A341450 -id:A341450 - cp's OEIS frontend

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.

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}]

A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

Original entry on oeis.org

1, 15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 195, 198, 201, 203, 204, 205, 207, 209, 210

Offset: 1

Gus Wiseman, Apr 15 2021

nonn

After 1, first differs from A318992 in lacking 390, with prime indices {1,2,3,6}.
First differs from A343337 in having 195, with prime indices {2,3,6}.
Alternative name: 1 and numbers where either the smallest prime index is not a divisor of all the other prime indices, or the greatest prime index is not divisible by all the other prime indices.
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.
Also Heinz numbers of partitions that either empty, have smallest part not dividing all the others, or have greatest part not divisible by all the others (counted by A343346). 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.

			The sequence of terms together with their prime indices begins:
      1: {}            90: {1,2,2,3}      141: {2,15}
     15: {2,3}         91: {4,6}          143: {5,6}
     30: {1,2,3}       93: {2,11}         145: {3,10}
     33: {2,5}         95: {3,8}          150: {1,2,3,3}
     35: {3,4}         99: {2,2,5}        153: {2,2,7}
     45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
     51: {2,7}        105: {2,3,4}        155: {3,11}
     55: {3,5}        110: {1,3,5}        161: {4,9}
     60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
     66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
     69: {2,9}        123: {2,13}         175: {3,3,4}
     70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
     75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
     77: {4,5}        138: {1,2,9}        182: {1,4,6}
     85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
For example, the prime indices of 90 are {1,2,2,3}, and, because 1 divides all the other parts, 90 is in the sequence, even though 3 is not divisible by all the other parts.

The partitions without these Heinz numbers are counted by A130714.
The first condition alone gives A342193.
The second condition alone gives A343337.
The "and" instead of "or" version is A343338.
The partitions with these Heinz numbers are counted by A343346.
A000005 counts divisors.
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).
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A083710, A257993, A338470, A339562, A341450, A343339, A343340, A343341, A343342, A343378, A343379, A343382.

Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)||!And@@IntegerQ/@(p/Min@@p)]&]

Equals the union of A342193 and A343337.

A343339 Numbers with no prime index dividing all the other prime indices, but with a prime index divisible by all the other prime indices.

Original entry on oeis.org

195, 555, 585, 915, 957, 975, 1295, 1335, 1665, 1695, 1755, 2193, 2265, 2343, 2535, 2585, 2715, 2745, 2775, 2871, 2925, 3115, 3345, 3367, 3729, 3765, 3885, 4005, 4209, 4215, 4575, 4755, 4875, 4995, 5085, 5265, 5285, 5385, 5457, 5467, 5709, 5955, 6205, 6215

Offset: 1

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Numbers > 1 whose smallest prime index does not divide all the other prime indices, but whose greatest prime index is divisible by all the other prime indices.
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.
Also Heinz numbers of partitions with greatest part divisible by all the others, but smallest part not dividing all the others (counted by A343344). 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.

			The sequence of terms together with their prime indices begins:
     195: {2,3,6}        2585: {3,5,15}       4575: {2,3,3,18}
     555: {2,3,12}       2715: {2,3,42}       4755: {2,3,66}
     585: {2,2,3,6}      2745: {2,2,3,18}     4875: {2,3,3,3,6}
     915: {2,3,18}       2775: {2,3,3,12}     4995: {2,2,2,3,12}
     957: {2,5,10}       2871: {2,2,5,10}     5085: {2,2,3,30}
     975: {2,3,3,6}      2925: {2,2,3,3,6}    5265: {2,2,2,2,3,6}
    1295: {3,4,12}       3115: {3,4,24}       5285: {3,4,36}
    1335: {2,3,24}       3345: {2,3,48}       5385: {2,3,72}
    1665: {2,2,3,12}     3367: {4,6,12}       5457: {2,7,28}
    1695: {2,3,30}       3729: {2,5,30}       5467: {4,5,20}
    1755: {2,2,2,3,6}    3765: {2,3,54}       5709: {2,5,40}
    2193: {2,7,14}       3885: {2,3,4,12}     5955: {2,3,78}
    2265: {2,3,36}       4005: {2,2,3,24}     6205: {3,7,21}
    2343: {2,5,20}       4209: {2,9,18}       6215: {3,5,30}
    2535: {2,3,6,6}      4215: {2,3,60}       6475: {3,3,4,12}

The first condition alone gives A342193.
The second condition alone gives the complement of A343337.
The partitions with these Heinz numbers are counted by A343344.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A130689, A130714, A257993, A338470, A339562, A341450, A343338, A343380.

Select[Range[2,1000],With[{p=PrimePi/@First/@FactorInteger[#]},And@@IntegerQ/@(Max@@p/p)&&!And@@IntegerQ/@(p/Min@@p)]&]

Complement of A343337 in A342193.

A339660 Number of strict 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, 1, 1, 2, 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 5, 2, 5, 5, 4, 5, 7, 3, 5, 6, 5, 5, 9, 4, 7, 6, 6, 9, 11, 6, 9, 10, 9, 10, 12, 6, 11, 12, 10, 12, 16, 9, 15, 16, 12, 14, 18, 14, 16, 18, 14, 15, 22, 11, 16, 20, 13, 21, 23, 15, 21, 24, 21, 21, 31, 14, 24

Offset: 0

Gus Wiseman, Apr 19 2021

nonn

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

			The a(n) partitions for n = 14, 12, 18, 24, 30, 39, 36:
  (14)     (12)    (18)      (24)        (30)        (39)          (36)
  (12,2)   (8,4)   (12,6)    (16,8)      (24,6)      (36,3)        (27,9)
  (8,4,2)  (9,3)   (15,3)    (18,6)      (25,5)      (26,13)       (30,6)
           (10,2)  (16,2)    (20,4)      (27,3)      (27,9,3)      (32,4)
                   (12,4,2)  (21,3)      (28,2)      (28,7,4)      (33,3)
                             (22,2)      (20,10)     (30,6,3)      (34,2)
                             (12,6,4,2)  (18,9,3)    (24,12,3)     (24,12)
                                         (24,4,2)    (24,8,4,3)    (24,8,4)
                                         (16,8,4,2)  (20,10,5,4)   (18,9,6,3)
                                                     (24,6,4,3,2)  (24,6,4,2)
                                                                   (20,10,4,2)

The dual version is A098965 (non-strict: A083711).
The non-strict version is A339619 (Heinz numbers: complement of A343337).
The version with 1's allowed is A343347 (non-strict: A130689).
The case without a part dividing all the other parts is A343380.
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.
Cf. A083710, A097986, A098743, A200745, A264401, A339563, A341450, A343337, A343341, A343344, A343378.

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

A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 0, 5, 2, 1, 0, 5, 3, 1, 0, 1, 7, 3, 1, 1, 0, 8, 4, 2, 1, 0, 10, 5, 2, 1, 0, 12, 6, 3, 1, 0, 15, 7, 3, 1, 0, 1, 17, 9, 4, 1, 1, 0, 21, 10, 4, 2, 1, 0, 25, 12, 6, 2, 1, 0, 29, 15, 6, 3, 1, 0, 35, 17, 8, 3, 1, 0

Offset: 0

Gus Wiseman, Apr 18 2021

The least gap (or mex) of a partition is the least positive integer that is not a part.

Row lengths are chosen to be consistent with the non-strict case A264401.

			Triangle begins:
   1
   0   1
   1   0
   1   0   1
   1   1   0
   2   1   0
   2   1   0   1
   3   1   1   0
   3   2   1   0
   5   2   1   0
   5   3   1   0   1
   7   3   1   1   0
   8   4   2   1   0
  10   5   2   1   0
  12   6   3   1   0
  15   7   3   1   0   1

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.

Row sums are A000009.
Row lengths are A002024.
Column k = 1 is A025147.
Column k = 2 is A025148.
The non-strict version is A264401.
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.
A257993 gives the least gap of the partition with Heinz number n.
A339564 counts factorizations with a selected factor.
A342050 ranks partitions with even least gap.
A342051 ranks partitions with odd least gap.
Cf. A003242, A083710, A083711, A097986, A098743, A098965, A130689, A200745, A341450, A343347, A343377.

mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}]

cp's OEIS Frontend

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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A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

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A343339 Numbers with no prime index dividing all the other prime indices, but with a prime index divisible by all the other prime indices.

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A339660 Number of strict integer partitions of n with no 1's and a part divisible by all the other parts.

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A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.

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