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

1,2

Comments

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)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000, first 1001 terms from Franklin T. Adams-Watters.

J. 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

Example

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...

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Cf. A324939.

Sequence in context: A128397 A120383 A324842 * A140067 A364157 A067946

Adjacent sequences: A055929 A055930 A055931 * A055933 A055934 A055935

Keyword

easy,nonn

Author

Leroy Quet, Jul 17 2000

Extensions

Edited by Daniel Forgues, Jan 24 2011

Status

approved