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Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.
4

%I #5 Jul 21 2022 07:40:35

%S 1,1,1,1,1,2,2,2,2,4,4,8,8,16,32,32,32,64,64,128,256,512,512,1024,

%T 1024,2048,2048,4096,4096,12288,12288,12288,24576,49152,98304,196608,

%U 196608,393216,786432,1572864,1572864,4718592,4718592,9437184,18874368,37748736

%N Number of ways to choose a sequence of prime factors, one of each integer from 2 to n.

%C Also partial products of A001221 without the first term 0, sum A013939.

%C For initial terms up to n = 29 we have a(n) = 2^A355538(n). The first non-power of 2 is a(30) = 12288.

%e The a(n) choices for n = 2, 6, 10, 12, with prime(k) replaced by k:

%e (1) (12131) (121314121) (12131412151)

%e (12132) (121314123) (12131412152)

%e (121324121) (12131412351)

%e (121324123) (12131412352)

%e (12132412151)

%e (12132412152)

%e (12132412351)

%e (12132412352)

%t Table[Times@@PrimeNu/@Range[2,m],{m,2,30}]

%Y The sum of the same integers is A000096.

%Y The product of the same integers is A000142, Heinz number A070826.

%Y The version for divisors instead of prime factors is A066843.

%Y The integers themselves are the rows of A131818.

%Y The version with multiplicity is A327486.

%Y Using prime indices instead of 2..n gives A355741, for multisets A355744.

%Y Counting sequences instead of multisets gives A355746.

%Y A001221 counts distinct prime factors, with sum A001414.

%Y A001222 counts prime factors with multiplicity.

%Y A003963 multiplies together the prime indices of n.

%Y A056239 adds up prime indices, row sums of A112798.

%Y Cf. A000005, A000040, A000720, A002110, A013939, A076610, A355538, A355731, A355733, A355745, A355747.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jul 20 2022