A384533 Inventory sequence starting a(1) = 1 and thereafter recording the number of terms which are the product of 0,1,2,... primes (taken with multiplicity). The count resets following a term = 0.

1, 1, 0, 2, 1, 0, 3, 2, 0, 3, 4, 1, 0, 4, 4, 3, 0, 4, 5, 4, 0, 4, 6, 7, 0, 4, 7, 8, 1, 0, 5, 9, 9, 1, 0, 6, 9, 12, 2, 0, 6, 10, 14, 2, 0, 6, 11, 16, 2, 1, 0, 7, 14, 17, 2, 1, 0, 8, 16, 17, 3, 2, 0, 8, 19, 17, 4, 2, 0, 8, 22, 19, 5, 2, 0, 8, 25, 20, 7, 2, 0, 8

Offset

1,4

Comments

When for some k, a(k) = 0, a(k+1) is the number 1's (empty products) in the sequence. The number of zero terms are never counted because there are no numbers whose product of prime factors = 0.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, replacing a(n) = 0 with 1/2 for visibility.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, replacing a(n) = 0 with 1/2 for visibility, with a color function showing counter c = 0 in black, c = 1 in red, c = 2 in orange, .. c = 14 in magenta.

Example

a(1) = 1, so a(2) = 1 since there is one term which is the product of no primes.
a(3) = 0 since there are no terms which are the product of one prime.
Now the count resets and a(4) = 2 since there are two empty products (a(1) = a(2) = 1).
a(5) = 1 since a(4) = 2 is the product of one prime.
a(6) = 0 since there are no terms which are the product of two primes.
The count resets again and a(7) = 3, the number of empty products.
a(8) = 2 because there are now two terms (a(4) = 2 and a(7) = 3) which are the product of one prime.
Since there are still no terms which are the product of two primes, a(9) = 0.
a(10) = 3, still the number of empty products and a(11) = 4 since there are now four prime terms. This is the first term with two prime divisors (counted with multiplicity), implying that a(12), the next count of two prime divisors must be = 1. And so on.

Mathematica

nn = 120; q[_] := 0; j = 1; c = 0; q[0]++; {j}~Join~Reap[Do[If[j == 0, c = 0]; j = Sow[q[c]]; c++; If[j > 0, q[PrimeOmega[j]]++], {n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Jun 02 2025 *)

Crossrefs

Cf. A342585.

Sequence in context: A276276 A157497 A257961 * A225310 A131358 A291895

Adjacent sequences: A384530 A384531 A384532 * A384534 A384535 A384536

Keyword

nonn,easy

Author

David James Sycamore, Jun 01 2025

Status

approved