A026794 -id:A026794 - cp's OEIS frontend

A185328 Number of partitions of n with parts >= 8.

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 21, 23, 27, 30, 36, 39, 46, 51, 60, 66, 77, 85, 99, 110, 126, 140, 162, 179, 205, 228, 260, 289, 329, 365, 415, 461, 521, 579, 655, 726, 818, 909, 1022, 1134, 1273, 1411

Offset: 0

Jason Kimberley, Jan 31 2012

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 8 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

By removing a single part of size 8, an A026801 partition of n becomes an A185328 partition of n - 8. Hence this sequence is essentially the same as A026801.

Robert Israel, Table of n, a(n) for n = 0..2000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), this sequence (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+8): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

N:= 100: # for a(0)..a(N)
g:= mul(1/(1-x^m),m=8..N):
S:= series(g,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Dec 19 2017

CoefficientList[Series[1/QPochhammer[x^8, x], {x,0,75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+8))) \\ G. C. Greubel, Nov 03 2019

def A185328_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+8)) for m in (0..80)) ).list()
A185328_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=8} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-8) - p(n-10) - p(n-11) - 2*p(n-12) + 2*p(n-16) + p(n-17) + p(n-18) - p(n-20) - p(n-21) - p(n-23) + p(n-26) + p(n-27) - p(n-28) where p(n)=A000041(n). - Shanzhen Gao
This sequence is the Euler transformation of A185118.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*Pi^7 / (18*sqrt(2)*n^(9/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(8*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
G.f.: 1 + Sum_{n >= 1} x^(n+7)/Product_{k = 0..n-1} (1 - x^(k+8)). - Peter Bala, Dec 01 2024

A026800 Number of partitions of n in which the least part is 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045

Offset: 0

Clark Kimberling

From Jason Kimberley, Feb 03 2011: (Start)
a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 7 (all such graphs are simple). The integer i corresponds to the i-cycle; the addition of integers corresponds to the disconnected union of cycles.
By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. (End)

			a(0)=0 because there does not exist a least part of the empty partition.
The  a(7)=1 partition is 7.
The a(14)=1 partition is 7+7.
The a(15)=1 partition is 7+8.
.............................
The a(20)=1 partition is 7+13.
The a(21)=2 partitions are 7+7+7 and 7+14.

Robert Israel, Table of n, a(n) for n = 0..10000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Cf. A185327 (Mathematica code)
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), this sequence (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 03 2011

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026800 := func< n | p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)- p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) >; // Jason Kimberley, Feb 03 2011

R:=PowerSeriesRing(Integers(), 75); [0,0,0,0,0,0,0] cat Coefficients(R!( x^7/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

N:= 100: # for a(0)..a(N)
S:= series(x^7/mul(1-x^i,i=7..N-7),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Jul 04 2019

CoefficientList[Series[x^7/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)
Join[{0},Table[Count[IntegerPartitions[n],?(#[[-1]]==7&)],{n,80}]] (* _Harvey P. Dale, Apr 05 2025 *)

my(x='x+O('x^75)); concat([0,0,0,0,0,0,0], Vec(x^7/prod(m=0,80, 1-x^(m+7)))) \\ G. C. Greubel, Nov 03 2019

def A026800_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^7/product((1-x^(m+7)) for m in (0..80)) ).list()
A026800_list(75) # G. C. Greubel, Nov 03 2019

G.f.: x^7 * Product_{m>=7} 1/(1-x^m).
a(n) = p(n-7) -p(n-8) -p(n-9) +p(n-12) +2*p(n-14) -p(n-16) -p(n-17) -p(n-18) -p(n-19) +2*p(n-21) +p(n-23) -p(n-26) -p(n-27) +p(n-28) where p(n)=A000041(n) including the implicit p(n)=0 for negative n. - Shanzhen Gao, Oct 28 2010; offset corrected / made explicit by Jason Kimberley, Feb 03 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(7*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A026801 Number of partitions of n in which the least part is 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 21, 23, 27, 30, 36, 39, 46, 51, 60, 66, 77, 85, 99, 110, 126, 140, 162, 179, 205, 228, 260, 289, 329, 365, 415, 461, 521, 579, 655, 726, 818, 909, 1022, 1134, 1273, 1411

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), this sequence (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0,0] cat Coefficients(R!( x^8/(&*[1-x^(m+8): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^8/mul(1-x^(m+8), m = 0..80), x, n+1), x, n), n = 1..70); # G. C. Greubel, Nov 03 2019

Rest@CoefficientList[Series[x^8/QPochhammer[x^8, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); concat(vector(7), Vec(x^8/prod(m=0,80, 1-x^(m+8)))) \\ G. C. Greubel, Nov 03 2019

def A026801_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^8/product((1-x^(m+8)) for m in (0..80)) ).list()
a=A026801_list(71); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^8 * Product_{m>=8} 1/(1-x^m).
a(n+8) = p(n) -p(n-1) -p(n-2) +p(n-5) +p(n-7) +p(n-8) -p(n-10) -p(n-11) -2*p(n-12) +2*p(n-16) +p(n-17) +p(n-18) -p(n-20) -p(n-21) -p(n-23) +p(n-26) +p(n-27) -p(n-28) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*Pi^7 / (18*sqrt(2)*n^(9/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(8*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A026802 Number of partitions of n in which the least part is 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 24, 26, 30, 34, 39, 43, 50, 55, 63, 71, 80, 89, 102, 113, 128, 143, 161, 179, 203, 225, 253, 282, 316, 351, 395, 437, 489, 544, 607, 673, 752, 832, 927, 1028, 1143

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), this sequence (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 80); [0,0,0,0,0,0,0,0] cat Coefficients(R!( x^9/(&*[1-x^(m+9): m in [0..85]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^9/mul(1-x^(m+9), m = 0..85), x, n+1), x, n), n = 1..80); # G. C. Greubel, Nov 03 2019

Table[Count[IntegerPartitions[n],?(Min[#]==9&)],{n,80}] (* _Harvey P. Dale, May 09 2013 *)
Rest@CoefficientList[Series[x^9/QPochhammer[x^9, x], {x,0,80}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); concat(vector(8), Vec(x^9/prod(m=0,85, 1-x^(m+9)))) \\ G. C. Greubel, Nov 03 2019

def A026802_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^9/product((1-x^(m+9)) for m in (0..85)) ).list()
a=A026802_list(81); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^9 * Product_{m>=9} 1/(1-x^m).
a(n+9) = p(n) -p(n-1) -p(n-2) +p(n-5) +p(n-7) +p(n-9) -p(n-11) -2*p(n-12) -p(n-13) -p(n-15) +p(n-16) +p(n-17) +2*p(n-18) +p(n-19) +p(n-20) -p(n-21) -p(n-23) -2*p(n-24) -p(n-25) +p(n-27) +p(n-29) +p(n-31) -p(n-34) -p(n-35) +p(n-36) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(9*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A026803 Number of partitions of n in which the least part is 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 27, 29, 34, 37, 43, 47, 54, 59, 68, 74, 85, 93, 106, 116, 132, 145, 164, 180, 203, 223, 252, 276, 310, 341, 382, 420, 470, 516, 576, 633, 706, 775, 863

Offset: 1

Clark Kimberling

In general, if g>=1 and g.f. = x^g * Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, Jun 02 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 80); [0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^10/(&*[1-x^(m+10): m in [0..85]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^10/mul(1-x^(m+10), m = 0..85), x, n+1), x, n), n = 1..80); # G. C. Greubel, Nov 03 2019

Rest@CoefficientList[Series[x^10/QPochhammer[x^10, x], {x,0,80}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^80)); concat(vector(9), Vec(x^10/prod(m=0,85, 1-x^(m+10)))) \\ G. C. Greubel, Nov 03 2019

def A026803_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^10/product((1-x^(m+10)) for m in (0..85)) ).list()
a=A026803_list(71); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^10 * Product_{m>=10} 1/(1-x^m).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*sqrt(2)*Pi^9 / (3*n^(11/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(10*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A238341 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the largest part, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 6, 1, 0, 1, 0, 12, 3, 0, 0, 1, 0, 23, 7, 1, 0, 0, 1, 0, 46, 13, 4, 0, 0, 0, 1, 0, 91, 25, 10, 1, 0, 0, 0, 1, 0, 183, 46, 21, 5, 0, 0, 0, 0, 1, 0, 367, 89, 39, 15, 1, 0, 0, 0, 0, 1, 0, 737, 175, 70, 35, 6, 0, 0, 0, 0, 0, 1, 0, 1478, 351, 125, 71, 21, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Columns k=0-10 give: A000007, A097979(n-1) for n>0, A243737, A243738, A243739, A243740, A243741, A243742, A243743, A243744, A243745.
T(n^2,n) gives A243746(n).
Row sums are A011782.

			Triangle starts:
00:  1;
01:  0,     1;
02:  0,     1,    1;
03:  0,     3,    0,   1;
04:  0,     6,    1,   0,   1;
05:  0,    12,    3,   0,   0,   1;
06:  0,    23,    7,   1,   0,   0,  1;
07:  0,    46,   13,   4,   0,   0,  0, 1;
08:  0,    91,   25,  10,   1,   0,  0, 0, 1;
09:  0,   183,   46,  21,   5,   0,  0, 0, 0, 1;
10:  0,   367,   89,  39,  15,   1,  0, 0, 0, 0, 1;
11:  0,   737,  175,  70,  35,   6,  0, 0, 0, 0, 0, 1;
12:  0,  1478,  351, 125,  71,  21,  1, 0, 0, 0, 0, 0, 1;
13:  0,  2962,  710, 229, 131,  56,  7, 0, 0, 0, 0, 0, 0, 1;
14:  0,  5928, 1443, 435, 230, 126, 28, 1, 0, 0, 0, 0, 0, 0, 1,
15:  0, 11858, 2926, 859, 395, 253, 84, 8, 0, 0, 0, 0, 0, 0, 0, 1;
...

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Cf. A026794 (the same for partitions), A238342 (the same for smallest part).

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_, k_] := Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]; a[0, 0] = 1; a[, 0] = 0; Table[a[n, k], {n, 0, 15}, {k, 0, n}] // Flatten (* _Jean-François Alcover, Jan 19 2015, after Maple code in A243737 *)

A342050 Numbers k which have an odd number of trailing zeros in their primorial base representation A049345(k).

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 30, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 60, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 90, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 120, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 150, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 180, 182, 184, 188, 190, 194, 196, 200, 202, 206, 208, 212

Offset: 1

Amiram Eldar, Feb 26 2021

Numbers k such that A276084(k) is odd.
All the terms are even since odd numbers have 0 trailing zeros, and 0 is not odd.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * Sum_{k=1..m}(-1)^k/A002110(k) = 1, 2, 11, 76, 837, 10880, 184961, ...
The asymptotic density of this sequence is Sum_{k>=1} (-1)^(k+1)/A002110(k) = 0.362306... (A132120).
Also Heinz numbers of partitions with even least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021
Numbers k such that A000720(A053669(k)) is even. Differences from the related A353531 seem to be terms that are multiples of 210, but not all of them, for example primorial 30030 (= 143*210) is in neither sequence. Consider also A038698. - Antti Karttunen, Apr 25 2022

			2 is a term since A049345(2) = 10 has 1 trailing zero.
4 is a term since A049345(2) = 20 has 1 trailing zero.
30 is a term since A049345(2) = 1000 has 3 trailing zeros.
From _Gus Wiseman_, Apr 23 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}             46: {1,9}             90: {1,2,2,3}
      4: {1,1}           50: {1,3,3}           92: {1,1,9}
      8: {1,1,1}         52: {1,1,6}           94: {1,15}
     10: {1,3}           56: {1,1,1,4}         98: {1,4,4}
     14: {1,4}           58: {1,10}           100: {1,1,3,3}
     16: {1,1,1,1}       60: {1,1,2,3}        104: {1,1,1,6}
     20: {1,1,3}         62: {1,11}           106: {1,16}
     22: {1,5}           64: {1,1,1,1,1,1}    110: {1,3,5}
     26: {1,6}           68: {1,1,7}          112: {1,1,1,1,4}
     28: {1,1,4}         70: {1,3,4}          116: {1,1,10}
     30: {1,2,3}         74: {1,12}           118: {1,17}
     32: {1,1,1,1,1}     76: {1,1,8}          120: {1,1,1,2,3}
     34: {1,7}           80: {1,1,1,1,3}      122: {1,18}
     38: {1,8}           82: {1,13}           124: {1,1,11}
     40: {1,1,1,3}       86: {1,14}           128: {1,1,1,1,1,1,1}
     44: {1,1,5}         88: {1,1,1,5}        130: {1,3,6}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A002110, A049345, A053669, A132120, A276084.
Complement of A342051.
A099800 is subsequence.
Analogous sequences: A001950 (Zeckendorf representation), A036554 (binary), A145204 (ternary), A217319 (base 4), A232745 (factorial base).
The version for reversed binary expansion is A079523.
Positions of even terms in A257993.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339662 gives greatest gap in prime indices.
Cf. A000720, A001223, A005408, A026794, A029707, A038698, A047235, A079068, A121539, A286469, A286470, A325351, A353525 (characteristic function).
Differs from A353531 for the first time at n=77, where a(77) = 212, as this sequence misses A353531(77) = 210.

seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100],EvenQ[Min@@Complement[Range[PrimeNu[#]+1],PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)

A353525(n) = { for(i=1,oo,if(n%prime(i),return((i+1)%2))); }
isA342050(n) = A353525(n);
k=0; n=0; while(k<77, n++; if(isA342050(n), k++; print1(n,", "))); \\ Antti Karttunen, Apr 25 2022

More terms added (to differentiate from A353531) by Antti Karttunen, Apr 25 2022

A342051 Numbers k which have an even number of trailing zeros in their primorial base representation A049345(k).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 102, 103

Offset: 1

Amiram Eldar, Feb 26 2021

Numbers k such that A276084(k) is even.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * (1 - Sum_{k=1..m}(-1)^k/A002110(k)) = 1, 4, 19, 134, 1473, 19150, 325549 ...
The asymptotic density of this sequence is Sum_{k>=0} (-1)^k/A002110(k) = 0.637693... = 1 - A132120.
Also Heinz numbers of partitions with odd least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021

			1 is a term since A049345(1) = 1 has 0 trailing zero.
6 is a term since A049345(6) = 100 has 2 trailing zeros.
From _Gus Wiseman_, Apr 23 2021: (Start)
The sequence of terms together with their prime indices begins:
     1: {}           25: {3,3}          51: {2,7}
     3: {2}          27: {2,2,2}        53: {16}
     5: {3}          29: {10}           54: {1,2,2,2}
     6: {1,2}        31: {11}           55: {3,5}
     7: {4}          33: {2,5}          57: {2,8}
     9: {2,2}        35: {3,4}          59: {17}
    11: {5}          36: {1,1,2,2}      61: {18}
    12: {1,1,2}      37: {12}           63: {2,2,4}
    13: {6}          39: {2,6}          65: {3,6}
    15: {2,3}        41: {13}           66: {1,2,5}
    17: {7}          42: {1,2,4}        67: {19}
    18: {1,2,2}      43: {14}           69: {2,9}
    19: {8}          45: {2,2,3}        71: {20}
    21: {2,4}        47: {15}           72: {1,1,1,2,2}
    23: {9}          48: {1,1,1,1,2}    73: {21}
    24: {1,1,1,2}    49: {4,4}          75: {2,3,3}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Cf. A002110, A049345, A132120, A276084.
Complement of A342050.
A099788 is subsequence.
Analogous sequences: A000201 (Zeckendorf representation), A003159 (binary), A007417 (ternary), A232744 (factorial base).
The version for reversed binary expansion is A121539.
Positions of odd terms in A257993.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A339662 gives greatest gap in prime indices.
Cf. A001223, A005408, A026794, A029707, A047235, A079068, A079523, A286469, A286470, A325351.

seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], EvenQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100],OddQ[Min@@Complement[Range[PrimeNu[#]+1],PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)

A238123 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k largest parts, n >= k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 7, 2, 0, 1, 0, 20, 5, 0, 0, 1, 0, 56, 14, 5, 0, 0, 1, 0, 182, 35, 14, 0, 0, 0, 1, 0, 589, 132, 28, 14, 0, 0, 0, 1, 0, 2088, 399, 90, 42, 0, 0, 0, 0, 1, 0, 7522, 1556, 285, 90, 42, 0, 0, 0, 0, 1, 0, 28820, 5346, 1232, 165, 132, 0, 0, 0, 0, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Also number of standard Young tableaux with last row of length k.

			Triangle starts:
00: 1;
01: 0,      1;
02: 0,      1,     1;
03, 0,      3,     0,     1;
04: 0,      7,     2,     0,    1;
05: 0,     20,     5,     0,    0,   1;
06: 0,     56,    14,     5,    0,   0,   1;
07: 0,    182,    35,    14,    0,   0,   0, 1;
08: 0,    589,   132,    28,   14,   0,   0, 0, 1;
09: 0,   2088,   399,    90,   42,   0,   0, 0, 0, 1;
10: 0,   7522,  1556,   285,   90,  42,   0, 0, 0, 0, 1;
11: 0,  28820,  5346,  1232,  165, 132,   0, 0, 0, 0, 0, 1;
12: 0, 113092, 21515,  4378,  737, 297, 132, 0, 0, 0, 0, 0, 1;
13: 0, 464477, 82940, 17082, 3003, 572, 429, 0, 0, 0, 0, 0, 0, 1;
...
The T(6,2)=14 ballot sequences of length 6 with 2 maximal elements are (dots for zeros):
01:  [ . . . . 1 1 ]
02:  [ . . . 1 . 1 ]
03:  [ . . . 1 1 . ]
04:  [ . . 1 . . 1 ]
05:  [ . . 1 . 1 . ]
06:  [ . . 1 1 . . ]
07:  [ . . 1 1 2 2 ]
08:  [ . . 1 2 1 2 ]
09:  [ . 1 . . . 1 ]
10:  [ . 1 . . 1 . ]
11:  [ . 1 . 1 . . ]
12:  [ . 1 . 1 2 2 ]
13:  [ . 1 . 2 1 2 ]
14:  [ . 1 2 . 1 2 ]
The T(8,4)=14 such ballot sequences of length 8 and 4 maximal elements are:
01:  [ . . . . 1 1 1 1 ]
02:  [ . . . 1 . 1 1 1 ]
03:  [ . . . 1 1 . 1 1 ]
04:  [ . . . 1 1 1 . 1 ]
05:  [ . . 1 . . 1 1 1 ]
06:  [ . . 1 . 1 . 1 1 ]
07:  [ . . 1 . 1 1 . 1 ]
08:  [ . . 1 1 . . 1 1 ]
09:  [ . . 1 1 . 1 . 1 ]
10:  [ . 1 . . . 1 1 1 ]
11:  [ . 1 . . 1 . 1 1 ]
12:  [ . 1 . . 1 1 . 1 ]
13:  [ . 1 . 1 . . 1 1 ]
14:  [ . 1 . 1 . 1 . 1 ]
These are the (reversed) Dyck words of semi-length 4.

Joerg Arndt and Alois P. Heinz, Rows n = 0..60, flattened
Wikipedia, Young tableau

The terms T(2*n,n) are the Catalan numbers (A000108).
Columns k=0-10 give: A000007, A238124, A244099, A244100, A244101, A244102, A244103, A244104, A244105, A244106, A244107.
Row sums give A000085.
Cf. A026794.

b:= proc(n, l) option remember; `if`(n<1, x^l[-1],
      b(n-1, [l[], 1]) +add(`if`(i=1 or l[i-1]>l[i],
      b(n-1, subsop(i=l[i]+1, l)), 0), i=1..nops(l)))
    end:
T:= n->`if`(n=0, 1, (p->seq(coeff(p, x, i), i=0..n))(b(n-1, [1]))):
seq(T(n), n=0..12);
# second Maple program (counting SYT):
h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, h([l[], 1$n])*x^`if`(n>0, 1,
       `if`(l=[], 0, l[-1])), g(n, i-1, l)+
       `if`(i>n, 0, g(n-i, i, [l[], i])))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n, n, [])):
seq(T(n), n=0..12);

b[n_, l_List] :=  b[n, l] = If[n<1, x^l[[-1]], b[n-1, Append[l, 1]] +  Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; T[n_] := If[n == 0, 1, Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, {1}]]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)

(A238123(n,k)=if(k, vecsum(apply(p->n!/Hook(Vecrev(p)), select(p->p[1]==k,partitions(n,[k,n])))), !n)); Hook(P,h=vector(P[1]),L=P[#P])={prod(i=1, L, h[i]=L-i+1)*prod(i=1,#P-1, my(D=-L+L=P[#P-i]); prod(k=0,L-1,h[L-k]+=min(k,D)+1))} \\  M. F. Hasler, Jun 03 2018

A322439 Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 33, 42, 82, 114, 195, 258, 466, 587, 954, 1317, 2021, 2637, 4124, 5298, 7995, 10565, 15075, 19665, 28798, 36773, 51509, 67501, 93060, 119299, 165589, 209967, 285535, 366488, 487536, 622509, 833998, 1048119, 1380410, 1754520, 2291406, 2876454

Offset: 0

Gus Wiseman, Dec 08 2018

nonn

			The a(5) = 15 pairs of integer partitions:
      (5)|(5)
     (41)|(5)
     (32)|(5)
    (311)|(5)
    (221)|(5)
    (221)|(32)
   (2111)|(5)
   (2111)|(32)
  (11111)|(5)
  (11111)|(41)
  (11111)|(32)
  (11111)|(311)
  (11111)|(221)
  (11111)|(2111)
  (11111)|(11111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A026794, A026820, A265947, A285573, A317144, A318915, A322435, A322436, A322440, A322441, A322442, A362051.

g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-1) +g(n-i, min(i, n-i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(g(n, i)*b(n-i, i), i=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 09 2018

Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]<=Min@@Last[#]&]],{n,20}]
(* Second program: *)
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, g[n, i - 1] + g[n - i, Min[i, n - i]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n == 0, 1, Sum[g[n, i]*b[n - i, i], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

a(n) = Sum_{k = 1..n} A026820(n,k) * A026794(n,k).

a(n) = A000041(2n) - A362051(n) for n>=1. - Alois P. Heinz, Apr 27 2023

cp's OEIS Frontend

A185328 Number of partitions of n with parts >= 8.

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A026800 Number of partitions of n in which the least part is 7.

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A026801 Number of partitions of n in which the least part is 8.

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A026802 Number of partitions of n in which the least part is 9.

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A026803 Number of partitions of n in which the least part is 10.

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A238341 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the largest part, n>=0, 0<=k<=n.

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