=1, 54, x^(3*k)/(1 + x^k)* prod(i=1, 54, 1 + x^i)))) \\ Indranil Ghosh, Apr 24 2017

G.f.: Sum_{k>=1} x^{3k}/(1+x^k)*Product_{i>=1} (1+x^i).

a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (2*log(2) - 1) / (4*Pi) = 0.040456547528... - Vaclav Kotesovec, May 24 2018

A325284 Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.

Original entry on oeis.org

3, 9, 10, 15, 20, 27, 40, 42, 45, 50, 70, 75, 80, 81, 84, 100, 105, 126, 135, 140, 160, 168, 200, 225, 243, 250, 252, 280, 294, 315, 320, 330, 336, 350, 375, 378, 400, 405, 462, 490, 500, 504, 525, 560, 588, 640, 660, 672, 675, 700, 729, 735, 756, 770, 800

Offset: 1

Gus Wiseman, Apr 19 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts form an initial interval with a single hole. The enumeration of these partitions by sum is given by A090858.

			The sequence of terms together with their prime indices begins:
    3: {2}
    9: {2,2}
   10: {1,3}
   15: {2,3}
   20: {1,1,3}
   27: {2,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   45: {2,2,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
  100: {1,1,3,3}
  105: {2,3,4}
  126: {1,2,2,4}
  135: {2,2,2,3}
  140: {1,1,3,4}

Cf. A055932, A056239, A061395, A090858, A112798, A124010, A127002, A130091, A325241, A325251, A325259, A325270.

Select[Range[100],Length[Complement[Range[PrimePi[FactorInteger[#][[-1,1]]]],PrimePi/@First/@FactorInteger[#]]]==1&]

A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 9, 14, 20, 29, 40, 55, 75, 100, 133, 175, 229, 296, 383, 489, 625, 791, 1000, 1254, 1573, 1957, 2434, 3009, 3716, 4564, 5602, 6841, 8347, 10142, 12308, 14882, 17975, 21636, 26013, 31184, 37336, 44582, 53172, 63260, 75173, 89133, 105556

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A080257.

Partitions with 2 distinct parts are in A002133(n). Partitions with at least 3 parts are in A004250(n). Some partitions are in both subsets, so A002133(n)+A004250(n) >= a(n). - R. J. Mathar, Dec 13 2022

			The a(1) = 1 through a(8) = 20 partitions:
  (21)   (31)    (32)     (42)      (43)       (53)
  (111)  (211)   (41)     (51)      (52)       (62)
         (1111)  (221)    (222)     (61)       (71)
                 (311)    (321)     (322)      (332)
                 (2111)   (411)     (331)      (422)
                 (11111)  (2211)    (421)      (431)
                          (3111)    (511)      (521)
                          (21111)   (2221)     (611)
                          (111111)  (3211)     (2222)
                                    (4111)     (3221)
                                    (22111)    (3311)
                                    (31111)    (4211)
                                    (211111)   (5111)
                                    (1111111)  (22211)
                                               (32111)
                                               (41111)
                                               (221111)
                                               (311111)
                                               (2111111)
                                               (11111111)

Cf. A001221, A001222, A001358, A001399, A007774, A008284, A060687, A080257, A090858, A116608, A325244.

Cf. A000041, A002133, A004250.

A325269 := proc(n)
    local a,p,s ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(p) >= 3 or nops(s) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A325269(n),n=0..40) ; # R. J. Mathar, Dec 13 2022

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2||Length[#]>2&]],{n,0,30}]

conjecture: a(n) = A000041(n) - A000034(n-1), n>0. - R. J. Mathar, Dec 13 2022

cp's OEIS Frontend

A265251 Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.

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A325284 Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.

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A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

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