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A337486 Numbers k such that b(k+1) divides b(k), where b() is Recamán's multiplicative sequence A008336.

%I A337486 #27 Apr 11 2024 15:54:31
%S A337486 1,6,10,12,14,18,20,22,26,28,30,34,36,38,42,44,45,46,50,52,54,58,60,
%T A337486 66,68,70,72,78,82,84,86,90,92,93,94,95,98,99,100,104,106,110,111,114,
%U A337486 116,118,119,122,124,126,130,132,134,135,136,142,146,147,148,150,154,156,158,161,162,164,165,166
%N A337486 Numbers k such that b(k+1) divides b(k), where b() is Recamán's multiplicative sequence A008336.
%C A337486 The old definition was: 1, together with the numbers formed by removing the required prime factors to form the number from a set which is initially empty and that has primes added via the addition of the prime factors of numbers which cannot be created from those currently in the set. Start by trying to create the number 2.
%C A337486 Consider an initially empty set of primes whose numbers are used to create a given number where each time a number is created those prime factors are removed from the set. If a number cannot be created as all its required prime factors are not currently in the set then all the prime factors of that number are instead added to the set. Start by trying to create the number 2 followed by all other integers. This sequence list the numbers that are created.
%C A337486 For the first 1 million terms the largest gap between terms is 15, between a(88018) = 189648 and a(88019) = 189663. The 1 millionth term is created after the addition of the number 2123404. The prime set at that point has 114323 primes, with the maximum number of entries for a single prime being eleven, for 887.
%C A337486 Note that if after the creation of a number the entire set of primes is cleared then the numbers created are those in A109895.
%H A337486 Michael De Vlieger, <a href="/A337486/b337486.txt">Table of n, a(n) for n = 1..10000</a>
%e A337486 a(2) = 6. As there are no primes initially in the set 2,3,4,5 cannot be created and instead these numbers add three 2's, one 3 and one 5 to the set. As there is now one 2 and one 3 the number 6 = 2*3 can be created. After 6 is created the set of primes now contains 2,2,5.
%e A337486 a(3) = 10. After 7,8,9, none of which can be created from the prime set, the prime set contains 2,2,2,2,2,3,3,5,7. As 2 and 5 are present 10 = 2*5 can be created, after which the set contains 2,2,2,2,3,3,7.
%t A337486 Block[{nn = 166, k = 1}, Reap[Do[If[Mod[k, i] == 0, k /= i; Sow[i], k *= i], {i, nn}]][[-1, 1]]] (* _Michael De Vlieger_, Sep 12 2020 *)
%o A337486 (Python)
%o A337486 from itertools import count, islice
%o A337486 def A337486_gen(): # generator of terms
%o A337486     c = 1
%o A337486     for n in count(1):
%o A337486         a, b = divmod(c,n)
%o A337486         if not b:
%o A337486             c = a
%o A337486             yield n
%o A337486         else:
%o A337486             c *= n
%o A337486 A337486_list = list(islice(A337486_gen(),30)) # _Chai Wah Wu_, Apr 11 2024
%Y A337486 Cf. A005132, A008336, A331276, A330430, A109895.
%Y A337486 First differences are A370969.
%K A337486 nonn
%O A337486 1,2
%A A337486 _Scott R. Shannon_, Aug 29 2020