A182850 -id:A182850 - cp's OEIS frontend

A353743 Least number with run-sum trajectory of length k; a(0) = 1.

1, 2, 4, 12, 84, 1596, 84588, 11081028, 3446199708, 2477817590052, 4011586678294188, 14726534696017964148, 120183249654202605411828, 2146833388573021140471483564, 83453854313999050793547980583372, 7011542477899258250521520684673165324

Offset: 0

Gus Wiseman, Jun 11 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832, A353847) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     12: {1,1,2}
     84: {1,1,2,4}
   1596: {1,1,2,4,8}
  84588: {1,1,2,4,8,16}

The ordered version is A072639, for run-lengths A333629.
The version for run-lengths is A325278, firsts in A182850 or A323014.
The run-sum trajectory is the iteration of A353832.
The first length-k row of A353840 has index a(k).
Other sequences pertaining to this trajectory are A353841-A353846.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.

Join[{1,2},Table[2*Product[Prime[2^k],{k,0,n}],{n,0,6}]]

a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).

a(n > 0) = 2 * A325782(n).

A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  120: {1,1,1,2,3}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  240: {1,1,1,1,2,3}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  420: {1,1,2,3,4}
  432: {1,1,1,1,2,2,2}
  480: {1,1,1,1,1,2,3}
  576: {1,1,1,1,1,1,2,2}
  768: {1,1,1,1,1,1,1,1,2}
  840: {1,1,1,2,3,4}
  864: {1,1,1,1,1,2,2,2}
  960: {1,1,1,1,1,1,2,3}

Cf. A001221, A001222, A001358, A005117, A007774, A007916, A055932, A071625, A112798, A181819, A182850, A182857, A275870, A304464, A304465, A304636, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,100}],2]

A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

Original entry on oeis.org

8, 27, 30, 42, 66, 70, 78, 102, 105, 110, 114, 125, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 343, 345, 354, 357, 360, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
     8: {1,1,1}
    30: {1,2,3}
   360: {1,1,1,2,2,3}
   720: {1,1,1,1,2,2,3}
   900: {1,1,2,2,3,3}
  1440: {1,1,1,1,1,2,2,3}
  2160: {1,1,1,1,2,2,2,3}
  2880: {1,1,1,1,1,1,2,2,3}
  4320: {1,1,1,1,1,2,2,2,3}
  5760: {1,1,1,1,1,1,1,2,2,3}
  8640: {1,1,1,1,1,1,2,2,2,3}
Starting with A112798(1801800) and repeatedly taking the multiset of multiplicities we have {1,1,1,2,2,3,3,4,5,6} -> {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so 1801800 belongs to the sequence.

Cf. A001221, A001222, A005117, A007916, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A304464, A304465, A304634, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,200}],3]

A316529 Heinz numbers of totally strong integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Jul 29 2018

nonn

First differs from A304678 at a(115) = 151, A304678(115) = 150.
The alternating version first differs from this sequence in having 150 and lacking 450.
An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			Starting with (3,3,2,1), which has Heinz number 150, and repeatedly taking run-lengths gives (3,3,2,1) -> (2,1,1) -> (1,2), so 150 is not in the sequence.
Starting with (3,3,2,2,1), which has Heinz number 450, and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2) -> (1), so 450 is in the sequence.

Cf. A056239, A181819, A182850, A242031, A296150, A305732, A317246.
The enumeration of these partitions by sum is A316496.
The complement is A316597.
The widely normal version is A332291.
The dual version is A335376.
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100883, A133808, A317257, A332275, A332293, A332297.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totstrQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totstrQ[Length/@Split[q]]]];
Select[Range[100],totstrQ[Reverse[primeMS[#]]]&]

Updated with corrected terminology by Gus Wiseman, Mar 08 2020

A317588 Number of uniformly normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 6, 3, 5, 6, 7, 5, 8, 5, 7, 10, 7, 6, 12, 7, 12, 14, 10, 11, 18, 11, 13, 16, 18, 15, 35, 16, 26, 24, 27, 26, 47, 33, 44, 48, 58, 48, 76, 63, 81, 79, 98, 94, 123, 109, 135, 131, 148, 140, 162, 149, 152, 162, 166, 175, 202, 191, 221, 232, 233

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

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.

			The a(6) = 6 uniformly normal integer partitions are (6), (33), (321), (222), (2211), (111111). Missing from this list are (51), (42), (411), (3111), (21111).
The a(21) = 14 uniformly normal integer partitions (n = 21):
  (n),
  (777),
  (654321),
  (4443321), (3333333),
  (44432211), (44333211), (44332221),
  (4432221111), (4333221111), (4332222111),
  (433322211),
  (22222221111111),
  (111111111111111111111).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A181819, A182850, A275870, A317081, A317088, A317246, A317246, A317491, A317589, A317590.

uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Table[Length[Select[IntegerPartitions[n],uninrmQ]],{n,0,30}]

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[#]]&]

A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

From Gus Wiseman, Jul 11 2023: (Start)
A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
For example:
- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).
- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).
(End)

			The a(8) = 16 partitions:
  (8),
  (44),
  (332), (422), (611),
  (2222), (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111),
  (11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).

These partitions have ranks A319161.
For distinct instead of relatively prime multiplicities we have A325329.
Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.

Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]

A325247 Numbers whose omega-sequence is strict (no repeated parts).

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, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 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 16 2019

nonn

First differs from A323306 in having 216.
We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).
Also Heinz numbers of integer partitions of whose omega-sequence is strict (counted by A325250). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			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}
    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}

Positions of squarefree numbers in A325248.
Cf. A056239, A112798, A118914, A181819, A323023, A325249, A325250, A325251, A325277.
Omega-sequence statistics: A001221 (second omega), A001222 (first omega), A071625 (third omega), A304465 (second-to-last omega), A182850 or A323014 (depth), A323022 (fourth omega), A325248 (Heinz number).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#1]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Select[Range[100],UnsameQ@@omseq[#]&]

A353842 Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 7, 13, 14, 15, 7, 17, 14, 19, 15, 21, 22, 23, 15, 13, 26, 13, 21, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 13, 41, 42, 43, 33, 35, 46, 47, 21, 19, 26, 51, 39, 53, 26, 55, 35, 57, 58, 59, 35, 61, 62, 19, 13, 65, 66, 67, 51

Offset: 1

Gus Wiseman, May 25 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The partition run-sum trajectory of 87780 is: 87780 -> 65835 -> 51205 -> 19855 -> 2915, so a(87780) = 2915.

The fixed points and image are A005117.
For run-lengths instead of sums we have A304464/A304465, counted by A325268.
These are the row-ends of A353840.
Other sequences pertaining to partition trajectory are A353841-A353846.
The version for compositions is A353855, run-ends of A353853.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A182850 and A323014 give frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A071625, A073093, A181819, A325239, A325277, A353834, A353838, A353847, A353865, A353867.

Table[NestWhile[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&],{n,100}]

A182858 Irregular table: Row n is the list of positive exponents in the canonical prime factorization of A182857(n), sorted in descending order, or 0 if there are no such exponents.

Original entry on oeis.org

0, 1, 2, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 7, 6, 5, 4, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 14, 13, 12, 11, 10, 10, 9, 9, 8, 8, 7, 7, 7, 6, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 42, 41, 40, 39, 38, 37, 36, 35, 35, 34, 34, 33, 33, 32, 32, 31, 31

Offset: 0

Matthew Vandermast, Jan 05 2011

Length of row 0 = length of row 1 = 1; for n > 1, length of row n = A011784(n-1).

For n > 2, row n = row (n-2) of A012257, reversed.

			Table begins:
0,
1,
2,
1,1,
2,1,
2,1,1,
3,2,1,1,...

Cf. A181819, A182850.

cp's OEIS Frontend

A353743 Least number with run-sum trajectory of length k; a(0) = 1.

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A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

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A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

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A316529 Heinz numbers of totally strong integer partitions.

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A317588 Number of uniformly normal integer partitions of n.

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A317589 Heinz numbers of uniformly normal integer partitions.

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A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

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A325247 Numbers whose omega-sequence is strict (no repeated parts).

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A353842 Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

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