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

%I A055932 #103 Aug 07 2025 20:25:53
%S A055932 1,2,4,6,8,12,16,18,24,30,32,36,48,54,60,64,72,90,96,108,120,128,144,
%T A055932 150,162,180,192,210,216,240,256,270,288,300,324,360,384,420,432,450,
%U A055932 480,486,512,540,576,600,630,648,720,750,768,810,840,864,900,960,972
%N A055932 Numbers all of whose prime divisors are consecutive primes starting at 2.
%C A055932 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
%C A055932 Squarefree kernels of these numbers are primorial numbers. See A080404. - _Labos Elemer_, Mar 19 2003
%C A055932 If u and v are terms then so is u*v. - _Reinhard Zumkeller_, Nov 24 2004
%C A055932 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
%C A055932 From _Daniel Forgues_, Jan 24 2011: (Start)
%C A055932 Least integer, in increasing order, of each ordered prime signature.
%C A055932 The least integer of each ordered prime signature are the smallest numbers with a given tuple of exponents of prime factors.
%C A055932 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)
%C A055932 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
%C A055932 Numbers n such that A053669(n) > A006530(n). - _Anthony Browne_, Jun 06 2016
%C A055932 From _David W. Wilson_, Dec 28 2018: (Start)
%C A055932 Numbers n such that for primes p > q, p | n => q | n.
%C A055932 Numbers n such that prime p | n => A034386(p) | n. (End)
%H A055932 Michael De Vlieger, <a href="/A055932/b055932.txt">Table of n, a(n) for n = 1..10000</a>, first 1001 terms from Franklin T. Adams-Watters.
%H A055932 Jon Awbrey, <a href="https://oeis.org/wiki/Riffs_and_Rotes">Riffs and Rotes</a>.
%H A055932 Michael De Vlieger, <a href="/A055932/a055932.txt">Extended table of n, a(n) for n = 1..100000</a>.
%H A055932 Robert Vajda, <a href="http://ceur-ws.org/Vol-2650/paper43.pdf">Computational Exploration of the Degree Sequence of the Malyshev Polynomials</a>, Proceedings of the 11th International Conference on Applied Informatics (Eger, Hungary, 2020).
%H A055932 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>.
%F A055932 Sum_{n>=1} 1/a(n) = Sum_{n>=0} 1/A005867(n) = 2.648101... (A345974). - _Amiram Eldar_, Jun 26 2025
%e A055932 60 is included because 60 = 2^2 * 3 * 5 and 2, 3 and 5 are consecutive primes beginning at 2.
%e A055932 Sequence A057335 begins
%e A055932 1..2..4..6..8..12..18..30..16..24..36..60..54..90..150..210... which is equal to
%e A055932 1..2..2..3..2...3...3...5...2...3...3...5...3...5....5....7... times
%e A055932 1..1..2..2..4...4...6...6...8...8..12..12..18..18...30...30...
%p A055932 isA055932 := proc(n)
%p A055932     local s,p ;
%p A055932     s := numtheory[factorset](n) ;
%p A055932     for p in s do
%p A055932         if p > 2 and not prevprime(p)  in s then
%p A055932             return false;
%p A055932         end if;
%p A055932     end do:
%p A055932     true ;
%p A055932 end proc:
%p A055932 for n from 2 to 100 do
%p A055932     if isA055932(n) then
%p A055932         printf("%d,",n) ;
%p A055932     end if;
%p A055932 end do: # _R. J. Mathar_, Oct 02 2012
%t A055932 Select[Range[1000], #==1||FactorInteger[ # ][[ -1, 1]]==Prime[Length[FactorInteger[ # ]]]&]
%t A055932 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 *)
%o A055932 (PARI) is(n)=my(f=factor(n)[,1]~);f==primes(#f) \\ _Charles R Greathouse IV_, Aug 22 2011
%o A055932 (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
%o A055932 (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
%o A055932 (Python)
%o A055932 from itertools import count, islice
%o A055932 from sympy import primepi, primefactors
%o A055932 def A055932_gen(startvalue=1): # generator of terms >= startvalue
%o A055932     for k in count(max(startvalue,1)):
%o A055932         p = list(map(primepi,primefactors(k)))
%o A055932         if k==1 or (min(p)==1 and max(p)==len(p)):
%o A055932             yield k
%o A055932 A055932_list = list(islice(A055932_gen(),40)) # _Chai Wah Wu_, Aug 07 2025
%Y A055932 Cf. A057335 (permuted), A056808, A025487, A007947, A002110, A080404, A034386, A106177, A124829, A124830, A124831, A124833, A080259 (complement), A215366.
%Y A055932 Cf. A005867, A324939, A345974.
%K A055932 easy,nonn
%O A055932 1,2
%A A055932 _Leroy Quet_, Jul 17 2000
%E A055932 Edited by _Daniel Forgues_, Jan 24 2011