A338900 Difference between the two prime indices of the n-th squarefree semiprime.

1, 2, 3, 1, 2, 4, 5, 3, 6, 1, 7, 4, 8, 5, 2, 6, 9, 10, 3, 7, 11, 1, 12, 4, 13, 8, 2, 9, 14, 5, 15, 10, 6, 16, 3, 17, 11, 12, 4, 18, 13, 19, 1, 7, 20, 8, 21, 14, 5, 22, 15, 23, 16, 9, 2, 24, 17, 25, 6, 10, 26, 3, 18, 27, 11, 7, 28, 19, 1, 29, 12, 20, 2, 21, 4

Offset: 1

Author

Gus Wiseman, Nov 16 2020

Keywords

• nonn

Comments

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

Is this sequence an anti-run, i.e., are there no adjacent equal parts? I have verified this conjecture up to n = 10^6. - Gus Wiseman, Nov 18 2020

Crossrefs

A176506 is the not necessarily squarefree version.

A338899 has row-differences equal to this sequence.

A338901 gives positions of first appearances.

A001221 counts distinct prime indices.

A001222 counts prime indices.

A001358 lists semiprimes.

A002100 and A338903 count partitions using squarefree semiprimes.

A004526 counts 2-part partitions, with strict case A140106 (shifted left).

A005117 lists squarefree numbers.

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

A065516 gives first differences of semiprimes.

A166237 gives first differences of squarefree semiprimes.

A270650 and A270652 give the prime indices of squarefree semiprimes.

A338912 and A338913 give the prime indices of semiprimes.

Cf. A000040, A056239, A112798, A167171, A320656, A320891, A320894, A320911, A338898, A338905, A338908.

Programs

Mathematica

-Subtract@@PrimePi/@First/@FactorInteger[#]&/@Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&]

Formula

If the n-th squarefree semiprime is prime(x) * prime(y) with x < y, then a(n) = y - x.

a(n) = A270652(n) - A270650(n).