A317090 -id:A317090 - cp's OEIS frontend

A307895 Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.

1, 2, 3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 45, 47, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 83, 89, 92, 97, 99, 101, 103, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164, 167, 171, 172, 173

Offset: 1

Gus Wiseman, May 04 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities, starting from the largest part through to the smallest, form an initial interval of positive integers. The enumeration of these partitions by sum is given by A179269.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}

Cf. A055932, A056239, A098859, A109298, A112798, A130091, A179269, A317090, A325326, A325337, A325460.

Select[Range[100],Last/@If[#==1,{},FactorInteger[#]]==Range[PrimeNu[#],1,-1]&]

A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A319161 in lacking 420.
The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
Numbers whose prime signature covers an initial interval are given by A317090.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities have multiplicities covering an initial interval of positive integers. The enumeration of these partitions by sum is given by A325330.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
For example, the prime indices of 1890 are {1,2,2,2,3,4}, whose multiplicities give the prime signature {1,1,1,3}, and since this does not cover an initial interval (2 is missing), 1890 is not in the sequence.

Cf. A000009, A055932, A056239, A112798, A118914, A317081, A317089, A317090, A319161, A325326, A325330, A325337, A325369, A325371.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&]

A317589 Heinz numbers of uniformly normal integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 150, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

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

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317492, A317588, A317590.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],uninrmQ[primeMS[#]]&]

A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, May 02 2019

nonn

The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities appear with distinct multiplicities. The enumeration of these partitions by sum is given by A325329.

			Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  80: {1,1,1,1,3}
  88: {1,1,1,5}
For example, the prime indices of 1260 are {1,1,2,2,3,4}, whose multiplicities give the prime signature {1,1,2,2}, and since 1 and 2 appear the same number of times, 1260 is not in the sequence.

Cf. A056239, A098859, A112798, A118914, A130091, A317090, A319161, A325326, A325329, A325331, A325337, A325370, A325371.

Select[Range[100],UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]

A361393 Positive integers k such that 2*omega(k) > bigomega(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Gus Wiseman, Mar 16 2023

nonn

First differs from A317090 in having 120 and lacking 360.

There are numbers like 1, 120, 168, 180, 252,... which are not in A179983 but in here, and others like 360, 504, 540, 600,... which are in A179983 but not in here. - R. J. Mathar, Mar 21 2023

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
The prime indices of 120 are {1,1,1,2,3}, with 3 distinct parts and 5 parts, and 2*3 > 5, so 120 is in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, with 3 distinct parts and 6 parts, and 2*3 is not greater than 6, so 360 is not in the sequence.

These partitions are counted by A237365.
The complement is A361204.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

isA361393 := proc(n)
    if 2*A001221(n) > numtheory[bigomega](n) then
        true;
    else
        false ;
    end if:
end proc:
for n from 1 to 100 do
    if isA361393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 21 2023

Select[Range[1000],2*PrimeNu[#]>PrimeOmega[#]&]

{k: 2*A001221(k) > A001222(k)}. - R. J. Mathar, Mar 21 2023

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

Original entry on oeis.org

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

Offset: 1

Gus Wiseman, Mar 24 2025

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, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Without the RHS we have A055932, counted by A000009.
Counting partitions by the RHS gives A091602, rank statistic A051903.
Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
Without the LHS we have A212166, counted by A239964.
Without the middle statistic we have A381542, counted by A240312.
Partitions of this type are counted by A382302.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A047993 counts balanced partitions, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).

A317092 Positive integers whose prime multiplicities are weakly decreasing and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 92

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A001222, A007916, A055932, A056239, A112798, A124010, A133808, A242031.

Cf. A317087, A317088, A317089, A317090, A317091.

normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,100],And[normalQ[FactorInteger[#][[All,2]]],OrderedQ[Reverse[FactorInteger[#][[All,2]]]]]&]

is(n) = my (f=factor(n), w=#f~); if (w==0 || f[w,2]!=1, return (0), for (k=1, w-1, if (f[k,2]!=f[k+1,2] && f[k,2]!=1+f[k+1,2], return (0))); return (1)) \\ Rémy Sigrist, Sep 05 2018

A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 44, 55, 77, 96, 127, 158, 208, 251, 329, 400, 501, 610, 766, 915, 1141, 1368, 1677, 2005, 2454, 2913, 3553, 4219, 5110, 6053, 7300, 8644, 10376, 12238, 14645, 17216, 20504, 24047, 28501, 33336, 39373, 45871, 53926, 62745

Offset: 0

Gus Wiseman, May 01 2019

nonn

Partitions whose parts cover an initial interval of positive integers are counted by A000009, with Heinz numbers A055932. Partitions whose multiplicities cover an initial interval of positive integers are counted by A317081, with Heinz numbers A317090. Partitions whose parts and multiplicities both cover an initial interval of positive integers are counted by A317088, with Heinz numbers A317089. Partitions whose multiplicities at every depth cover an initial interval of positive integers are counted by A317245, with Heinz numbers A317246.

The Heinz numbers of these partitions are given by A325370.

			The a(0) = 1 through a(8) = 16 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (221)    (33)      (322)      (44)
                        (211)   (311)    (222)     (331)      (332)
                        (1111)  (2111)   (411)     (511)      (422)
                                (11111)  (3111)    (2221)     (611)
                                         (21111)   (3211)     (2222)
                                         (111111)  (4111)     (3221)
                                                   (22111)    (4211)
                                                   (31111)    (5111)
                                                   (211111)   (22211)
                                                   (1111111)  (32111)
                                                              (41111)
                                                              (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)
For example, the partition (5,5,4,3,3,3,2,2) has multiplicities (2,1,3,2) with multiplicities (1,2,1) which cover the initial interval {1,2}, so (5,5,4,3,3,3,2,2) is counted under a(27).

Cf. A000837, A055932, A317081, A317088, A317089, A317090, A317245, A320348, A325331, A325333, A325337, A325370.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&]],{n,0,30}]

A339886 Numbers whose prime indices cover an interval of positive integers starting with 2.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 75, 81, 105, 135, 225, 243, 315, 375, 405, 525, 675, 729, 735, 945, 1125, 1155, 1215, 1575, 1875, 2025, 2187, 2205, 2625, 2835, 3375, 3465, 3645, 3675, 4725, 5145, 5625, 5775, 6075, 6561, 6615, 7875, 8085, 8505, 9375, 10125, 10395, 10935

Offset: 1

Gus Wiseman, Apr 20 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.

			The sequence of terms together with their prime indices begins:
    3: {2}
    9: {2,2}
   15: {2,3}
   27: {2,2,2}
   45: {2,2,3}
   75: {2,3,3}
   81: {2,2,2,2}
  105: {2,3,4}
  135: {2,2,2,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  315: {2,2,3,4}
  375: {2,3,3,3}
  405: {2,2,2,2,3}
  525: {2,3,3,4}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  735: {2,3,4,4}
  945: {2,2,2,3,4}

The version starting at 1 is A055932.
The partitions with these Heinz numbers are counted by A264396.
Positions of 1's in A339662.
A000009 counts partitions covering an initial interval.
A000070 counts partitions with a selected part.
A016945 lists numbers with smallest prime index 2.
A034296 counts gap-free (or flat) partitions.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A107428 counts gap-free compositions (initial: A107429).
A286469 and A286470 give greatest difference for Heinz numbers.
A325240 lists numbers with smallest prime multiplicity 2.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A006128, A007052, A124010, A257989, A257993, A264401, A317090, A317589, A339737.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[primeMS[#]-1]&]

A317091 Positive integers whose prime multiplicities are weakly increasing and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 101, 102

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A007916, A055932, A056239, A112798, A124010, A133808, A304678.

Cf. A317087, A317088, A317089, A317090, A317092.

normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[2,150],And[normalQ[FactorInteger[#][[All,2]]],OrderedQ[FactorInteger[#][[All,2]]]]&]

cp's OEIS Frontend

A307895 Numbers whose prime exponents, starting from the largest prime factor through to the smallest, form an initial interval of positive integers.

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A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.

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A317589 Heinz numbers of uniformly normal integer partitions.

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A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.

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A361393 Positive integers k such that 2*omega(k) > bigomega(k).

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A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

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A317092 Positive integers whose prime multiplicities are weakly decreasing and span an initial interval of positive integers.

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A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

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Mathematica

A339886 Numbers whose prime indices cover an interval of positive integers starting with 2.

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