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
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.
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).
FORMULA
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...
From David A. Corneth, Jan 24 2026: (Start)
150 is in the sequence as follows:
150 is not 1 so we need to investigate more.
2 is a divisor of 150, so it could still be that 150 is in the sequence. We divide out all factors of 2 and get 75. The next prime larger than 2 is 3.
3 is a divisor of 75. It could still be that 150 is in the sequence. We divide out all factors of 3 and get 25. The next prime larger than 3 is 5.
5 is a divisor of 25. It could still be that 150 is in the sequence. We divide out all factors of 5 and get 1. As we get 1, 150 is in the sequence.
20 is not in the sequence as follows:
20 is not 1 so we need to investigate more.
2 is a divisor of 20, so it could still be that 20 is in the sequence. We divide out all factors of 2 and get 5. The next prime larger than 2 is 3.
3 is no divisor of 5. 5 is not 1, therefore 20 is not in the sequence. (End)
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
(PARI) is(n) = {if(n == 1, return(1)); forprime(p = 2, oo, v = valuation(n, p); if(v == 0, return(0)); n\=p^v; if(n==1, return(1)))} \\ David A. Corneth, Jan 24 2026
(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
(Python)
from functools import lru_cache
from itertools import count
from sympy import prime, integer_log, primorial
from oeis_sequences.OEISsequences import bisection
def A055932(n):
@lru_cache(maxsize=None)
def g(x, m): return sum(g(x//(prime(m)**i), m-1) for i in range(1, integer_log(x, prime(m))[0]+1)) if m-1 else x.bit_length()-1
def f(x):
c = n-1+x
for k in count(1):
if primorial(k)>x:
break
c -= g(x, k)
return c
return bisection(f, n, n) # Chai Wah Wu, Mar 19 2026
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import nextprime
def A055932_gen(): # generator of terms if the first n terms are desired.
h, hset = [(1, (1, ))], {1}
while True:
m, ps = heappop(h)
yield m
for p in ps:
mp = m*p
if mp not in hset:
heappush(h, (mp, ps))
hset.add(mp)
q = nextprime(max(ps, default=1))
mp = m*q
if mp not in hset:
heappush(h, (mp, (ps+(q, ))))
hset.add(mp)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Leroy Quet, Jul 17 2000
EXTENSIONS
Edited by Daniel Forgues, Jan 24 2011
STATUS
approved