A338470 -id:A338470 - cp's OEIS frontend

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

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}]

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.

cp's OEIS Frontend

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

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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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Mathematica

Formula