A133808 -id:A133808 - cp's OEIS frontend

A304678 Numbers with weakly increasing prime multiplicities.

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 16 2018

nonn

Complement of A112769.

			12 = 2*2*3 has prime multiplicities (2,1) so is not in the sequence.
36 = 2*2*3*3 has prime multiplicities (2,2) so is in the sequence.
150 = 2*3*5*5 has prime multiplicities (1,1,2) so is in the sequence.

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

Cf. A001221, A001222, A071365, A112769, A130091, A133808, A133811, A242031, A304465, A304679, A304687.

q:= n-> (l-> (t-> andmap(i-> l[i, 2]<=l[i+1, 2],
        [$1..t-1]))(nops(l)))(sort(ifactors(n)[2])):
select(q, [$1..120])[];  # Alois P. Heinz, Nov 11 2019

Select[Range[200],OrderedQ[FactorInteger[#][[All,2]]]&]
Select[Range[90],Min[Differences[FactorInteger[#][[;;,2]]]]>=0&] (* Harvey P. Dale, Jan 28 2024 *)

isok(n) = my(vm = factor(n)[,2]); vm == vecsort(vm); \\ Michel Marcus, May 17 2018

A304660 A run-length describing inverse to A181819. The multiplicity of prime(k) in a(n) is the k-th smallest prime index of n, which is A112798(n,k).

Original entry on oeis.org

1, 2, 4, 6, 8, 18, 16, 30, 36, 54, 32, 150, 64, 162, 108, 210, 128, 450, 256, 750, 324, 486, 512, 1470, 216, 1458, 900, 3750, 1024, 2250, 2048, 2310, 972, 4374, 648, 7350, 4096, 13122, 2916, 10290, 8192, 11250, 16384, 18750, 4500, 39366, 32768, 25410, 1296

Offset: 1

Gus Wiseman, May 16 2018

nonn

A permutation of A133808. a(n) is the smallest member m of A133808 such that A181819(m) = n.

			Sequence of normalized prime multisets together with the normalized prime multisets of their images begins:
   1:        {} -> {}
   2:       {1} -> {1}
   3:       {2} -> {1,1}
   4:     {1,1} -> {1,2}
   5:       {3} -> {1,1,1}
   6:     {1,2} -> {1,2,2}
   7:       {4} -> {1,1,1,1}
   8:   {1,1,1} -> {1,2,3}
   9:     {2,2} -> {1,1,2,2}
  10:     {1,3} -> {1,2,2,2}
  11:       {5} -> {1,1,1,1,1}
  12:   {1,1,2} -> {1,2,3,3}
  13:       {6} -> {1,1,1,1,1,1}
  14:     {1,4} -> {1,2,2,2,2}
  15:     {2,3} -> {1,1,2,2,2}
  16: {1,1,1,1} -> {1,2,3,4}
  17:       {7} -> {1,1,1,1,1,1,1}
  18:   {1,2,2} -> {1,2,2,3,3}

Cf. A055932, A056239, A112798, A130091, A133808, A181819, A181821, A182850, A182857, A275870, A304455.

Table[With[{y=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},Times@@Power[Array[Prime,Length[y]],y]],{n,100}]

a(n) = Product_{i = 1..Omega(n)} prime(i)^A112798(n,i).

A317090 Positive integers whose prime multiplicities span an initial interval of positive integers.

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, Jul 21 2018

nonn

The first term in this sequence but absent from A179983 is 180.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 6, 78, 820, 8379, 84440, 846646, 8473868, 84763404, 847714834, 8477408261, ... . Apparently, the asymptotic density of this sequence exists and equals 0.8477... . - Amiram Eldar, Aug 04 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Cf. A001222, A007916, A055932, A056239, A069799, A112798, A124010, A133808.
Cf. A317087, A317088, A317089, A317091, A317092.
Subsequences: A129912\{1}, A179983\{1}.
Subsequence of A337533.

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

is(k) = {my(e = Set(factor(k)[,2])); k > 1 && vecmax(e) == #e;} \\ Amiram Eldar, Aug 04 2024

A087980 Numbers with strictly decreasing prime exponents.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 72, 96, 128, 144, 192, 256, 288, 360, 384, 432, 512, 576, 720, 768, 864, 1024, 1152, 1440, 1536, 1728, 2048, 2160, 2304, 2592, 2880, 3072, 3456, 4096, 4320, 4608, 5184, 5760, 6144, 6912, 8192, 8640, 9216, 10368, 10800

Offset: 1

Rainer Rosenthal, Oct 27 2003

This representation provides a natural ordering between strictly decreasing sequences of natural numbers. Let f and g be such sequences with f(1) > f(2) > ... > f(m) and g(1) > g(2) > ... > g(n). Define f < g iff p^f < p^g, where p^f is short for Product(i=1..m) p_i^f(i) and p^g is defined likewise as Product(i=1..n) p_i^g(i).
Note that "strictly decreasing sequences of natural numbers" is another way to say "partitions into distinct parts".
Also products of primorial numbers p_1#^k_1 * p_2#^k_2 * ... * p_n#^k_n where all k_i > 0.
A124010(a(n),k+1) < A124010(a(n),k), 1 <= k < A001221(a(n)). - Reinhard Zumkeller, Apr 13 2015
Numbers whose prime indices cover an initial interval of positive integers with strictly decreasing multiplicities. Intersection of A055932 and A304686. First differs from A181818 in having 72. - Gus Wiseman, Oct 21 2022

			The sequence starts with a(1)=1, a(2)=2, a(3)=4 and a(4)=8. The next term is a(5)=12 = 2^2*3^1 = p_1^k_1 * p_2^k_2 with k_1=2 > k_2=1.

T. D. Noe, Table of n, a(n) for n = 1..1000

The weak (weakly decreasing) version is A025487.
The weak opposite (weakly increasing) version is A133808.
The opposite (strictly increasing) version is A133809.
For strictly decreasing prime signature we have A304686.
Cf. A000009, A000040, A002110, A027748, A055932, A124010, A133813, A317791, A328524, A357864.

import Data.List (isPrefixOf)
a087980 n = a087980_list !! (n-1)
a087980_list = 1 : filter f [2..] where
   f x = isPrefixOf ps a000040_list && all (< 0) (zipWith (-) (tail es) es)
         where ps = a027748_row x; es = a124010_row x
-- Reinhard Zumkeller, Apr 13 2015

selQ[k_] := Module[{n = k, e = IntegerExponent[k, 2], t}, n /= 2^e; For[p = 3, True, p = NextPrime[p], t = IntegerExponent[n, p]; If[t == 0, Return[n == 1]]; If[t >= e, Return[False]]; e = t; n /= p^e]];
Select[Range[12000], selQ] (* Jean-François Alcover, Mar 27 2020, after first PARI program *)

is(n)=my(e=valuation(n,2),t); n>>=e; forprime(p=3,, t=valuation(n,p); if(t==0, return(n==1)); if(t>=e, return(0)); e=t; n/=p^e) \\ Charles R Greathouse IV, Jun 25 2017

list(lim)=my(v=[],u=powers(2,logint(lim\=1,2)),w,p=2,t); forprime(q=3,, w=List(); for(i=1,#u, t=u[i]; for(e=1,valuation(u[i],p)-1, t*=q; if(t>lim, break); listput(w,t))); v=concat(v,Vec(u)); if(#w==0, break); u=w; p=q); Set(v) \\ Charles R Greathouse IV, Jun 25 2017

The numbers of the form Product(i=1..n) p_i^k_i where p_i = A000040(i) is the i-th prime and k_1 > k_2 > ... > k_n are positive natural numbers.

Compute x = 2^k_1 * 3^k_2 * 5^k_3 * 7^k_4 * 11^k_5 for k_1 > ... > k_5 allowing k_i = 0 for i > 1 and k_i = k_(i+1) in that case. Discard all x > 174636000 = 2^5*3^4*5^3*7^2*11 and enumerate those below. For more members take higher primes into account.

Edited by Franklin T. Adams-Watters, Apr 25 2006

Offset change to 1 by T. D. Noe, May 24 2010

A304686 Numbers with strictly decreasing prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121

Offset: 1

Gus Wiseman, May 16 2018

nonn

			10 = 2*5 has prime multiplicities (1,1) so is not in the sequence.
20 = 2*2*5 has prime multiplicities (2,1) so is in the sequence
90 = 2*3*3*5 has prime multiplicities (1,2,1) so is not in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A001597, A005117, A055932, A071365, A112769, A130091, A133808, A133811, A181819, A242031, A304678, A304679, A283050.

Select[Range[200],Greater@@FactorInteger[#][[All,2]]&]

isok(n) = my(vm = factor(n)[,2]); vm == vecsort(vm,,4) && (#vm == #Set(vm)); \\ Michel Marcus, May 17 2018

list(lim)=my(v=List()); forfactored(n=1,lim\1, if(n[2][,2]==vecsort(n[2][,2],,8), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Oct 28 2021

a(n) ~ n log n. - Charles R Greathouse IV, Oct 28 2021

A316496 Number of totally strong integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 13, 18, 20, 27, 27, 38, 41, 52, 56, 73, 77, 99, 105, 129, 145, 176, 186, 229, 253, 300, 329, 395, 427, 504, 555, 648, 716, 836, 905, 1065, 1173, 1340, 1475, 1703, 1860, 2140, 2349, 2671, 2944, 3365, 3666, 4167, 4582, 5160, 5668

Offset: 0

Gus Wiseman, Jul 29 2018

nonn

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 a(1) = 1 through a(8) = 12 totally strong partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.

Cf. A181819, A182850, A182857, A304660, A305563, A316597.
The Heinz numbers of these partitions are A316529.
The version for compositions is A332274.
The dual version is A332275.
The version for reversed partitions is (also) A332275.
The narrowly normal version is A332297.
The alternating version is A332339 (see also A317256).
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100882, A133808, A317245, A317491, A332278, A332291, A332336.

totincQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]

Updated with corrected terminology by Gus Wiseman, Mar 07 2020

A317089 Numbers whose prime factors span an initial interval of prime numbers and whose prime multiplicities span an initial interval of positive integers.

Original entry on oeis.org

2, 6, 12, 18, 30, 60, 90, 150, 180, 210, 300, 360, 420, 450, 540, 600, 630, 1050, 1260, 1350, 1470, 1500, 2100, 2250, 2310, 2520, 2940, 3150, 3780, 4200, 4410, 4620, 5880, 6300, 6930, 7350, 8820, 9450, 10500, 11550, 12600, 13230, 13860, 14700, 15750, 16170

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

			The sequence of rows of A296150 indexed by the terms of this sequence begins: (1), (21), (211), (221), (321), (3211), (3221), (3321), (32211), (4321), (33211), (322111), (43211).

Andrew Howroyd, Table of n, a(n) for n = 1..300

Cf. A001222, A007916, A055932, A056239, A124010, A133808, A242414, A296150.

Cf. A317087, A317088, A317090, A317091, A317092.

normalQ[m_]:=Union[m]==Range[Max[m]];
Select[Range[10000],And[normalQ[PrimePi/@FactorInteger[#][[All,1]]],normalQ[FactorInteger[#][[All,2]]]]&]

ok(n)={my(f=factor(n), p=f[,1], e=vecsort(f[,2],,8)); n > 1 && #p==primepi(p[#p]) && #e==e[#e]} \\ Andrew Howroyd, Aug 26 2018

A317087 Numbers whose prime factors span an initial interval of prime numbers and whose sequence of prime multiplicities is a palindrome.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 30, 32, 36, 64, 90, 128, 210, 216, 256, 270, 300, 512, 810, 900, 1024, 1296, 2048, 2310, 2430, 2700, 2940, 3000, 3150, 4096, 7290, 7776, 8100, 8192, 9000, 11550, 16384, 21870, 24300, 27000, 30000, 30030, 32768, 41160, 44100, 46656, 47250, 48510

Offset: 1

Gus Wiseman, Jul 21 2018

nonn

3^m*10^k for k, m > 0 are terms of this sequence. - Chai Wah Wu, Jun 23 2020

			The sequence of rows of A296150 indexed by the terms of this sequence begins: (1), (11), (21), (111), (1111), (321), (11111), (2211), (111111), (3221), (1111111), (4321), (222111), (11111111), (32221), (33211), (111111111), (322221), (332211).

Giovanni Resta, Table of n, a(n) for n = 1..10000
Wikipedia, Palindrome

Cf. A025065, A055932, A124010, A133808, A242414, A296150, A317085, A317086.

nrmpalQ[n_]:=With[{f=If[n==1,{},FactorInteger[n]]}, And[PrimePi/@ Sort[First/@f] == Range[ Length[f]], Reverse[Last/@f] == Last/@f]]; Select[Range[100],nrmpalQ]
upto = 10^20; pL[n_] := Block[{p = Prime@Range@n, h = Ceiling[n/2]}, Take[p, h] Reverse@ If[n == 2 h, Take[p, -h], Prepend[ Take[p, 1-h], 1]]]; ric[v_, p_] := If[p == {}, AppendTo[L, v], Block[{w = v}, While[w <= upto, ric[w, Rest@ p]; w *= First@ p]]]; np = 1; L = {1}; While[(b = Times @@ Prime[Range@ np]) <= upto, ric[b, pL[np++]]]; Sort[L] (* Giovanni Resta, Jun 23 2020 *)

from sympy import factorint, primepi
A317087_list = [1]
for n in range(1,10**5):
    d = factorint(n)
    k, l = sorted(d.keys()), len(d)
    if l > 0 and l == primepi(max(d)):
        for i in range(l//2):
            if d[k[i]] != d[k[l-i-1]]:
                break
        else:
            A317087_list.append(n) # Chai Wah Wu, Jun 23 2020

A332289 Number of widely alternately co-strongly normal integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 13 2020

nonn

An integer partition is widely alternately co-strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

			The a(1) = 1, a(3) = 2, and a(10) = 3 partitions:
  (1)  (21)   (4321)
       (111)  (322111)
              (1111111111)
For example, starting with y = (4,3,2,2,1,1,1) and repeatedly taking run-lengths and reversing gives y -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal, have weakly increasing run-lengths, and the last is all 1's, so y is counted a(14).

Normal partitions are A000009.
Dominated by A317245.
The non-co-strong version is A332277.
The total (instead of alternate) version is A332278.
The Heinz numbers of these partitions are A332290.
The strong version is A332292.
The case of reversed partitions is (also) A332292.
The generalization to compositions is A332340.
Cf. A100883, A107429, A133808, A316496, A317081, A317256, A317491, A329746, A332291, A332295, A332297, A332337, A332338, A332339.

totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[IntegerPartitions[n],totnQ]],{n,0,30}]

A332278 Number of widely totally co-strongly normal integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 05 2020

A sequence of integers is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal sequence.

Is this sequence bounded?

			The a(1) = 1 through a(20) = 2 partitions:
   1: (1)
   2: (11)
   3: (21),(111)
   4: (211),(1111)
   5: (11111)
   6: (321),(111111)
   7: (1111111)
   8: (11111111)
   9: (32211),(111111111)
  10: (4321),(322111),(1111111111)
  11: (11111111111)
  12: (111111111111)
  13: (1111111111111)
  14: (11111111111111)
  15: (54321),(111111111111111)
  16: (1111111111111111)
  17: (11111111111111111)
  18: (111111111111111111)
  19: (1111111111111111111)
  20: (4332221111),(11111111111111111111)

Not requiring co-strength gives A332277.
The strong version is A332297(n) - 1 for n > 1.
The narrow version is a(n) - 1 for n > 1.
The alternating version is A332289.
The Heinz numbers of these partitions are A332293.
The case of compositions is A332337.
Cf. A000009, A100883, A107429, A133808, A181819, A316496, A317245, A317491, A329746, A332279, A332290, A332291, A332292, A332296, A332576.

totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n],totnQ]],{n,0,30}]

a(71)-a(78) from Jinyuan Wang, Jun 26 2020

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A304678 Numbers with weakly increasing prime multiplicities.

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A304660 A run-length describing inverse to A181819. The multiplicity of prime(k) in a(n) is the k-th smallest prime index of n, which is A112798(n,k).

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A317090 Positive integers whose prime multiplicities span an initial interval of positive integers.

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A087980 Numbers with strictly decreasing prime exponents.

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A304686 Numbers with strictly decreasing prime multiplicities.

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A316496 Number of totally strong integer partitions of n.

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A317089 Numbers whose prime factors span an initial interval of prime numbers and whose prime multiplicities span an initial interval of positive integers.

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