A124829 -id:A124829 - 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

A124832 Table of exponents of prime factorizations in A025487.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 4, 3, 1, 1, 1, 1, 5, 2, 2, 4, 1, 2, 1, 1, 6, 3, 2, 5, 1, 3, 1, 1, 7, 4, 2, 2, 2, 1, 6, 1, 1, 1, 1, 1, 3, 3, 4, 1, 1, 8, 5, 2, 3, 2, 1, 7, 1, 2, 1, 1, 1, 4, 3, 5, 1, 1, 9, 6, 2, 4, 2, 1, 8, 1, 3, 1, 1, 1, 5, 3, 2, 2, 2, 6, 1, 1, 10, 3, 3, 1, 7, 2, 2, 2, 1, 1, 4, 4, 5, 2, 1, 9, 1, 4, 1, 1, 1, 6

Offset: 2

Franklin T. Adams-Watters, Nov 09 2006

This is an enumeration of all partitions.

			From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts as follows:
  n : signature   (A025487(n) = factorization)
  1 : []          (1 = empty product)
  2 : [1]         (2 = 2^1)
  3 : [2]         (4 = 2^2)
  4 : [1, 1]      (6 = 2^1 * 3^1)
  5 : [3]         (8 = 2^3)
  6 : [2, 1]      (12 = 2^2 * 3^1)
  7 : [4]         (16 = 2^4)
  8 : [3,1]       (24 = 2^3 * 3^1)
  9 : [1, 1, 1]   (30 = 2^1 * 3^1 * 5^1)
  etc. (End)

Michael De Vlieger, Table of n, a(n) for n = 2..10820 (rows 2 <= n <= 2500, first 106 terms from Ray Chandler).

Cf. A025487, A036041 (row sums), A061394 (row lengths), A124829, A036036, A080577.

Map[FactorInteger[#][[All, -1]] &, Import["https://oeis.org/A025487/b025487.txt", "Data"][[2 ;; 48, -1]] ] // Flatten (* Michael De Vlieger, Feb 06 2020 *)

A124832_row(n)=factor(A025487(n))[,2] \\ M. F. Hasler, Oct 12 2018

A025487(n) = Product_{k=1..A061394(n)} prime(k)^T(n,k). [Edited by M. F. Hasler, Oct 12 2018]

Erroneous explanations in cross-references corrected by M. F. Hasler, Oct 12 2018

A124830 Number of distinct prime factors of A055932(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 3, 2, 4, 2, 3, 1, 3, 2, 3, 2, 3, 2, 4, 2, 3, 3, 2, 1, 3, 2, 3, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 2, 1, 4, 3, 2, 3, 4, 2, 3, 3, 2, 4, 3, 2, 3, 4, 2, 3, 4, 3, 2, 1, 4, 3, 3, 2, 5, 3, 3, 4, 2, 3, 3, 2, 4, 3, 2, 4, 3, 4, 2, 3, 3, 4, 3, 2, 3, 1, 4

Offset: 1

Franklin T. Adams-Watters, Nov 09 2006

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 1000 terms from G. C. Greubel.)

Cf. A055932, A001221, A124829, A124831, A061394.

PrimeNu /@ Select[Range[4000], ! MemberQ[Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #, 0] &] (* Michael De Vlieger, Feb 02 2017 *)
A055932[n_] := Module[{f = Transpose[FactorInteger[n]][[1]]}, f == {1} || f == Prime[Range[Length[f]]]]; PrimeNu[Select[Range[2000], A055932]] (* G. C. Greubel, May 11 2017 *)

from sympy import nextprime, primefactors
def a053669(n):
    p = 2
    while True:
        if n%p!=0: return p
        else: p=nextprime(p)
def ok(n): return True if n==1 else a053669(n)>max(primefactors(n))
print([len(primefactors(n)) for n in range(1, 10001) if ok(n)]) # Indranil Ghosh, May 11 2017

a(n) = A001221(A055932(n)).

A124831 Number of prime factors of A055932(n) with repetition.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 3, 5, 4, 5, 4, 4, 6, 5, 4, 6, 5, 5, 7, 6, 4, 5, 5, 7, 4, 6, 6, 8, 5, 7, 5, 6, 6, 8, 5, 7, 5, 7, 6, 9, 6, 8, 6, 5, 7, 7, 5, 9, 6, 6, 8, 6, 8, 7, 10, 5, 7, 9, 7, 6, 8, 6, 8, 7, 5, 6, 10, 7, 7, 9, 7, 6, 9, 8, 11, 6, 8, 6, 10, 5, 8, 7, 7, 9, 7, 9, 8, 6, 7, 11, 6, 8, 8, 10, 8, 6, 7, 10

Offset: 0

Franklin T. Adams-Watters, Nov 09 2006

nonn

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

Cf. A055932, A001222, A124829, A124831, A036041.

Map[PrimeOmega, {1}~Join~Select[Range[10^4], Last[#] == Length[#] &@ PrimePi@ FactorInteger[#][[All, 1]] &]] (* Michael De Vlieger, Feb 06 2020 *)

a(n) = A001222(A055932(n)).

A331049 Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 4, 7, 5, 7, 9, 12, 7, 11, 11, 16, 11, 19, 16, 21, 15, 29, 11, 12, 26, 30, 15, 31, 38, 22, 21, 47, 26, 29, 52, 45, 36, 57, 26, 64, 19, 30, 52, 77, 52, 36, 57, 98, 21, 67, 38, 74, 97, 66, 105, 47, 42, 36, 109, 118, 98, 92, 109, 52, 171, 30

Offset: 1

Gus Wiseman, Jan 10 2020

nonn

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. Factorizations are counted by A001055.

The unsorted prime signature of A055932(n) is given by row n of A124829.

			The a(1) = 1 through a(11) = 7 factorizations:
  {}  2  4    6    8      12     16       18     24       30     32
         2*2  2*3  2*4    2*6    2*8      2*9    3*8      5*6    4*8
                   2*2*2  3*4    4*4      3*6    4*6      2*15   2*16
                          2*2*3  2*2*4    2*3*3  2*12     3*10   2*2*8
                                 2*2*2*2         2*2*6    2*3*5  2*4*4
                                                 2*3*4           2*2*2*4
                                                 2*2*2*3         2*2*2*2*2

The sorted-signature version is A050322.
This sequence has range A045782.
Factorizations are A001055.
Cf. A025487, A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length@*facs/@First/@GatherBy[Range[1500],If[#==1,{},Last/@FactorInteger[#]]&]

a(n) = A001055(A055932(n)).

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

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A124832 Table of exponents of prime factorizations in A025487.

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A124830 Number of distinct prime factors of A055932(n).

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A124831 Number of prime factors of A055932(n) with repetition.

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A331049 Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

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