A304678 -id:A304678 - cp's OEIS frontend

A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 9, 0, 7, 0, 12, 0, 1, 0, 38, 0, 1, 1, 18, 0, 38, 0, 32, 0, 1, 0, 90, 0, 1, 0, 71, 0, 78, 0, 33, 10, 1, 0, 228, 0, 31, 0, 42, 0, 156, 0, 123, 0, 1, 0, 447, 0, 1, 16, 146, 0, 222, 0, 63, 0, 102, 0, 811, 0, 1, 29, 75, 0, 334, 0

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
  (211)  (3111)  (422)     (511111)  (633)        (71111111)  (844)
                 (41111)             (6222)                   (82222)
                 (221111)            (33222)                  (442222)
                                     (4221111)                (44221111)
                                     (6111111)                (422221111)
                                     (33111111)               (811111111)
                                     (222111111)              (4411111111)
                                                              (42211111111)
                                                              (222211111111)

These partitions are ranked by A383015, positions of terms > 1 in A382877.
For run-lengths instead of sums we have A383090, ranks A383089, unique A383094.
The complement is A383095 + A383096, ranks A383099 \/ A383100.
For any positive number of permutations we have A383098, ranks A383110.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A381717, A382076.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Apr 20 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:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   37: {12}
   41: {13}

For distinct instead of equal run-sums we have A000961, counted by A000005.
These are the positions of 1 in A382877.
For more than one choice we have A383015.
Partitions of this type are counted by A383095.
For no choices we have A383100, counted by A383096.
For at least one choice we have A383110, counted by A383098, see A383013.
For run-lengths instead of sums we have A383112 = positions of 1 in A382857.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A000720, A001221, A001222, A351294, A353832, A353838, A353932, A354584, A382876, A383091.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==1&]

The complement is A383015 \/ A383100, for run-lengths A382879 \/ A383089.

A112769 Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is less than the previous exponent when read from left to right.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212, 220

Offset: 1

Ray Chandler, Sep 23 2005, based on a suggestion from Leroy Quet

nonn

This sequence lists the integers x such that A085079(x) > x. - Michel Marcus, Jun 25 2025 and Jul 30 2015

			90 = 2^1 * 3^2 * 5^1 and 2 > 1, so 90 is in sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A085079, A097318, A097319, A097320, A304678.

mopfQ[n_]:=Module[{e=FactorInteger[n][[All,2]]},Length[e]>1&&Min[ Differences[ e]]<0]; Select[Range[300],mopfQ] (* Harvey P. Dale, May 30 2018 *)

isok(n) = {f = factor(n)[,2]; if (#f > 1, for (k=2, #f, if (f[k] < f[k-1], return (1)););); return (0);} \\ Michel Marcus, Jul 30 2015

A334965 Numbers with strictly increasing prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 151, 157, 162, 163, 167, 169

Offset: 1

Gus Wiseman, May 18 2020

nonn

First differs from A329131 in lacking 150.

Also numbers whose unsorted prime signature is strictly increasing.

			The sequence of terms together with their prime indices begins:
    1: {}            25: {3,3}           64: {1,1,1,1,1,1}
    2: {1}           27: {2,2,2}         67: {19}
    3: {2}           29: {10}            71: {20}
    4: {1,1}         31: {11}            73: {21}
    5: {3}           32: {1,1,1,1,1}     75: {2,3,3}
    7: {4}           37: {12}            79: {22}
    8: {1,1,1}       41: {13}            81: {2,2,2,2}
    9: {2,2}         43: {14}            83: {23}
   11: {5}           47: {15}            89: {24}
   13: {6}           49: {4,4}           97: {25}
   16: {1,1,1,1}     50: {1,3,3}         98: {1,4,4}
   17: {7}           53: {16}           101: {26}
   18: {1,2,2}       54: {1,2,2,2}      103: {27}
   19: {8}           59: {17}           107: {28}
   23: {9}           61: {18}           108: {1,1,2,2,2}

These are the Heinz numbers of the partitions counted by A100471.
Partitions with strictly decreasing run-lengths are A100881.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
The weakly decreasing version is A242031.
The weakly increasing version is A304678.
The strictly decreasing version is A304686.
Compositions with strictly increasing or decreasing run-lengths are A333191.
Cf. A000837, A032020, A098859, A100883, A329744, A332835, A333147, A333190.

Select[Range[100],Less@@Last/@FactorInteger[#]&]

A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

Original entry on oeis.org

1, 2, 12, 36, 40, 112, 352, 832, 960, 1296, 2176, 2880, 4864, 5376, 11776, 12544, 16128, 29696, 33792, 34560, 38400, 63488, 64000, 101376, 115200, 143360, 151552, 159744, 335872, 479232, 704512, 835584, 1540096, 1658880, 1802240

Offset: 1

Gus Wiseman, May 17 2022

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. A number's prime signature (row n A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   352: {1,1,1,1,1,5}
   832: {1,1,1,1,1,1,6}
   960: {1,1,1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  2176: {1,1,1,1,1,1,1,7}
  2880: {1,1,1,1,1,1,2,2,3}
  4864: {1,1,1,1,1,1,1,1,8}
  5376: {1,1,1,1,1,1,1,1,2,4}

For shadows instead of exponents we get A003586, counted by A008619.
The LHS (product of prime indices) is A003963, counted by A339095.
The RHS (product of prime exponents) is A005361, counted by A266477.
The version for shadows instead of indices is A353399, counted by A353398.
These partitions are counted by A353506.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A008480, A085629, A097318, A109297, A304678, A318871, A320325, A325131, A325755, A353500, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==Times@@Last/@FactorInteger[#]&]

from itertools import count, islice
from math import prod
from sympy import primepi, factorint
def A353503_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or prod((f:=factorint(n)).values()) == prod(primepi(p)**e for p,e in f.items()), count(max(startvalue,1)))
A353503_list = list(islice(A353503_gen(),20)) # Chai Wah Wu, May 20 2022

A003963(a(n)) = A005361(a(n)).

A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 6, 2, 4, 5, 6, 2, 12, 2, 6, 8, 5, 2, 20, 2, 12, 8, 6, 2, 20, 5, 6, 12, 12, 2, 34, 2, 6, 8, 6, 8, 45, 2, 6, 8, 20, 2, 34, 2, 12, 28, 6, 2, 30, 5, 20, 8, 12, 2, 52, 8, 20, 8, 6, 2, 78, 2, 6, 28, 7, 8, 34, 2, 12, 8, 34, 2, 80, 2, 6, 28, 12, 8, 34, 2, 30, 25

Offset: 0

Gus Wiseman, Apr 16 2025

nonn

			The partition (2,2,1,1) has permutation (2,1,1,2) so is counted under a(6).
The a(1) = 1 through a(10) = 6 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      33111      22222
                           2211             11111111  3111111    2221111
                           21111                      111111111  22111111
                           111111                                1111111111

For distinct instead of equal run-sums we have A000005.
For run-lengths instead of sums we have A383094.
The complement is counted by A383096 + A383097, ranks A383100 \/ A383015.
These partitions are ranked by A383099 = positions of 1 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A329738, A353839, A353832, A353932, A354584, A381636, A381871, A382076, A382876.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Total/@Split[#]&]]==1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              211          222              422       33111      22222
              1111         2211             2222      3111111    511111
                           3111             41111     111111111  2221111
                           21111            221111               22111111
                           111111           11111111             1111111111

For distinct instead of equal run-sums we appear to have A382427.
For run-lengths instead of sums we have A383013, ranked by complement of A382879.
The case of a unique choice is A383095, ranks A383099 = positions of 1 in A382877.
The complement is counted by A383096, ranks A383100 = positions of 0 in A382877.
These partitions are ranked by A383110.
The case of more than one choice is A383097, ranks A383015.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A382076, A382857, A382876, A383094, A383112.

Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Total/@Split[#]&]!={}&]],{n,0,15}]

a(n) = A383097(n) + A383095(n), ranks A383015 \/ A383099.

More terms from Bert Dobbelaere, Apr 26 2025

A383110 Numbers whose prime indices have a permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173

Offset: 1

Gus Wiseman, Apr 20 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 prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), so 144 is in the sequence.
The terms together with their prime indices begin:
   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}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}

For distinct run-sums we appear to have complement of A381636 (counted by A381717).
These are the positions of positive terms in A382877.
For run-lengths instead of sums we have complement of A382879, counted by A383013.
For more than one choice we have A383015.
Partitions of this type are counted by A383098.
For a unique choice we have A383099, counted by A383095.
The complement is A383100, counted by A383096.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]>0&]

Equals A383015 \/ A383099, counted by A353851 \/ A383095.

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]==0&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000

Offset: 1

Gus Wiseman, May 17 2022

nonn

All terms are highly powerful (A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.

			The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Highly powerful number.

These are the positions of first appearances in A005361, counted by A266477.
This is the sorted version of A085629.
The version for shadows instead of exponents is A353397, firsts in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime exponents, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
Subsequence of A181800.

nn=1000;
d=Table[Times@@Last/@FactorInteger[n],{n,nn}];
Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]
lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]], q] (* Amiram Eldar, Sep 29 2024, using the function f by T. D. Noe at A162247 *)

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A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

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A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

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A112769 Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is less than the previous exponent when read from left to right.

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A334965 Numbers with strictly increasing prime multiplicities.

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A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

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A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.

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A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

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A383110 Numbers whose prime indices have a permutation with all equal run-sums.

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