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Lesser prime index of the n-th semiprime.
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%I #11 Sep 07 2024 08:54:19

%S 1,1,2,1,1,2,2,1,3,1,2,1,3,1,2,1,4,2,3,2,1,1,3,2,1,4,1,3,1,2,4,2,1,3,

%T 1,2,3,1,4,5,1,2,2,4,1,2,1,5,3,1,3,1,2,4,1,6,2,1,2,3,5,1,2,1,4,3,1,5,

%U 2,1,3,4,1,2,6,1,3,2,6,2,5,1,4,1,3,2,1

%N Lesser prime index of the n-th semiprime.

%C A semiprime is a product of any two prime numbers. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%H Amiram Eldar, <a href="/A338912/b338912.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000720(A084126(n)).

%e The semiprimes are:

%e 2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...

%e so the lesser prime factors are:

%e 2, 2, 3, 2, 2, 3, 3, 2, 5, 2, ...

%e with indices:

%e 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, ...

%t Table[Min[PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}]

%Y A084126 is the lesser prime factor (not index).

%Y A084127 is the greater factor, with index A338913.

%Y A115392 lists positions of ones.

%Y A128301 lists positions of first appearances of each positive integer.

%Y A270650 is the squarefree case, with greater part A270652.

%Y A338898 has this as first column.

%Y A001221 counts distinct prime indices.

%Y A001222 counts prime indices.

%Y A001358 lists semiprimes, with odds A046315 and evens A100484.

%Y A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.

%Y A087794/A176504/A176506 are product/sum/difference of semiprime indices.

%Y A338910/A338911 list products of pairs of odd/even-indexed primes.

%Y Cf. A037143, A056239, A065516, A112798, A289182, A320732, A320892, A338900, A338904, A338909.

%K nonn

%O 1,3

%A _Gus Wiseman_, Nov 20 2020