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A333978 Numbers of the form b_1 * b_2 * ... * b_t, where b_1 = 1 and b_(i + 1) - b_i = 0 or 1.

%I A333978 #90 Sep 24 2022 15:45:17
%S A333978 1,2,4,6,8,12,16,18,24,32,36,48,54,64,72,96,108,120,128,144,162,192,
%T A333978 216,240,256,288,324,360,384,432,480,486,512,576,600,648,720,768,864,
%U A333978 960,972,1024,1080,1152,1200,1296,1440,1458,1536,1728,1800,1920,1944,2048
%N A333978 Numbers of the form b_1 * b_2 * ... * b_t, where b_1 = 1 and b_(i + 1) - b_i = 0 or 1.
%C A333978 This sequence gives the distinct values in A284001, sorted.
%C A333978 If m and k are in this sequence, then so is their product m*k.
%C A333978 If a prime p divides a(n), then so does p!.
%C A333978 A001013 is a subsequence.
%C A333978 Define a set S of polynomials by: (i) 1 is in S; (ii) if P is in S, then x*P and dP/dx are in S; (iii) if the repeated application of (i) and (ii) fails to prove that P is in S then P is not in S. This sequence enumerates the elements of S of degree 0. - _Luc Rousseau_, Aug 20 2022
%C A333978 Numbers k divisible by A102068(k)  (or in other words, numbers k divisible by h(k)! where h(k) is the largest prime factor of k). - _David A. Corneth_, Aug 20 2022
%H A333978 Michael S. Branicky, <a href="/A333978/b333978.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Peter Kagey)
%e A333978 The first 11 terms can be written as
%e A333978    1 = 1
%e A333978    2 = 1 * 2
%e A333978    4 = 1 * 2 * 2
%e A333978    6 = 1 * 2 * 3
%e A333978    8 = 1 * 2 * 2 * 2
%e A333978   12 = 1 * 2 * 2 * 3
%e A333978   16 = 1 * 2 * 2 * 2 * 2
%e A333978   18 = 1 * 2 * 3 * 3
%e A333978   24 = 1 * 2 * 3 * 4 or 1 * 2 * 2 * 2 * 3
%e A333978   32 = 1 * 2 * 2 * 2 * 2 * 2
%e A333978   36 = 1 * 2 * 2 * 3 * 3
%o A333978 (SWI-Prolog)
%o A333978 main :- iter(1).
%o A333978 iter(K) :-
%o A333978     (legal(K * x ^ 0) -> (maplist(write, [K, ', ']), flush_output) ; true),
%o A333978     KK is K + 1, iter(KK).
%o A333978 legal(1 * x ^ 0).
%o A333978 legal(K * x ^ N) :-
%o A333978     NN is N + 1, 0 is K mod NN, KK is K / NN,
%o A333978     legal(KK * x ^ NN).
%o A333978 legal(K * x ^ N) :-
%o A333978     ((K = 1, N = 1) ; (N > 1)), NN is N - 1,
%o A333978     legal(K * x ^ NN).
%o A333978 % _Luc Rousseau_, Aug 20 2022
%o A333978 (Python)
%o A333978 import heapq
%o A333978 from math import factorial
%o A333978 from sympy import nextprime
%o A333978 from itertools import islice
%o A333978 def agen(): # generator of terms
%o A333978     oldv, h, primes, nextp, nextfact = 0, [(1, 1)], [], 0, 0
%o A333978     while True:
%o A333978         v, maxp = heapq.heappop(h)
%o A333978         if v != oldv:
%o A333978             yield v; oldv = v
%o A333978             while nextfact < v:
%o A333978                 nextp = nextprime(nextp); nextfact = factorial(nextp)
%o A333978                 primes.append(nextp); heapq.heappush(h, (nextfact, nextp))
%o A333978             for p in primes:
%o A333978                 if p <= maxp: heapq.heappush(h, (v*p, max(maxp, p)))
%o A333978                 else: break
%o A333978 print(list(islice(agen(), 60))) # _Michael S. Branicky_, Aug 20 2022
%o A333978 (PARI) is(n) = if(n==1, return(1)); my(f = factor(n), p = f[#f~, 1]); n%p! == 0 \\ _David A. Corneth_, Sep 05 2022
%Y A333978 Cf. A001013, A003586, A006530, A102068, A284001, A334636.
%K A333978 nonn,easy
%O A333978 1,2
%A A333978 _Peter Kagey_, Sep 20 2020