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Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.
4

%I #9 Jul 07 2024 19:12:53

%S 0,0,0,0,0,1,1,1,1,2,2,3,3,4,5,5,5,6,6,7,8,9,9,10,10,11,11,12,12,14,

%T 14,14,15,16,17,18,18,19,20,21,21,23,23,24,25,26,26,27,27,28,29,30,30,

%U 31,32,33,34,35,35,37,37,38,39,39,40,42,42,43,44,46,46

%N Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

%C For initial terms up to 30 we have a(n) = Log_2 A355537(n).

%F a(n) = A013939(n) - n + 1.

%t Table[Total[(PrimeNu[#]-1)&/@Range[2,n]],{n,1,100}]

%Y The sum of the same range is A000096.

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

%Y For divisors (not just prime factors) we get A002541, also A006218, A077597.

%Y A shifted variation is A013939.

%Y The unshifted version is A022559, product A327486, w/o multiplicity A355537.

%Y The ranges themselves are the rows of A131818 (shifted).

%Y Partial sums of A297155 (shifted).

%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 A066843 gives partial sums of A000005.

%Y Cf. A000720, A002110, A076610, A305054, A355538, A355731, A355733, A355741, A355744, A355745, A355746, A355747.

%K nonn

%O 1,10

%A _Gus Wiseman_, Jul 23 2022