A375205 PrimePi(greatest prime < sqrt(Q)) - PrimePi(greatest prime factor(Q) < sqrt(Q)), with Q = A082686(n).

0, 0, 1, 1, 0, 1, 0, 2, 1, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 3, 2, 3, 1, 2, 3, 0, 2, 1, 3, 2, 3, 1, 0, 4, 2, 4, 4, 3, 1, 2, 0, 4, 2, 3, 4, 1, 4, 3, 2, 4, 0, 1, 3, 4, 4, 2, 0, 4, 1, 3, 2, 4, 3, 4, 0, 1, 4, 3, 2, 5, 4, 2, 1, 3, 5, 4, 5, 3

Offset: 1

Author

Bill McEachen, Oct 15 2024

Keywords

• nonn

Comments

New records appear to be in consecutive numerical order, suggesting every integer should be seen in this infinite sequence. Considering a(n)=0, empirically a power fit Y=k*x^c correlates well with the "x-th" occurrence. For example, the 491st 0 value is at n=99808.

Examples

			A082686(8)=51, with square root = 7.14... so the greatest prime < 7.14 is 7, while the greatest prime factor of 51 < 7.14 is 3. The prime count from 3 to 7 is 2, so a(8)=2.
A082686(999)=2883 with square root = 53.69... so the greatest prime < 53.69 is 53, while the greatest prime factor of 2883 < 53.69 is 31. The prime count from 31 to 53 is 5, so a(999)=5.
		

Links

• Bill McEachen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A082686.

Programs

Mathematica

nmax=85;q={};m=15;Until[Length[q]==nmax,If[ !PrimeQ[m]&&EvenQ[DivisorSigma[0, m]],AppendTo[q,m]];m=m+2];Table[PrimePi[NextPrime[Sqrt[q[[n]]],-1]]-PrimePi[Select[First/@FactorInteger[q[[n]]],#James C. McMahon, Dec 06 2024 *)