A057334 -id:A057334 - 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

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

cp's OEIS Frontend

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

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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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Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions