A343340 -id:A343340 - cp's OEIS frontend

A083710 Number of integer partitions of n with a part dividing all the other parts.

1, 1, 2, 3, 5, 6, 11, 12, 20, 25, 37, 43, 70, 78, 114, 143, 196, 232, 330, 386, 530, 641, 836, 1003, 1340, 1581, 2037, 2461, 3127, 3719, 4746, 5605, 7038, 8394, 10376, 12327, 15272, 17978, 22024, 26095, 31730, 37339, 45333, 53175, 64100, 75340, 90138

Offset: 0

N. J. A. Sloane, Jun 16 2003

Since the summand (part) which divides all the other summands is necessarily the smallest, an equivalent definition is: "Number of partitions of n such that smallest part divides every part." - Joerg Arndt, Jun 08 2009
The first few partitions that fail the criterion are 5=3+2, 7=5+2=4+3=3+2+2. So a(5) = A000041(5) - 1 = 6, a(7) = A000041(7) - 3 = 12. - Vladeta Jovovic, Jun 17 2003
Starting with offset 1 = inverse Mobius transform (A051731) of the partition numbers, A000041. - Gary W. Adamson, Jun 08 2009

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (331)
                    (211)   (311)    (51)      (421)
                    (1111)  (2111)   (222)     (511)
                            (11111)  (321)     (2221)
                                     (411)     (3211)
                                     (2211)    (4111)
                                     (3111)    (22111)
                                     (21111)   (31111)
                                     (111111)  (211111)
                                               (1111111)
(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.

Cf. A018783, A137587.
Cf. A000041, A051731. - Gary W. Adamson, Jun 08 2009
The case with no 1's is A083711.
The strict case is A097986.
The version for "divisible by" instead of "dividing" is A130689.
The case where there is also a part divisible by all the others is A130714.
The complement of these partitions is counted by A338470.
The Heinz numbers of these partitions are dense, complement of A342193.
The case where there is also no part divisible by all the others is A343345.
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).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A001792, A098965, A264401, A339563, A343340, A343341, A343378.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-0 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 0 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11, ...). - Gary W. Adamson, Jan 27 2008
G.f.: 1 + Sum_{n>=1} x^n/eta(x^n). The g.f. for partitions into parts that are a multiple of n is x^n/eta(x^n), now sum over n. - Joerg Arndt, Jun 08 2009
Gary W. Adamson's comment is equivalent to the formula a(n) = Sum_{d|n} p(d-1) where p(i) = number of partitions of i (A000041(i)). Hence A083710 has g.f. Sum_{d>=1} p(d-1)*x^d/(1-x^d), - N. J. A. Sloane, Jun 08 2009

More terms from Vladeta Jovovic, Jun 17 2003

Name shortened by Gus Wiseman, Apr 18 2021

A343337 Numbers with 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, 198, 201, 203, 204, 205, 207, 209, 210, 215

Offset: 1

Gus Wiseman, Apr 13 2021

nonn

Alternative name: 1 and numbers whose greatest prime index is not divisible by all the other prime indices.
First differs from A318992 in lacking 195.
First differs from A343343 in lacking 195.
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 not divisible by all the others (counted by A343341). 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, 195 has prime indices {2,3,6}, and 6 is divisible by both 2 and 3, so 195 does not belong to the sequence.

The complement is counted by A130689.
The dual version is A342193.
The case with smallest prime index not dividing all the others is A343338.
The case with smallest prime index dividing by all the others is A343340.
These are the Heinz numbers of the partitions counted by A343341.
Including the dual version gives A343343.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
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, A341450, A343342, A343345, A343346, A343377, A343379, A343381, A343382.

Select[Range[1000],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)]&]

A339563 Squarefree numbers > 1 whose smallest prime index divides all the other prime indices.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 37, 38, 39, 41, 42, 43, 46, 47, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 74, 78, 79, 82, 83, 86, 87, 89, 94, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113, 114, 115, 118, 122, 127

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

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 strict integer partitions whose smallest part divides all the others (counted by A097986). 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:
      2: {1}       29: {10}        59: {17}
      3: {2}       30: {1,2,3}     61: {18}
      5: {3}       31: {11}        62: {1,11}
      6: {1,2}     34: {1,7}       65: {3,6}
      7: {4}       37: {12}        66: {1,2,5}
     10: {1,3}     38: {1,8}       67: {19}
     11: {5}       39: {2,6}       70: {1,3,4}
     13: {6}       41: {13}        71: {20}
     14: {1,4}     42: {1,2,4}     73: {21}
     17: {7}       43: {14}        74: {1,12}
     19: {8}       46: {1,9}       78: {1,2,6}
     21: {2,4}     47: {15}        79: {22}
     22: {1,5}     53: {16}        82: {1,13}
     23: {9}       57: {2,8}       83: {23}
     26: {1,6}     58: {1,10}      86: {1,14}

These partitions are counted by A097986 (non-strict: A083710).
The case with no 1's is counted by A098965 (non-strict: A083711).
The squarefree complement is A339562, ranked by A341450.
The complement of the not necessarily squarefree version is A342193.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A005117 lists squarefree numbers.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices, row sums of A112798.
A338470 counts partitions with no dividing part.
Cf. A130714, A253249, A257993, A299702, A339564, A343340, A343378.

Select[Range[2,100],SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]},And@@IntegerQ/@(p/Min@@p)]&]

A343338 Numbers with no prime index dividing or divisible by all the other prime indices.

Original entry on oeis.org

1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299, 301

Offset: 1

Gus Wiseman, Apr 13 2021

nonn

Alternative name: 1 and numbers whose smallest prime index does not divide all the other prime indices, nor whose greatest prime index is divisible by all the other prime indices.
First differs from A302697 in having 91.
First differs from A337987 in having 91.
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 not divisible by all the others and smallest part not dividing all the others (counted by A343342). 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: {}         105: {2,3,4}      203: {4,10}
     15: {2,3}      119: {4,7}        205: {3,13}
     33: {2,5}      123: {2,13}       207: {2,2,9}
     35: {3,4}      135: {2,2,2,3}    209: {5,8}
     45: {2,2,3}    141: {2,15}       215: {3,14}
     51: {2,7}      143: {5,6}        217: {4,11}
     55: {3,5}      145: {3,10}       219: {2,21}
     69: {2,9}      153: {2,2,7}      221: {6,7}
     75: {2,3,3}    155: {3,11}       225: {2,2,3,3}
     77: {4,5}      161: {4,9}        231: {2,4,5}
     85: {3,7}      165: {2,3,5}      245: {3,4,4}
     91: {4,6}      175: {3,3,4}      247: {6,8}
     93: {2,11}     177: {2,17}       249: {2,23}
     95: {3,8}      187: {5,7}        253: {5,9}
     99: {2,2,5}    201: {2,19}       255: {2,3,7}
For example, the prime indices of 975 are {2,3,3,6}, all of which divide 6, but not all of which are multiples of 2, so 975 is not in the sequence.

The first condition alone gives A342193.
The second condition alone gives A343337.
The half-opposite versions are A343339 and A343340.
The partitions with these Heinz numbers are counted by A343342.
The opposite version is the complement of A343343.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
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, A130689, A338470, A339562, A341450, A343341, A343346, A343347, A343348, A343377, A343379, A343382.

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

Intersection of A342193 and A343337.

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}]

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.

cp's OEIS Frontend

A083710 Number of integer partitions of n with a part dividing all the other parts.

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A343337 Numbers with no prime index divisible by all the other prime indices.

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A339563 Squarefree numbers > 1 whose smallest prime index divides all the other prime indices.

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A343338 Numbers with no prime index dividing or divisible by all the other prime indices.

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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.

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