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

1,2

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

Links

Table of n, a(n) for n=1..75.

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).

Mathematica

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

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.

Sequence in context: A181803 A144962 A227542 * A244567 A254112 A249111

Adjacent sequences: A338897 A338898 A338899 * A338901 A338902 A338903

Keyword

nonn

Author

Gus Wiseman, Nov 16 2020

Status

approved