A026811 -id:A026811 - cp's OEIS frontend

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

1, 3, 5, 9, 10, 15, 25, 27, 30, 45, 50, 75, 81, 90, 100, 125, 135, 150, 225, 243, 250, 270, 300, 375, 405, 450, 500, 625, 675, 729, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is 5-smooth if its prime divisors are all <= 5.

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 sequence of terms together with their prime indices begins:
       1: {}            125: {3,3,3}
       3: {2}           135: {2,2,2,3}
       5: {3}           150: {1,2,3,3}
       9: {2,2}         225: {2,2,3,3}
      10: {1,3}         243: {2,2,2,2,2}
      15: {2,3}         250: {1,3,3,3}
      25: {3,3}         270: {1,2,2,2,3}
      27: {2,2,2}       300: {1,1,2,3,3}
      30: {1,2,3}       375: {2,3,3,3}
      45: {2,2,3}       405: {2,2,2,2,3}
      50: {1,3,3}       450: {1,2,2,3,3}
      75: {2,3,3}       500: {1,1,3,3,3}
      81: {2,2,2,2}     625: {3,3,3,3}
      90: {1,2,2,3}     675: {2,2,2,3,3}
     100: {1,1,3,3}     729: {2,2,2,2,2,2}

Allowing any number of parts and sum gives A051037, counted by A001399.
These are Heinz numbers of the partitions counted by A266755.
Allowing parts > 5 gives A344291, counted by A110618.
The non-3-smooth case is A344294, counted by A325691.
Requiring the sum of prime indices to be even gives A344295.
A000070 counts non-multigraphical partitions, ranked by A344292.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A300061 ranks partitions of even numbers, with 5-smooth case A344297.
Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]

Intersection of A051037 and A344291.

A060024 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 5.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 0, -3, -3, -8, -10, -16, -20, -29, -35, -47, -56, -72, -85, -105, -122, -148, -171, -202, -231, -270, -306, -353, -397, -453, -507, -573, -637, -715, -791, -881, -970, -1075, -1178, -1298, -1417, -1554, -1691, -1846, -2001, -2177, -2353, -2550, -2748, -2969

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+4 into 4 parts and the number of partitions of n+4 into 5 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

Cf. A026810, A026811.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), this sequence (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

CoefficientList[Series[(1-x-x^5)/(Times@@(1-x^Range[5])),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1},{1,0,1,1,2,1,2,1,2,0,0,-3,-3,-8,-10},60] (* Harvey P. Dale, Dec 21 2015 *)

Vec((1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(0)=1, a(1)=0, a(2)=1, a(3)=1, a(4)=2, a(5)=1, a(6)=2, a(7)=1, a(8)=2, a(9)=0, a(10)=0, a(11)=-3, a(12)=-3, a(13)=-8, a(14)=-10, a(n) = a(n-1)+ a(n-2)-a(n-5)-a(n-6)-a(n-7)+a(n-8)+a(n-9)+a(n-10)-a(n-13)- a(n-14)+ a(n-15). - Harvey P. Dale, Dec 21 2015
a(n) = A026810(n+4) - A026811(n+4). - Wesley Ivan Hurt, Apr 16 2019
G.f.: (1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Apr 17 2019

A060025 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 6.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 2, 3, -1, -1, -6, -9, -17, -22, -35, -43, -61, -76, -100, -121, -155, -185, -229, -271, -328, -383, -458, -529, -622, -715, -830, -946, -1090, -1233, -1407, -1584, -1794, -2008, -2261, -2517, -2816, -3124, -3476, -3838, -4253, -4677, -5159, -5656, -6213

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+5 into 5 parts and the number of partitions of n+5 into 6 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1).

Cf. A026811, A026812.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), this sequence (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

m:=6; R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x-x^m)/( (&*[1-x^j: j in [1..m]]) ) )); // G. C. Greubel, Apr 17 2019

With[{nn=6},CoefficientList[Series[(1-x-x^nn)/Times@@(1-x^Range[nn]),{x,0,60}],x]] (* Harvey P. Dale, May 15 2016 *)

Vec((1 - x - x^6) / ((1 - x)^6*(1 + x)^3*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Apr 17 2019

m=6; ((1-x-x^m)/( product(1-x^j for j in (1..m)) )).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Apr 17 2019

a(n) = A026811(n+5) - A026812(n+5). - Wesley Ivan Hurt, Apr 16 2019

G.f.: (1 - x - x^6) / ((1 - x)^6*(1 + x)^3*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Apr 17 2019

A211860 Number of partitions of n into parts <= 5 with the property that all parts have distinct multiplicities.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 9, 11, 12, 16, 22, 21, 33, 37, 39, 51, 65, 63, 86, 85, 105, 118, 149, 148, 185, 198, 238, 251, 304, 304, 381, 388, 454, 478, 565, 576, 679, 704, 819, 842, 978, 1013, 1168, 1195, 1377, 1415, 1616, 1668, 1874, 1937, 2197, 2246, 2512, 2625

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

			For n=3 the a(3)=2 partitions are {3} and {1,1,1}. Note that {2,1} does not count, as 1 and 2 appear with the same nonzero multiplicity.

Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.

Cf. A026811, A098859.

Cf. A105637, A211858, A211859, A211861, A211862, A211863.

a211860 n = p 0 [] [1..5] n where
   p m ms _      0 = if m `elem` ms then 0 else 1
   p    []     _ = 0
   p m ms ks'@(k:ks) x
     | x < k       = 0
     | m == 0      = p 1 ms ks' (x - k) + p 0 ms ks x
     | m `elem` ms = p (m + 1) ms ks' (x - k)
     | otherwise   = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x
-- Reinhard Zumkeller, Dec 27 2012

A308822 Sum of all the parts in the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 6, 14, 24, 45, 70, 110, 156, 234, 322, 450, 592, 799, 1026, 1330, 1680, 2121, 2618, 3243, 3936, 4800, 5746, 6885, 8148, 9657, 11310, 13237, 15360, 17820, 20502, 23590, 26928, 30747, 34884, 39546, 44600, 50266, 56364, 63167, 70488, 78615

Offset: 0

Wesley Ivan Hurt, Jun 26 2019

nonn

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     70         110         156         234         322        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019

Index entries for sequences related to partitions

Cf. A026811, A308823, A308824, A308825, A308826, A308827.

Table[n*Sum[Sum[Sum[Sum[1, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]

a(n) = n * A026811(n).

a(n) = A308823(n) + A308824(n) + A308825(n) + A308826(n) + A308827(n).

A308823 Sum of the smallest parts of the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 48, 62, 78, 98, 122, 149, 181, 219, 262, 314, 370, 436, 510, 595, 691, 797, 916, 1050, 1198, 1365, 1545, 1747, 1968, 2212, 2480, 2771, 3089, 3437, 3814, 4227, 4669, 5151, 5670, 6232, 6838, 7487, 8185, 8936

Offset: 0

Wesley Ivan Hurt, Jun 26 2019

nonn

			Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      8          11          15          21          28        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019

Index entries for sequences related to partitions

Cf. A026811, A308822, A308824, A308825, A308826, A308827.

Table[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n,{5}][[;;,5]]],{n,0,60}] (* Harvey P. Dale, Nov 20 2024 *)

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} l.
a(n) = A308822(n) - A308824(n) - A308825(n) - A308826(n) - A308827(n).
Conjectures from Colin Barker, Jun 30 2019: (Start)
G.f.: x^5 / ((1 - x)^6*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = a(n-1) + a(n-2) - 2*a(n-6) - 2*a(n-7) + a(n-8) + a(n-9) + 2*a(n-10) + a(n-11) + a(n-12) - 2*a(n-13) - 2*a(n-14) + a(n-18) + a(n-19) - a(n-20) for n>19.
(End)

A308824 Sum of the fourth largest parts in the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 13, 18, 27, 36, 50, 64, 86, 109, 140, 175, 220, 269, 331, 399, 486, 577, 689, 811, 959, 1119, 1305, 1508, 1747, 2003, 2300, 2617, 2984, 3376, 3821, 4300, 4839, 5415, 6060, 6749, 7521, 8337, 9243, 10207, 11273, 12404, 13641

Offset: 0

Wesley Ivan Hurt, Jun 26 2019

nonn

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |      9          13          18          27          36        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019

Index entries for sequences related to partitions

Cf. A026811, A308822, A308823, A308825, A308826, A308827.

Table[Sum[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} k.
a(n) = A308822(n) - A308823(n) - A308825(n) - A308826(n) - A308827(n).
Conjectures from Colin Barker, Jun 30 2019: (Start)
G.f.: x^5*(1 + x^3 + x^6) / ((1 - x)^6*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = a(n-1) + a(n-2) - a(n-3) + 2*a(n-4) - 4*a(n-6) + a(n-8) - 3*a(n-9) + 4*a(n-10) + 4*a(n-11) - 3*a(n-12) + a(n-13) - 4*a(n-15) + 2*a(n-17) - a(n-18) + a(n-19) + a(n-20) - a(n-21) for n>20.
(End)

A308825 Sum of the third largest parts of the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 17, 24, 36, 50, 69, 91, 123, 158, 204, 259, 326, 403, 499, 606, 739, 886, 1060, 1256, 1489, 1745, 2041, 2371, 2750, 3166, 3643, 4160, 4750, 5393, 6112, 6897, 7774, 8720, 9772, 10910, 12168, 13518, 15006, 16601, 18352, 20229

Offset: 0

Wesley Ivan Hurt, Jun 26 2019

nonn

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     11          17          24          36          50        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 11 2019

Index entries for sequences related to partitions

Cf. A026811, A308822, A308823, A308824, A308826, A308827.

Table[Sum[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]
Table[Total[IntegerPartitions[n,{5}][[;;,3]]],{n,0,50}] (* Harvey P. Dale, Oct 01 2024 *)

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} j.

a(n) = A308822(n) - A308823(n) - A308824(n) - A308826(n) - A308827(n).

A308826 Sum of the second largest parts in the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 25, 35, 54, 74, 105, 138, 189, 242, 317, 400, 509, 628, 783, 950, 1164, 1394, 1677, 1985, 2361, 2765, 3246, 3768, 4382, 5043, 5815, 6640, 7596, 8621, 9789, 11043, 12465, 13981, 15689, 17513, 19554, 21723, 24139, 26704

Offset: 0

Wesley Ivan Hurt, Jun 26 2019

nonn

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     15          25          35          54          74        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 12 2019

Index entries for sequences related to partitions

Cf. A026811, A308822, A308823, A308824, A308825, A308827.

Table[Sum[Sum[Sum[Sum[i, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} i.

a(n) = A308822(n) - A308823(n) - A308824(n) - A308825(n) - A308827(n).

A308827 Sum of the largest parts of the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 9, 17, 27, 44, 64, 96, 134, 188, 251, 339, 439, 571, 724, 917, 1137, 1411, 1719, 2097, 2519, 3023, 3586, 4253, 4990, 5848, 6797, 7891, 9092, 10467, 11966, 13670, 15526, 17612, 19880, 22417, 25159, 28209, 31502, 35145, 39061, 43375

Offset: 0

Wesley Ivan Hurt, Jun 26 2019

nonn

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     27          44          64          96         134        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 12 2019

Index entries for sequences related to partitions

Cf. A026811, A308822, A308823, A308824, A308825, A308826.

Table[Sum[Sum[Sum[Sum[n - i - j - k - l, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 100}]

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} (n-i-j-k-l).

a(n) = A308822(n) - A308823(n) - A308824(n) - A308825(n) - A308826(n).

cp's OEIS Frontend

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

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A060024 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 5.

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A060025 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 6.

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A211860 Number of partitions of n into parts <= 5 with the property that all parts have distinct multiplicities.

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A308822 Sum of all the parts in the partitions of n into 5 parts.

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A308823 Sum of the smallest parts of the partitions of n into 5 parts.

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A308824 Sum of the fourth largest parts in the partitions of n into 5 parts.

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A308825 Sum of the third largest parts of the partitions of n into 5 parts.

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