A015723 -id:A015723 - cp's OEIS frontend

A325506 Product of Heinz numbers over all strict integer partitions of n.

1, 2, 3, 30, 70, 2310, 180180, 21441420, 6401795400, 200984366583000, 41615822944675980000, 10515527757483671302380000, 4919824049783476260137727416400000, 5158181210492841550866520676965246284000000, 29776760895364738730693151196801613158042403043600000000

Offset: 0

Gus Wiseman, May 07 2019

nonn

a(n) is the product of row n of A246867 (squarefree numbers arranged by sum of prime indices).

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with Heinz numbers {13,22,21,30}, with product 13*22*21*30 = 180180, so a(6) = 180180.
The sequence of terms together with their prime indices begins:
                     1: {}
                     2: {1}
                     3: {2}
                    30: {1,2,3}
                    70: {1,3,4}
                  2310: {1,2,3,4,5}
                180180: {1,1,2,2,3,4,5,6}
              21441420: {1,1,2,2,3,4,4,5,6,7}
            6401795400: {1,1,1,2,2,3,3,4,5,5,6,7,8}
       200984366583000: {1,1,1,2,2,2,3,3,3,4,4,5,5,6,6,7,8,9}
  41615822944675980000: {1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,6,6,7,7,8,9,10}

Cf. A003963, A006128, A015723, A022629, A056239, A112798, A147655, A215366, A246867, A325501, A325504, A325505, A325512, A325513.

Table[Times@@Prime/@(Join@@Select[IntegerPartitions[n],UnsameQ@@#&]),{n,0,15}]

a(n) = Product_{i = 1..A000009(n)} A246867(n,i).
A001222(a(n)) = A015723(n).
A056239(a(n)) = A066189(n).
A003963(a(n)) = A325504(n).
a(n) = A003963(A325505(n)).

A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 8, 8, 32, 144, 432, 2160, 27000, 582120, 7623000, 336936600, 6740402760, 543454231320, 57619849046760, 4683793138766280, 412882704970215480, 88171665744392750520, 12780536107937124847320, 2685589660883755945879560, 942036670625665177379096280

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also the Heinz number of row n of A015716 (with zeros removed).

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), so a(6) = prime(1)^4*prime(2)^2 = 144.
The sequence of terms together with their prime indices begins:
               1: {}
               2: {1}
               2: {1}
               8: {1,1,1}
               8: {1,1,1}
              32: {1,1,1,1,1}
             144: {1,1,1,1,2,2}
             432: {1,1,1,1,2,2,2}
            2160: {1,1,1,1,2,2,2,3}
           27000: {1,1,1,2,2,2,3,3,3}
          582120: {1,1,1,2,2,2,3,4,4,5}
         7623000: {1,1,1,2,2,3,3,3,4,5,5}
       336936600: {1,1,1,2,2,3,3,4,5,5,6,7}
      6740402760: {1,1,1,2,2,3,4,4,4,6,6,7,8}
    543454231320: {1,1,1,2,2,3,4,4,5,6,7,8,9,10}
  57619849046760: {1,1,1,2,2,3,4,5,5,6,8,9,10,11,12}

Alois P. Heinz, Table of n, a(n) for n = 0..172

Cf. A000009, A006128, A015716, A015723, A022629, A056239, A066633, A112798, A246867, A325500, A325504, A325506.

b:= proc(n, i) option remember;
      `if`(n>(i*(i+1)/2), 0, `if`(n=0, [1, 0], b(n, i-1)+
          (p-> p+[0, p[1]*x^i])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> (p-> mul((c-> `if`(c=0, 1, ithprime(c)))(
    coeff(p, x, i)), i=1..degree(p)))(b(n$2)[2]):
seq(a(n), n=0..21);  # Alois P. Heinz, Feb 23 2024

Table[Times@@Prime/@Length/@Split[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]]],{n,0,15}]

a(n) = A181819(A003963(A325505(n))).

A056239(a(n)) = A015723(n).

A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 5, 1, 6, 4, 6, 7, 15, 6, 16, 15, 20, 17, 36, 18, 43, 36, 46, 48, 72, 45, 93, 82, 103, 88, 152, 104, 179, 158, 191, 194, 285, 202, 328, 292, 373, 348, 502, 391, 576, 519, 659, 634, 864, 665

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of integer partitions of n with no part dividing all the others, but with a part divisible by all the others.

			The a(18) = 1  through a(23) = 15 partitions (A..E = 10..14):
  633222   C43       C332      C432       C64        E72
           A522      66332     A5222      A552       F53
           C322      633332    C3222      C433       I32
           66322     6332222   663222     C3322      C443
           633322              6333222    663322     C632
           6322222             63222222   6333322    66632
                                          63322222   C3332
                                                     C4322
                                                     663332
                                                     A52222
                                                     C32222
                                                     6333332
                                                     6632222
                                                     63332222
                                                     632222222

The second condition alone gives A130689.
The half-opposite versions are A130714 and A343342.
The first condition alone gives A338470.
The Heinz numbers of these partitions are 1 and A343339.
The opposite version is A343345.
The strict case is A343380.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
Cf. A083710, A097986, A264401, A339562, A341450, A342193, A343346.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A066637 Total number of elements in all factorizations of n with all factors > 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 8, 1, 3, 3, 12, 1, 8, 1, 8, 3, 3, 1, 17, 3, 3, 6, 8, 1, 10, 1, 20, 3, 3, 3, 22, 1, 3, 3, 17, 1, 10, 1, 8, 8, 3, 1, 34, 3, 8, 3, 8, 1, 17, 3, 17, 3, 3, 1, 27, 1, 3, 8, 35, 3, 10, 1, 8, 3, 10, 1, 46, 1, 3, 8, 8, 3, 10, 1, 34, 12, 3, 1, 27, 3, 3, 3, 17, 1, 27, 3, 8, 3, 3, 3

Offset: 1

Amarnath Murthy, Dec 28 2001

nonn

From Gus Wiseman, Apr 18 2021: (Start)
Number of ways to choose a factor index or position in a factorization of n. The version selecting a factor value is A339564. For example, the factorizations of n = 2, 4, 8, 12, 16, 24, 30 with a selected position (in parentheses) are:
  ((2))  ((4))    ((8))      ((12))     ((16))       ((24))       ((30))
         ((2)*2)  ((2)*4)    ((2)*6)    ((2)*8)      ((3)*8)      ((5)*6)
         (2*(2))  (2*(4))    (2*(6))    (2*(8))      (3*(8))      (5*(6))
                  ((2)*2*2)  ((3)*4)    ((4)*4)      ((4)*6)      ((2)*15)
                  (2*(2)*2)  (3*(4))    (4*(4))      (4*(6))      (2*(15))
                  (2*2*(2))  ((2)*2*3)  ((2)*2*4)    ((2)*12)     ((3)*10)
                             (2*(2)*3)  (2*(2)*4)    (2*(12))     (3*(10))
                             (2*2*(3))  (2*2*(4))    ((2)*2*6)    ((2)*3*5)
                                        ((2)*2*2*2)  (2*(2)*6)    (2*(3)*5)
                                        (2*(2)*2*2)  (2*2*(6))    (2*3*(5))
                                        (2*2*(2)*2)  ((2)*3*4)
                                        (2*2*2*(2))  (2*(3)*4)
                                                     (2*3*(4))
                                                     ((2)*2*2*3)
                                                     (2*(2)*2*3)
                                                     (2*2*(2)*3)
                                                     (2*2*2*(3))
(End)

			a(12) = 8: there are 4 factorizations of 12: (12), (6*2), (4*3), (3*2*2) having 1, 2, 2, 3 elements respectively, a total of 8.

Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy, Length and extent of Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

The version for normal multisets is A001787.
The version for compositions is A001792.
The version for partitions is A006128 (strict: A015723).
Choosing a value instead of position gives A339564.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A002033 and A074206 count ordered factorizations.
A067824 counts strict chains of divisors starting with n.
A336875 counts compositions with a selected part.
Cf. A000005, A000041, A045778, A050336, A066186, A162247, A264401, A281116, A292504, A292886, A322794.

# Return a list of lists which are factorizations (product representations)
# of n. Within each sublist, the factors are sorted. A minimum factor in
# each element of sublists returned can be specified with 'mincomp'.
# If mincomp=2, the number of sublists contained in the list returned is A001055(n).
# Example:
# n=8 and mincomp=2 return [[2,2,2],[4,8],[8]]
listProdRep := proc(n,mincomp)
    local dvs,resul,f,i,j,rli,tmp ;
    resul := [] ;
    # list returned is empty if n < mincomp
    if n >= mincomp then
        if n = 1 then
            RETURN([1]) ;
        else
            # compute the divisors, and take each divisor
            # as a head element (minimum element) of one of the
            # sublists. Example: for n=8 use {1,2,4,8}, and consider
            # (for mincomp=2) sublists [2,...], [4,...] and [8].
            dvs := numtheory[divisors](n) ;
            for i from 1 to nops(dvs) do
                # select the head element 'f' from the divisors
                f := op(i,dvs) ;
                # if this is already the maximum divisor n
                # itself, this head element is the last in
                # the sublist
                if f =n and f >= mincomp then
                    resul := [op(resul),[f]] ;
                elif f >= mincomp then
                    # if this is not the maximum element
                    # n itself, produce all factorizations
                    # of the remaining factor recursively.
                    rli := procname(n/f,f) ;
                    # Prepend all the results produced
                    # from the recursion with the head
                    # element for the result.
                    for j from 1 to nops(rli) do
                        tmp := [f,op(op(j,rli))] ;
                        resul := [op(resul),tmp] ;
                    od ;
                fi ;
            od ;
        fi ;
    fi ;
    resul ;
end:
A066637 := proc(n)
    local f,d;
    a := 0 ;
    for d in listProdRep(n,2) do
        a := a+nops(d) ;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 11 2013
# second Maple program:
with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, [1$2])+
      `if`(isprime(n), 0, (p-> p+[0, p[1]])(add(
      `if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n})))
    end:
a:= n-> `if`(n<2, 0, b(n$2)[2]):
seq(a(n), n=1..120); # Alois P. Heinz, Feb 12 2019

g[1, r_] := g[1, r]={1, 0}; g[n_, r_] := g[n, r]=Module[{ds, i, val}, ds=Select[Divisors[n], 1<#<=r&]; val={0, 0}+Sum[g[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]; val+{0, val[[1]]}]; a[n_] := g[n, n][[2]]; a/@Range[95] (* g[n, r] = {c, f}, where c is the number of factorizations of n with factors <= r and f is the total number of factors in them. - Dean Hickerson, Oct 28 2002 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];Table[Sum[Length[fac],{fac,facs[n]}],{n,50}] (* Gus Wiseman, Apr 18 2021 *)

A097910 Number of parts in all compositions of n into distinct parts.

Original entry on oeis.org

1, 1, 5, 5, 9, 27, 31, 49, 71, 185, 207, 339, 457, 685, 1421, 1745, 2577, 3615, 5143, 6877, 13439, 15965, 23823, 31983, 45553, 59425, 83549, 139013, 173769, 244803, 330391, 452257, 597935, 810929, 1052559, 1692723, 2074321, 2890333, 3783821, 5178041, 6658377

Offset: 1

Vladeta Jovovic, Sep 04 2004

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Cf. A001792, A008289, A015723, A032020, A072574, A336875.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i*(i+1)/2, [][], zip((x, y)->x+y, [b(n, i-1)],
      `if`(i>n, [], [0, b(n-i, i-1)]), 0)[]))
    end:
a:= n-> (l-> add(i*l[i+1]*i!, i=1..nops(l)-1))([b(n$2)]):
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2012
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 10 2020

Drop[ CoefficientList[ Series[ Sum[ k*k!*x^((k^2 + k)/2)/Product[1 - x^j, {j, 1, k}], {k, 1, 45}], {x, 0, 40}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

G.f.: Sum(k >= 0; k*k! x^((k^2+k)/2) / Prod(1<=j<=k; 1-x^j)).

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} k! * k * A008289(n,k). - Alois P. Heinz, Aug 10 2020

More terms from Robert G. Wilson v and John W. Layman, Sep 08 2004

A116676 Number of odd parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 4, 5, 8, 10, 14, 16, 22, 26, 34, 43, 54, 64, 80, 96, 116, 142, 170, 202, 242, 288, 340, 404, 474, 556, 652, 762, 886, 1034, 1198, 1389, 1606, 1852, 2132, 2454, 2814, 3224, 3690, 4214, 4804, 5478, 6228, 7072, 8028, 9094, 10290, 11635, 13134

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

a(n) = Sum(k*A116675(n,k), k>=0).

			a(9) = 10 because in the partitions of 9 into distinct parts, namely, [9], [81], [72], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2], we have a total of 10 odd parts.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A116675, A305123, A305124.

Cf. A305082, A015723, A090867, A116680, A067588.

f:=product(1+x^j,j=1..64)*sum(x^(2*j-1)/(1+x^(2*j-1)),j=1..35): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..56);
# second Maple program:
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0] elif i<1 then [0, 0]
    else f:=b(n, i-1); g:=`if`(i>n, [0, 0], b(n-i, min(n-i, i-1)));
         [f[1]+g[1], f[2]+g[2] +irem(i, 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which [n == 0, {1, 0}, i<1 , {0, 0}, True, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, Min[n-i, i-1]]]; {f[[1]] + g[[1]],       f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 0, 60}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

G.f.: product(1+x^j, j=1..infinity)*sum(x^(2j-1)/(1+x^(2j-1)), j=1..infinity).
For n > 0, a(n) = A015723(n) - A116680(n). - Vaclav Kotesovec, May 26 2018
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018

A116680 Number of even parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 4, 5, 5, 8, 11, 14, 18, 23, 29, 37, 44, 55, 69, 83, 102, 124, 148, 178, 213, 253, 300, 356, 421, 494, 582, 680, 793, 926, 1074, 1246, 1446, 1668, 1922, 2215, 2545, 2918, 3345, 3823, 4366, 4982, 5668, 6445, 7321, 8300, 9401, 10639, 12021, 13566

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

			a(9)=8 because in the partitions of 9 into distinct parts, namely, [9], [8,1], [7,2], [6,3], [6,2,1], [5,4], [5,3,1], and [4,3,2], we have a total of 8 even parts. [edited by _Rishi Advani_, Jun 07 2019]

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
D. Herden, M. R. Sepanski, J. Stanfill, C. C. Hammon, J. Henningsen, H. Ickes, J. M. Menendez, T. Poe, I. Ruiz, and E. L. Smith, Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k, arXiv:2010.02788 [math.CO], 2020. See also Integers (2022) Vol. 22, #A49.
Runqiao Li and Andrew Y. Z. Wang, On the combinatorics of the number of even parts in all partitions with distinct parts, The Raman. J. 56 (2021) 712-727

Cf. A116679, A305121, A305122.

Cf. A305082, A015723, A090867, A067588, A116676.

m:=25; R:=PowerSeriesRing(Integers(), 3*m); [0,0] cat Coefficients(R!( (&*[1+x^j: j in [1..4*m]])*(&+[x^(2*k)/(1+x^(2*k)): k in [1..2*m]]) )); // G. C. Greubel, Jun 07 2019

f:=product(1+x^j,j=1..70)*sum(x^(2*j)/(1+x^(2*j)),j=1..40): fser:=series(f,x=0,65): seq(coeff(fser,x,n),n=0..60);
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 p+`if`(i::odd, 0, [0, p[1]]))(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, May 24 2022

With[{m = 25}, CoefficientList[Series[Product[1+x^j, {j,1,4*m}]* Sum[x^(2*k)/(1+x^(2*k)), {k,1,2*m}], {x,0,3*m}], x]] (* G. C. Greubel, Jun 07 2019 *)

my(m=25); my(x='x+O('x^(3*m))); concat([0, 0], Vec( prod(j=1, 4*m, 1+x^j)*sum(k=1, 2*m, x^(2*k)/(1+x^(2*k))) )) \\ G. C. Greubel, Jun 07 2019

m = 25
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(3*m)
s = product(1+x^j for j in (1..4*m))*sum(x^(2*k)/(1+x^(2*k)) for k in (1..2*m))
[0, 0] + s.coefficients() # G. C. Greubel, Jun 07 2019

a(n) = Sum_{k >= 0} k*A116679(n,k).
G.f.: (Product_{j >= 1} (1+x^j)) * (Sum_{k >= 1} x^(2*k)/(1+x^(2*k))).
For n > 0, a(n) = A015723(n) - A116676(n). - Vaclav Kotesovec, May 26 2018
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018

A305102 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 4, 10, 23, 46, 88, 158, 274, 459, 748, 1190, 1858, 2846, 4292, 6384, 9373, 13602, 19536, 27782, 39158, 54740, 75928, 104562, 143036, 194423, 262704, 352988, 471778, 627382, 830352, 1093994, 1435132, 1874920, 2439832, 3163020, 4085825, 5259602, 6748136

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A006128 and A000009.
Convolution of A305082 and A000041.
Convolution of A000005 and A015128.
a(n) is the number of non-overlined parts in all overpartitions of n. - Joerg Arndt, Jun 18 2020

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

Cf. A006128, A015723, A209423, A305082, A305101.

Cf. A335651 and A335666.

nmax = 40; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, 1*q^k/(1-q^k)) ) \\ Joerg Arndt, Jun 18 2020

a(n) ~ exp(Pi*sqrt(n)) * (2*gamma + log(4*n/Pi^2)) / (8*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A339619 Number of integer partitions of n with no 1's and a part divisible by all the other parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 5, 3, 7, 2, 13, 2, 13, 9, 17, 6, 27, 7, 33, 19, 35, 16, 58, 22, 58, 39, 75, 37, 108, 44, 117, 75, 132, 88, 190, 94, 199, 147, 250, 153, 322, 180, 363, 271, 405, 286, 544, 339, 601, 458, 699, 503, 868, 608, 990, 777, 1113, 865, 1422, 993

Offset: 0

Gus Wiseman, Apr 18 2021

nonn

Alternative name: Number of integer partitions of n with no 1's that are empty or have greatest part divisible by all the other parts.

			The a(6) = 4 through a(16) = 17 partitions (A..G = 10..16):
  6    7  8     9    A      B    C       D     E        F      G
  33      44    63   55     632  66      6322  77       A5     88
  42      62    333  82          84            C2       C3     C4
  222     422        442         93            662      555    E2
          2222       622         A2            842      663    844
                     4222        444           A22      933    C22
                     22222       633           4442     6333   4444
                                 822           6332     33333  6622
                                 3333          8222     63222  8422
                                 4422          44222           A222
                                 6222          62222           44422
                                 42222         422222          63322
                                 222222        2222222         82222
                                                               442222
                                                               622222
                                                               4222222
                                                               22222222

The dual version is A083711.
The version with 1's allowed is A130689.
The strict case is A339660.
The Heinz numbers of these partitions are the odd complement of A343337.
The strict case with 1's allowed is A343347.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
Cf. A066186, A083710, A083711, A097986, A130714, A338470, A343341, A343342, A343346, A343377, A343382.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Or@@And@@IntegerQ/@(Max@@#/#)&]]],{n,0,30}]

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

cp's OEIS Frontend

A325506 Product of Heinz numbers over all strict integer partitions of n.

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A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

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A343344 Number of integer partitions of n that are either empty, or do not have smallest part dividing all the others, but do have greatest part divisible by all the others.

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A066637 Total number of elements in all factorizations of n with all factors > 1.

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A097910 Number of parts in all compositions of n into distinct parts.

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A116676 Number of odd parts in all partitions of n into distinct parts.

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A116680 Number of even parts in all partitions of n into distinct parts.

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