A304678 -id:A304678 - cp's OEIS frontend

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

Members m of A025487 such that A181819(m) is also a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.
Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - Gus Wiseman, May 21 2022

			The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From _Gus Wiseman_, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Conjugate Partition.

Intersection of A025487 and A179983.
Subsequence of A129912 and A181826.
Includes all members of A182862.
Positions of first appearances in A353742, unordered version A238747.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];
Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

A329142 Numbers whose prime signature is not a necklace.

Original entry on oeis.org

1, 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

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

After a(1) = 1, first differs from A112769 in lacking 1350.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   1: ()
  12: (2,1)
  20: (2,1)
  24: (3,1)
  28: (2,1)
  40: (3,1)
  44: (2,1)
  45: (2,1)
  48: (4,1)
  52: (2,1)
  56: (3,1)
  60: (2,1,1)
  63: (2,1)
  68: (2,1)
  72: (3,2)
  76: (2,1)
  80: (4,1)
  84: (2,1,1)
  88: (3,1)
  90: (1,2,1)
  92: (2,1)

Complement of A329138.
Binary necklaces are A000031.
Non-necklace compositions are A329145.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is periodic are A329140.
Cf. A001037, A008965, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329139.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],#==1||!neckQ[Last/@FactorInteger[#]]&]

A362619 One and all numbers whose greatest prime factor is a mode, meaning it appears at least as many times as each of the others.

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, May 09 2023

nonn

First differs from A304678 in having 300.

			The prime factorization of 300 is 2*2*3*5*5, with modes {2,5} and maximum 5, so 300 is in the sequence.

Wikipedia, Mode (statistics).

Partitions of this type are counted by A171979.
The case of a unique mode is A362616, counted by A362612.
The complement is A362620, counted by A240302.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with a more than one mode, counted by A362607.
A362606 ranks partitions with a more than one co-mode, counted by A362609.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
A362621 ranks partitions with median equal to maximum, counted by A053263.
Cf. A000040, A002865, A237824, A237984, A327473, A327476, A359908.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],MemberQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

A304679 A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).

Original entry on oeis.org

3, 4, 6, 18, 450, 205439850, 241382525361273331926149714645357743772646450

Offset: 0

Gus Wiseman, May 16 2018

nonn

The first entry 3 is optional so has offset 0.

			The list of multisets with Heinz numbers in the sequence is A014643. The number of k's in row n + 1 is equal to the k-th term of row n. The length of row n is A014644(n).
        3: {2}
        4: {1,1}
        6: {1,2}
       18: {1,2,2}
      450: {1,2,2,3,3}
205439850: {1,2,2,3,3,4,4,4,5,5,5}

Cf. A001462, A014643, A014644, A055932, A056239, A112798, A130091, A181819, A181821, A182850-A182858, A296150, A275870, A304455, A304678.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{#[[i]]}],{i,Length[#]}]&,{2},6]

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

A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-up partitions.
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 first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 60 are {1,1,2,3}, with first quotients (1,2,3/2), so 60 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}

Robert Price, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A304678.
For differences instead of quotients we have A325360 (count: A240026).
These partitions are counted by A342523 (strict: A342516, ordered: A342492).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342086 counts strict chains of divisors with strictly increasing quotients.
Cf. A000005, A002843, A056239, A067824, A112798, A124010, A130091, A238710, A253249, A325351, A325352, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],LessEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

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]]]]&]

A317082 Number of integer partitions of n whose multiplicities are weakly decreasing and span an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 8, 9, 13, 17, 22, 26, 35, 42, 53, 66, 81, 96, 122, 143, 174, 210, 251, 293, 358, 417, 493, 582, 686, 793, 941, 1087, 1267, 1471, 1709, 1961, 2285, 2615, 3013, 3460, 3976, 4523, 5204, 5914, 6747, 7681, 8745, 9884, 11262, 12714, 14393, 16261

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

			The a(7) = 8 integer partitions are (7), (61), (52), (511), (43), (421), (322), (3211).

Cf. A000041, A000837, A055932, A304678, A317081, A317084, A317085, A317087, A317090.

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],And[normalQ[Length/@Split[#]],OrderedQ[Length/@Split[#]]]&]],{n,60}]

A353745 Number of runs in the ordered prime signature of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1

Offset: 1

Gus Wiseman, May 20 2022

nonn

First differs from A071625 at a(90) = 3.
First differs from A331592 at a(90) = 3.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The prime indices of 630 are {1,2,2,3,4}, with multiplicities {1,2,1,1}, with runs {{1},{2},{1,1}}, so a(630) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are A354233.
A001222 counts prime factors, distinct A001221.
A005361 gives product of prime signature, firsts A353500/A085629.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A182850/A323014 give frequency depth, counted by A225485/A325280.
Cf. A005811, A097318, A130091, A304678, A333755, A353503, A353507, A353742.
Cf. also A329747.

Table[Length[Split[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i < #lista) && (lista[i]==lista[i+1]), rl++, listput(runs,rl); rl=1)); (runs)));
A353745(n) = #runlengths(runlengths(pis_to_runs(n))); \\ Antti Karttunen, Jan 20 2025

A328870 Numbers whose lengths of runs of 1's in their reversed binary expansion are not weakly increasing.

Original entry on oeis.org

11, 19, 22, 23, 35, 38, 39, 43, 44, 45, 46, 47, 55, 67, 70, 71, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 103, 107, 110, 111, 131, 134, 135, 139, 140, 141, 142, 143, 147, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 163, 166, 167

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

			The sequence of terms together with their reversed binary expansions begins:
  11: (1101)
  19: (11001)
  22: (01101)
  23: (11101)
  35: (110001)
  38: (011001)
  39: (111001)
  43: (110101)
  44: (001101)
  45: (101101)
  46: (011101)
  47: (111101)
  55: (111011)
  67: (1100001)
  70: (0110001)
  71: (1110001)
  75: (1101001)
  76: (0011001)
  77: (1011001)
  78: (0111001)

Complement of A328869.
The version for prime indices is A112769.
The binary expansion of n has A069010(n) runs of 1's.
Cf. A000120, A003714, A014081, A164707, A245563, A304678, A328592.

Select[Range[100],!LessEqual@@Length/@Split[Join@@Position[Reverse[IntegerDigits[#,2]],1],#2==#1+1&]&]

cp's OEIS Frontend

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

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A329142 Numbers whose prime signature is not a necklace.

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A362619 One and all numbers whose greatest prime factor is a mode, meaning it appears at least as many times as each of the others.

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A304679 A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).

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A316529 Heinz numbers of totally strong integer partitions.

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A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

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

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A317082 Number of integer partitions of n whose multiplicities are weakly decreasing and span an initial interval of positive integers.

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A353745 Number of runs in the ordered prime signature of n.

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