A264401 -id:A264401 - cp's OEIS frontend

A083711 a(n) = A083710(n) - A000041(n-1).

1, 1, 1, 2, 1, 4, 1, 5, 3, 7, 1, 14, 1, 13, 8, 20, 1, 33, 1, 40, 14, 44, 1, 85, 6, 79, 25, 117, 1, 181, 1, 196, 45, 233, 17, 389, 1, 387, 80, 545, 1, 750, 1, 839, 165, 1004, 1, 1516, 12, 1612, 234, 2040, 1, 2766, 48, 3142, 388, 3720, 1, 5295, 1, 5606, 663, 7038, 83, 9194, 1, 10379, 1005

Offset: 1

N. J. A. Sloane, Jun 16 2003

Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 18 2021

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(6) = 4 through a(12) = 13 partitions:
  (6)      (7)  (8)        (9)      (10)         (11)  (12)
  (3,3)         (4,4)      (6,3)    (5,5)              (6,6)
  (4,2)         (6,2)      (3,3,3)  (8,2)              (8,4)
  (2,2,2)       (4,2,2)             (4,4,2)            (9,3)
                (2,2,2,2)           (6,2,2)            (10,2)
                                    (4,2,2,2)          (4,4,4)
                                    (2,2,2,2,2)        (6,3,3)
                                                       (6,4,2)
                                                       (8,2,2)
                                                       (3,3,3,3)
                                                       (4,4,2,2)
                                                       (6,2,2,2)
                                                       (4,2,2,2,2)
                                                       (2,2,2,2,2,2)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Allowing 1's gives A083710.
The strict case is A098965.
The complement (except also without 1's) is counted by A338470.
The dual version is A339619.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A001787, A001792, A015723, A097986, A098743, A130689, A130714, A264401, A339563, A342193.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d,`,a(j)) od: # James Sellers, Jun 21 2003
# second Maple program:
a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
seq(a(n), n=1..69);  # Alois P. Heinz, Feb 15 2023

a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 15 2023 *)

a(n) = Sum_{ d|n, dA000041(d-1).

More terms from James Sellers, Jun 21 2003

A343377 Number of strict integer partitions of n with no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 9, 13, 18, 21, 26, 32, 38, 47, 57, 66, 80, 95, 110, 132, 157, 181, 211, 246, 282, 327, 379, 435, 500, 570, 648, 743, 849, 963, 1094, 1241, 1404, 1592, 1799, 2025, 2282, 2568, 2882, 3239, 3634, 4066, 4554, 5094, 5686, 6346

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part not divisible by all the others.

			The a(5) = 1 through a(12) = 9 partitions:
  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)      (7,5)
                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)      (5,4,3)
                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)      (6,4,2)
                                  (5,3,1)  (5,4,1)    (9,2)      (6,5,1)
                                           (7,2,1)    (5,4,2)    (7,3,2)
                                           (4,3,2,1)  (6,4,1)    (7,4,1)
                                                      (7,3,1)    (8,3,1)
                                                      (5,3,2,1)  (9,2,1)
                                                                 (5,4,2,1)

The dual strict complement is A097986.
The dual version is A341450.
The non-strict version is A343341 (Heinz numbers: A343337).
The strict complement is counted by A343347.
The case with smallest part not divisible by all the others is A343379.
The case with smallest part divisible by all the others is A343381.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A130689, A200745, A264401, A338470, A339562, A343338, A343342, A343345, A343346, A343382.

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

A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 3, 9, 9, 12, 12, 18, 18, 27, 27, 36, 41, 51, 51, 73, 80, 96, 105, 132, 137, 177, 188, 230, 253, 303, 320, 398, 431, 508, 550, 659, 705, 847, 913, 1063, 1165, 1359, 1452, 1716, 1856, 2134, 2329, 2688, 2894, 3345, 3622, 4133

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either empty, or (1) have smallest part not dividing all the others and (2) have greatest part not divisible by all the others.

			The a(5) = 1 through a(13) = 9 partitions (empty column indicated by dot):
  (3,2)  .  (4,3)  (5,3)  (5,4)    (6,4)    (6,5)    (7,5)    (7,6)
            (5,2)         (7,2)    (7,3)    (7,4)    (5,4,3)  (8,5)
                          (4,3,2)  (5,3,2)  (8,3)    (7,3,2)  (9,4)
                                            (9,2)             (10,3)
                                            (5,4,2)           (11,2)
                                                              (6,4,3)
                                                              (6,5,2)
                                                              (7,4,2)
                                                              (8,3,2)

The first condition alone gives A341450.
The non-strict version is A343342  (Heinz numbers: A343338).
The second condition alone gives A343377.
The opposite version is A343378.
The half-opposite versions are A343380 and A343381.
The version for "or" instead of "and" is A343382.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A200745, A264401, A338470, A339562, A342193, A343337, A343341, A343343, A343346, A343347.

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

The Heinz numbers for the non-strict version are A343338 = A342193 /\ A343337.

A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 0, 1, 2, 0, 5, 0, 0, 1, 1, 4, 0, 8, 0, 0, 0, 1, 2, 4, 0, 10, 0, 0, 0, 2, 1, 2, 7, 0, 16, 0, 0, 0, 0, 2, 1, 3, 8, 0, 20, 0, 0, 0, 0, 2, 2, 2, 4, 12, 0, 31, 0, 0, 0, 0, 0, 2, 2, 2, 5, 14, 0

Offset: 0

Gus Wiseman, Sep 30 2023

			The partition (6,2,1,1) has subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 so is counted under T(10,5).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  0  1  2  0
   4  0  0  1  2  0
   5  0  0  1  1  4  0
   8  0  0  0  1  2  4  0
  10  0  0  0  2  1  2  7  0
  16  0  0  0  0  2  1  3  8  0
  20  0  0  0  0  2  2  2  4 12  0
  31  0  0  0  0  0  2  2  2  5 14  0
  39  0  0  0  0  0  4  2  2  3  6 21  0
  55  0  0  0  0  0  0  4  2  4  3  9 24  0
  71  0  0  0  0  0  0  5  4  2  4  5 10 34  0
Row n = 8 counts the following partitions:
  (4211)      .  .  .  (521)   (611)  (71)   (8)     .
  (41111)              (5111)         (431)  (62)
  (3311)                                     (53)
  (3221)                                     (44)
  (32111)                                    (422)
  (311111)                                   (332)
  (22211)                                    (2222)
  (221111)
  (2111111)
  (11111111)

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Row sums are A000041.
Diagonal k = n-1 is A002865.
Column k = 1 is A126796 (complete partitions), ranks A325781.
Central diagonal n = 2k is A126796 also.
For parts instead of sums we have A339737, rank stat A339662, min A257993.
This is the triangle for the rank statistic A365920.
Latter row sums are A365924 (incomplete partitions), ranks A365830.
Column sums are A366127.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
A366128 gives the least non-subset-sum of prime indices.
Cf. A001522, A079068, A098743, A264401, A286469 or A286470, A339886.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Max@@Prepend[nmz[#],0]==k&]],{n,0,10},{k,0,n}]

A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 5, 1, 7, 1, 8, 4, 6, 1, 15, 2, 9, 5, 14, 1, 22, 1, 20, 7, 18, 4, 36, 1, 26, 10, 40, 1, 51, 1, 48, 18, 49, 1, 86, 3, 73, 19, 86, 1, 117, 7, 120, 27, 120, 1, 196, 1, 160, 42, 201, 10, 259, 1, 258, 50, 292, 1, 407, 1, 357, 81, 431, 8, 548, 1, 577

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume this part is the smallest. - Gus Wiseman, Apr 18 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

The non-strict version with 1's allowed is A083710.
The non-strict version is A083711.
The version with 1's allowed is A097986.
The Heinz numbers of these partitions are the odd terms of A339563.
The non-strict dual is A339619.
The strict complement is counted by A341450.
A000005 counts divisors.
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.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A130689, A130714, A200745, A264401, A343345, A343347, A343378, A343381.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 92}], {k, 2, 92}]], x], {2, 81}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[If[n==0,0,Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&]]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

a(n) = Sum_{d|n, dA025147(d-1).

G.f.: Sum_{k>=2} (x^k*Product_{i>=2}(1 + x^(k*i))).

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021

A343347 Number of strict integer partitions of n with a part divisible by all the others.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 5, 4, 6, 6, 6, 8, 7, 7, 10, 9, 9, 12, 10, 8, 11, 11, 10, 14, 13, 11, 13, 12, 15, 20, 17, 15, 19, 19, 19, 22, 18, 17, 23, 22, 22, 28, 25, 24, 31, 28, 26, 32, 32, 30, 34, 32, 29, 37, 33, 27, 36, 33, 34, 44, 38, 36, 45, 45

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   632  93    841  D1   C3
                                 621  631  821  A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The dual version is A097986 (non-strict: A083710).
The non-strict version is A130689 (Heinz numbers: complement of A343337).
The strict complement is counted by A343377.
The case with smallest part divisible by all the others is A343378.
The case with smallest part not divisible by all the others is A343380.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.

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

seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021

A339662 Greatest gap in the partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 3, 1, 0, 6, 0, 7, 2, 3, 4, 8, 0, 2, 5, 1, 3, 9, 0, 10, 0, 4, 6, 2, 0, 11, 7, 5, 2, 12, 3, 13, 4, 1, 8, 14, 0, 3, 2, 6, 5, 15, 0, 4, 3, 7, 9, 16, 0, 17, 10, 3, 0, 5, 4, 18, 6, 8, 2, 19, 0, 20, 11, 1, 7, 3, 5, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Apr 20 2021

nonn

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also the index of the greatest prime, up to the greatest prime index of n, not dividing n. A prime index of n is a number m such that prime(m) divides n.

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
FindStat, Dyson's crank of a partition.
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)

Positions of first appearances are A000040.
Positions of 0's are A055932.
The version for positions of 1's in reversed binary expansion is A063250.
The prime itself (not just the index) is A079068.
The version for crank is A257989.
The minimal instead of maximal version is A257993.
The version for greatest difference is A286469 or A286470.
Positive integers by Heinz weight and image are counted by A339737.
Positions of 1's are A339886.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[maxgap[primeMS[n]],{n,100}]

a(n) = A000720(A079068(n)).

A343378 Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 6, 5, 4, 6, 6, 4, 8, 6, 7, 9, 8, 5, 12, 9, 8, 9, 11, 6, 14, 10, 10, 11, 10, 10, 20, 12, 12, 15, 18, 10, 21, 13, 15, 19, 17, 11, 27, 19, 20, 20, 25, 13, 27, 22, 26, 23, 24, 15, 34, 23, 21, 27, 30, 19, 38, 24, 26, 27, 37

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n with a part dividing all the others and a part divisible by all the others.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   821  93    841  D1   C3
                                 621  631       A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

The first condition alone gives A097986.
The non-strict version is A130714 (Heinz numbers are complement of A343343).
The second condition alone gives A343347.
The opposite version is A343379.
The half-opposite versions are A343380 and A343381.
The strict complement is counted by A343382.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A130689, A264401, A339562, A339563, A341450, A343346, A343377.

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

A343345 Number of integer partitions of n that are empty, or have smallest part dividing all the others, but do not have greatest part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 29, 36, 59, 79, 115, 149, 216, 270, 379, 473, 634, 793, 1063, 1292, 1689, 2079, 2667, 3241, 4142, 4982, 6291, 7582, 9434, 11321, 14049, 16709, 20545, 24490, 29860, 35380, 43004, 50741, 61282, 72284, 86680, 101906, 121990

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

First differs from A343346 at a(14) = 79, A343346(14) = 80.

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

			The a(6) = 1 through a(11) = 16 partitions:
  (321)  (3211)  (431)    (531)     (541)      (641)
                 (521)    (3321)    (721)      (731)
                 (3221)   (4311)    (4321)     (4331)
                 (32111)  (5211)    (5221)     (5321)
                          (32211)   (5311)     (5411)
                          (321111)  (32221)    (7211)
                                    (33211)    (33221)
                                    (43111)    (43211)
                                    (52111)    (52211)
                                    (322111)   (53111)
                                    (3211111)  (322211)
                                               (332111)
                                               (431111)
                                               (521111)
                                               (3221111)
                                               (32111111)

The first condition alone gives A083710.
The half-opposite versions are A130714 and A343342.
The Heinz numbers of these partitions are 1 and A343340.
The second condition alone gives A343341.
The opposite version is A343344.
The strict case is A343381.
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.
A018818 counts partitions into divisors (strict: A033630).
Cf. A083711, A097986, A098743, A098965, A130689, A264401, A339563, A343337.

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

A343381 Number of strict integer partitions of n with a part dividing all the others but no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 4, 9, 9, 14, 14, 20, 20, 30, 30, 39, 44, 59, 59, 77, 85, 106, 114, 145, 150, 191, 205, 247, 267, 328, 345, 418, 455, 544, 582, 699, 745, 886, 962, 1117, 1209, 1430, 1523, 1778, 1932, 2225, 2406, 2792, 3001, 3456, 3750

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or (1) have smallest part dividing all the others and (2) have greatest part not divisible by all the others.

			The a(6) = 1 through a(16) = 14 partitions (empty column indicated by dot, A..D = 10..13):
  321   .  431   531   541    641    642    751    761    861     862
           521         721    731    651    5431   851    951     871
                       4321   5321   741    6421   941    A41     961
                                     831    7321   A31    B31     A42
                                     921           B21    6531    B41
                                     5421          6431   7431    D21
                                                   6521   7521    6541
                                                   7421   9321    7531
                                                   8321   54321   7621
                                                                  8431
                                                                  8521
                                                                  9421
                                                                  A321
                                                                  64321

The first condition alone gives A097986.
The non-strict version is A343345 (Heinz numbers: A343340).
The second condition alone gives A343377.
The half-opposite versions are A343378 and A343379.
The opposite (and dual) version is A343380.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A130689, A200745, A264401, A339563, A341450, A343337, A343341, A343347, A343382.

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

cp's OEIS Frontend

A083711 a(n) = A083710(n) - A000041(n-1).

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A343377 Number of strict integer partitions of n with no part divisible by all the others.

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A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

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A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

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A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

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A343347 Number of strict integer partitions of n with a part divisible by all the others.

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A339662 Greatest gap in the partition with Heinz number n.

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A343378 Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.

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