A056808 -id:A056808 - cp's OEIS frontend

A055932 Numbers all of whose prime divisors are consecutive primes starting at 2.

1, 2, 4, 6, 8, 12, 16, 18, 24, 30, 32, 36, 48, 54, 60, 64, 72, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192, 210, 216, 240, 256, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 512, 540, 576, 600, 630, 648, 720, 750, 768, 810, 840, 864, 900, 960, 972

Offset: 1

Leroy Quet, Jul 17 2000

a(n) is also the sorted version of A057335 which is generated recursively using the formula A057335 = A057334 * A057335(repeated), where A057334 = A000040(A000120). - Alford Arnold, Nov 11 2001
Squarefree kernels of these numbers are primorial numbers. See A080404. - Labos Elemer, Mar 19 2003
If u and v are terms then so is u*v. - Reinhard Zumkeller, Nov 24 2004
Except for the initial value a(1) = 1, a(n) gives the canonical primal code of the n-th finite sequence of positive integers, where n = (prime_1)^c_1 * ... * (prime_k)^c_k is the code for the finite sequence c_1, ..., c_k. See examples of primal codes at A106177. - Jon Awbrey, Jun 22 2005
From Daniel Forgues, Jan 24 2011: (Start)
Least integer, in increasing order, of each ordered prime signature.
The least integer of each ordered prime signature are the smallest numbers with a given tuple of exponents of prime factors.
The ordered prime signature (where the order of exponents matters) of n corresponds to a given composition of Omega(n), as opposed to the prime signature of n, which corresponds to a given partition of Omega(n). (End)
Except for the initial entry 1, the entries of the sequence are the Heinz numbers of all partitions that contain all parts 1,2,...,k, where k is the largest part. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,2,4,10] the Heinz number is 2*2*3*7*29 = 2436. The number 150 (= 2*3*5*5) is in the sequence because it is the Heinz number of the partition [1,2,3,3]. - Emeric Deutsch, May 22 2015
Numbers n such that A053669(n) > A006530(n). - Anthony Browne, Jun 06 2016
From David W. Wilson, Dec 28 2018: (Start)
Numbers n such that for primes p > q, p | n => q | n.
Numbers n such that prime p | n => A034386(p) | n. (End)

			60 is included because 60 = 2^2 * 3 * 5 and 2, 3 and 5 are consecutive primes beginning at 2.
Sequence A057335 begins
1..2..4..6..8..12..18..30..16..24..36..60..54..90..150..210... which is equal to
1..2..2..3..2...3...3...5...2...3...3...5...3...5....5....7... times
1..1..2..2..4...4...6...6...8...8..12..12..18..18...30...30...

Michael De Vlieger, Table of n, a(n) for n = 1..10000, first 1001 terms from Franklin T. Adams-Watters.
Jon Awbrey, Riffs and Rotes.
Michael De Vlieger, Extended table of n, a(n) for n = 1..100000.
Robert Vajda, Computational Exploration of the Degree Sequence of the Malyshev Polynomials, Proceedings of the 11th International Conference on Applied Informatics (Eger, Hungary, 2020).
Index entries for sequences related to prime signature.

Cf. A057335 (permuted), A056808, A025487, A007947, A002110, A080404, A034386, A106177, A124829, A124830, A124831, A124833, A080259 (complement), A215366.

Cf. A005867, A324939, A345974.

[1] cat [k:k in[2..1000 by 2]|forall{i:i in [1..#PrimeDivisors(k)-1]|NextPrime(pd[i]) in pd where pd is PrimeDivisors(k)}]; // Marius A. Burtea, Feb 01 2020

isA055932 := proc(n)
    local s,p ;
    s := numtheory[factorset](n) ;
    for p in s do
        if p > 2 and not prevprime(p)  in s then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 2 to 100 do
    if isA055932(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 02 2012

Select[Range[1000], #==1||FactorInteger[ # ][[ -1, 1]]==Prime[Length[FactorInteger[ # ]]]&]
cpQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},f=={1}||f==Prime[ Range[Length[f]]]]; Select[Range[1000],cpQ] (* Harvey P. Dale, Jul 14 2012 *)

is(n)=my(f=factor(n)[,1]~);f==primes(#f) \\ Charles R Greathouse IV, Aug 22 2011

list(lim,p=2)=my(v=[1],q=nextprime(p+1),t=1);while((t*=p)<=lim,v=concat(v,t*list(lim\t,q))); vecsort(v) \\ Charles R Greathouse IV, Oct 02 2012

from itertools import count, islice
from sympy import primepi, primefactors
def A055932_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        p = list(map(primepi,primefactors(k)))
        if k==1 or (min(p)==1 and max(p)==len(p)):
            yield k
A055932_list = list(islice(A055932_gen(),40)) # Chai Wah Wu, Aug 07 2025

Sum_{n>=1} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

Edited by Daniel Forgues, Jan 24 2011

A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 21, 49, 106, 226, 470, 968, 1971, 3995, 8057, 16208, 32537, 65239, 130687, 261654, 523661, 1047784, 2096150, 4193049, 8387033, 16775258, 33551996, 67105854, 134214010, 268430891, 536865308, 1073734982, 2147475299, 4294957153, 8589922282

Offset: 0

Alford Arnold, Aug 29 2000

Previous name was: Counts members of A056808 by number of factors.
A056808 relates to least prime signatures (cf. A025487)
a(n) is also the number of compositions of n that are not partitions of n. - Omar E. Pol, Jan 31 2009, Oct 14 2013
a(n) is the number of compositions of n into positive parts containing pattern [1,2]. - Bob Selcoe, Jul 08 2014

			A011782 begins     1 1 2 4 8 16 32 64 128 256 ...;
A000041 begins     1 1 2 3 5  7 11 15  22  30 ...;
so sequence begins 0 0 0 1 3  9 21 49 106 226 ... .
For n = 3 the factorizations are 8=2*2*2, 12=2*2*3, 18=2*3*3 and 30=2*3*5.
a(5) = 9: {[1,1,1,2], [1,1,2,1], [1,1,3], [1,2,1,1], [1,2,2], [1,3,1], [1,4], [2,1,2], [2,3]}. - _Bob Selcoe_, Jul 08 2014

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A025487, A056808, A117989.
The version for patterns is A002051.
(1,2)-avoiding compositions are just partitions A000041.
The (1,1)-matching version is A261982.
The version for prime indices is A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns matched by compositions are counted by A335456.
Cf. A011782, A056986, A106356, A144300, A238279, A269134, A333755.

a:= n-> ceil(2^(n-1))-combinat[numbpart](n):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 30 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!GreaterEqual@@#&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)
a[n_] := If[n == 0, 0, 2^(n-1) - PartitionsP[n]];
a /@ Range[0, 37] (* Jean-François Alcover, May 23 2021 *)

a(n) = A011782(n) - A000041(n).
a(n) = 2*a(n-1) + A117989(n-1). - Bob Selcoe, Apr 11 2014
G.f.: (1 - x) / (1 - 2*x) - Product_{k>=1} 1 / (1 - x^k). - Ilya Gutkovskiy, Jan 30 2020

More terms from James Sellers, Aug 31 2000

New name from Joerg Arndt, Sep 02 2013

A057335 a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 60, 54, 90, 150, 210, 32, 48, 72, 120, 108, 180, 300, 420, 162, 270, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 240, 216, 360, 600, 840, 324, 540, 900, 1260, 1500, 2100, 2940, 4620, 486, 810, 1350, 1890, 2250, 3150, 4410

Offset: 0

Alford Arnold, Aug 27 2000

Note that for n>0 the prime divisors of a(n) are consecutive primes starting with 2. All of the least prime signatures (A025487) are included; with the other values forming A056808.
Using the formula, terms of b(n)= a(n)/A057334(n) are: 1, 1, 2, 2, 4, 4, 6, 6, 8, ..., indeed a(n) repeated. - Michel Marcus, Feb 09 2014
a(n) is the unique normal number whose unsorted prime signature is the k-th composition in standard order (graded reverse-lexicographic). This composition (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. A number is normal if its prime indices cover an initial interval of positive integers. Unsorted prime signature is the sequence of exponents in a number's prime factorization. - Gus Wiseman, Apr 19 2020

			From _Gus Wiseman_, Apr 19 2020: (Start)
The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     12: {1,1,2}
     18: {1,2,2}
     30: {1,2,3}
     16: {1,1,1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     60: {1,1,2,3}
     54: {1,2,2,2}
     90: {1,2,2,3}
    150: {1,2,3,3}
    210: {1,2,3,4}
     32: {1,1,1,1,1}
     48: {1,1,1,1,2}
For example, the 27th composition in standard order is (1,2,1,1), and the normal number with prime signature (1,2,1,1) is 630 = 2*3*3*5*7, so a(27) = 630.
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000120, A057334, A055932 and A056808.
Cf. A324939.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
The reversed version is A334031.
A partial inverse is A334032.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
Cf. A005867, A029931, A048793, A052409, A056239, A066099, A112798, A124767, A228351, A233249, A333220, A345974.
Related to A019565 via A122111 and to A000079 via A336321.

Table[Times @@ Map[If[# == 0, 1, Prime@ #] &, Accumulate@ IntegerDigits[n, 2]], {n, 0, 54}] (* Michael De Vlieger, May 23 2017 *)

mg(n) = if (n==0, 1, prime(hammingweight(n))); \\ A057334
lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); v;} \\ Michel Marcus, Feb 09 2014

A057335(n) = if(0==n,1,prime(hammingweight(n))*A057335(n\2)); \\ Antti Karttunen, Jul 20 2020

a(n) = A057334(n) * a (repeated).
A334032(a(n)) = n; a(A334032(n)) = A071364(n). - Gus Wiseman, Apr 19 2020
a(n) = A122111(A019565(n)); A019565(n) = A122111(a(n)). - Peter Munn, Jul 18 2020
a(n) = A336321(2^n). - Peter Munn, Mar 04 2022
Sum_{n>=0} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - Amiram Eldar, Jun 26 2025

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

New primary name from Antti Karttunen, Jul 20 2020

A369361 Numbers that are not products of primorials.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Michael De Vlieger, Jan 21 2024

Differs from A080259 in that this sequence includes 18.
Differs from A093373 in that this sequence does not include 30.
Primorials P(n) = A002110(n) are not in this sequence.
Does not contain powers of 2.
Contains odd numbers k > 1.
Numbers that are not the first with their prime signature, complement of A025487. - Peter Munn, Jan 24 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000079, A002110, A005408, A025487, A046523, A056808, A080259, A093373.

Select[Range[210], Nor[# == 1, IntegerQ@ Log2[#], And[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, AllTrue[Differences@ #2[[All, -1]], # <= 0 &]]] & @@ {#, FactorInteger[#]} &]

{a(n)} = A000027 \ A025487.
Union of A080259 and A056808, superset of both.
Superset of A005408 \ {1}.
{a(n)} = {k : A046523(k) < k}. - Peter Munn, Jan 24 2024

A363814 Intersection of A126706 and A055932.

Original entry on oeis.org

12, 18, 24, 36, 48, 54, 60, 72, 90, 96, 108, 120, 144, 150, 162, 180, 192, 216, 240, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 540, 576, 600, 630, 648, 720, 750, 768, 810, 840, 864, 900, 960, 972, 1050, 1080, 1152, 1200, 1260, 1296, 1350, 1440, 1458

Offset: 1

Michael De Vlieger, Dec 18 2023

Products m*P(i) of primorials P(i) = A002110(i) such that rad(m) | P(i), i > 1, m > 1, where rad(m) = A007947(m).

			Sequence contains terms k > 1 in {6 * A003586} since all are divisible by P(2) = 6 and by no prime q that does not divide 6. Therefore 12, 18, 24, etc. are in the sequence.
Sequence does not contain k > 1 in {10 * A003592} since such k are divisible by 5 but not 3. Hence, 20, 40, etc. are not in this sequence.
Sequence does not contain k > 1 in {15 * A003593} since such k are odd. Hence, 45, 135, etc. are not in this sequence, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002110, A007947, A013929, A024619, A055932, A056808, A126706, A364710.

Cf. A003586, A003592, A003593.

Select[Range[12, 1080, 2], And[AnyTrue[#2, # > 1 &], Length[#1] > 1, Union@ Differences@ PrimePi[#1] == {1}] & @@ Transpose@ FactorInteger[#] &]

Union of A056808 and A364710. - Michael De Vlieger, Jan 31 2024

A369419 Numbers k that are neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is not a primorial.

Original entry on oeis.org

18, 90, 150, 630, 1050, 1470, 1890, 2100, 6930, 11550, 16170, 20790, 23100, 25410, 90090, 150150, 210210, 270270, 300300, 330330, 390390, 420420, 450450, 1531530, 2552550, 3573570, 4594590, 5105100, 5615610, 6636630, 7147140, 7657650, 8678670, 9189180, 29099070

Offset: 1

Michael De Vlieger, Mar 10 2024

nonn

			Seen as an irregular triangle T(n,k) of rows n where T(n,k) = P(n)*k, and k < prime(n+1) is in A369361.
n\k    3       5       7       9      10      11
------------------------------------------------
2:    18;
3:    90,    150;
4:   630,   1050,   1470,   1890,   2100;
5:  6930,  11550,  16170,  20790,  23100,  25410;
    ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002110, A003557, A007947, A025487, A053669, A055932, A056808, A060735, A119288, A364998, A369361, A369540, A369541.

P = 2; nn = 8;
s = Select[Range[3, Prime[nn+1]],
  Nor[IntegerQ@ Log2[#],
      And[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},
          AllTrue[Differences@ #2[[All, -1]], # <= 0 &]]] & @@
    {#, FactorInteger[#]} &];
Table[P *= Prime[n]; P*TakeWhile[s, # < Prime[n + 1] &], {n, 2, nn}]

This sequence is { k = m*P(i) : 3 <= m < prime(i), i > 1, m in A369361 }.

Intersection of A364998 and A056808.

A369420 Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.

Original entry on oeis.org

108, 324, 648, 972, 1944, 2700, 2916, 3888, 4500, 5832, 8100, 8748, 9000, 11664, 13500, 16200, 17496, 18000, 22500, 23328, 24300, 26244, 34992, 36000, 40500, 45000, 48600, 52488, 67500, 69984, 72000, 72900, 78732, 81000, 90000, 97200, 104976, 112500, 121500, 132300

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1, prime powers p^m | k are such that m > 1, rad(k) is a primorial, but k is not a product of primorials, where Omega = A001222 and omega = A001221.
Contains no odd numbers as a consequence of being a proper subset of A055932.
Proper subset of A369419, which is in turn a proper subset of A126706.

			36 = 2^2 * 3^2 is a product of primorials, therefore not in the sequence.
72 = 2^3 * 3^2 is not a term because it is a product of primorials.
100 = 2^2 * 5^2 is not in the sequence since it does not have a primorial kernel.
108 = 2^2 * 3*3 is in the sequence since it is not a product of primorials, but its squarefree kernel is 6, a primorial.
144 = 2^4 * 3^2 is not in the sequence since it is a product of primorials, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A002110, A025487, A055932, A056808, A126706, A286708, A364930, A369374.

With[{nn = 2^20},
 Select[
   Select[
     Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
     Not@*PrimePowerQ],
   And[EvenQ[#1],
     Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, !
     AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
     {#, FactorInteger[#]} &] ]

{a(n)} = {A369374 \ A364930}.

Intersection of A056808 and A286708.

A386224 Nonsquarefree weak numbers k that are not products of primorials, whose squarefree kernel is a primorial.

Original entry on oeis.org

18, 54, 90, 150, 162, 270, 300, 450, 486, 540, 600, 630, 750, 810, 1050, 1200, 1350, 1458, 1470, 1500, 1620, 1890, 2100, 2250, 2400, 2430, 2940, 3000, 3150, 3240, 3750, 3780, 4050, 4200, 4374, 4410, 4800, 4860, 5250, 5670, 5880, 6000, 6750, 6930, 7290, 7350, 7500

Offset: 1

Michael De Vlieger, Jul 15 2025

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    18   2 * 3^2
 2    54   2 * 3^3
 3    90   2 * 3^2 * 5
 4   150   2 * 3 * 5^2
 5   162   2 * 3^4
 6   270   2 * 3^3 * 5
 7   300   2^2 * 3 * 5^2
 8   450   2 * 3^2 * 5^2
 9   486   2 * 3^5
10   540   2^2 * 3^3 * 5
11   600   2^3 * 3 * 5^2
12   630   2 * 3^2 * 5 * 7

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A056808, A126706, A286708, A332785, A369420, A380543, A386223.

(* Load May 19 2018 function f at A025487, then run the following: *)
fQ[x_] :=
 Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &,
   Nest[Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &,
     If[x == 1, {0},
       Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@
         Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ x], 2] ] == x;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
Select[Union@ Flatten@ f[6][[3 ;; -1, 2 ;; -1]], Nor[Divisible[#, rad[#]^2], fQ[#]] &]

{a(n)} = A380543 \ A386223.
Intersection of A056808 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694, and A056808 = A055932 \ A025487.
The union of this sequence and A369420 is A126706.

A084918 Numbers n >= 1000, such that if prime P divides n, then so does each smaller prime.

Original entry on oeis.org

1024, 1050, 1080, 1152, 1200, 1260, 1296, 1350, 1440, 1458, 1470, 1500, 1536, 1620, 1680, 1728, 1800, 1890, 1920, 1944, 2048, 2100, 2160, 2250, 2304, 2310, 2400, 2430, 2520, 2592, 2700, 2880, 2916, 2940, 3000, 3072, 3150, 3240, 3360, 3456, 3600, 3750

Offset: 0

Alford Arnold, Jul 15 2003

A055932 lists terms below 1000.

Cf. A055932, A056808, A057335, A066099, A071364, A080259, A080404.

espQ[n_]:=Module[{f=FactorInteger[n][[All,1]]},Prime[Range[ PrimePi[ Max[f]]]] == f]; Select[Range[1000,4000],espQ] (* Harvey P. Dale, Mar 09 2019 *)

Edited by Don Reble, Nov 03 2003

A096011 Arrange numbers n such that A071364(n) <> A046523(n) by prime signature, cf. A071365.

Original entry on oeis.org

18, 54, 50, 90, 250, 75, 108, 126, 375, 98, 150, 500, 198, 686, 147, 162, 294, 1125, 234, 1029, 242, 270, 1250, 490, 1372, 306, 1715, 245, 300, 378, 1875, 726, 3087, 342, 2662, 338, 324, 588, 594, 4802, 735, 5324, 350, 3993, 363, 450, 2500, 980, 702

Offset: 0

Alford Arnold, Jul 20 2004

In the example, the first column is sequence A056808; the first row is A095990.

			The array begins
18 50 75 98 147 ...
54 250 375 686
90 126 198
108 500
150
...

Extended by Ray Chandler, Jul 31 2004

cp's OEIS Frontend

A055932 Numbers all of whose prime divisors are consecutive primes starting at 2.

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A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

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A057335 a(0) = 1, and for n > 0, a(n) = A000040(A000120(n)) * a(floor(n/2)); essentially sequence A055932 generated using A000120, hence sorted by number of factors.

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A369361 Numbers that are not products of primorials.

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A363814 Intersection of A126706 and A055932.

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A369419 Numbers k that are neither squarefree nor prime powers such that A119288(k) <= k/A007947(k) < A053669(k) and A007947(k) is not a primorial.

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A369420 Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.

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