A324385 Distance from the n-th highly composite number, A002182(n), from the largest prime <= A002182(n).

0, 1, 1, 1, 1, 5, 1, 1, 7, 1, 1, 1, 1, 1, 1, 11, 17, 1, 1, 1, 13, 11, 11, 19, 17, 13, 1, 23, 1, 1, 13, 17, 17, 13, 17, 1, 17, 1, 1, 23, 17, 17, 17, 1, 19, 83, 37, 23, 17, 23, 1, 43, 19, 1, 19, 43, 19, 31, 23, 19, 31, 19, 19, 1, 1, 1, 1, 47, 1, 31, 47, 23, 53, 23, 83, 37, 31, 1, 31, 1, 23, 61, 1, 41, 47, 61, 41, 29, 41, 29, 43, 73, 29, 47, 31, 31

Offset: 2

Author

Antti Karttunen, Feb 26 2019

Keywords

• nonn

Comments

Like in A141345 it appears (or is conjectured) that no composite numbers ever occur here. Taken together, this leads to McEachen's conjecture given in A117825. Here in range 2..10000 term 1 occurs for 313 times.

The arithmetic mean of a(n)/log(A002182(n)) for the terms 3..10000 is 1.513, i.e., a rough approximation is given by a(n) ~ log(A002182(n)^(3/2)). - A.H.M. Smeets, Dec 02 2020

Examples

			A002182(2) = 2, the largest prime <= 2 is 2 itself, thus a(2) = 2-2 = 0.
A002182(7) = 36, the largest prime <= 36 is 31, thus a(7) = 36-31 = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A002182, A007917, A060270, A117825, A141345.

Programs

Mathematica

With[{s = Array[DivisorSigma[0, #] &, 10^6]}, {0}~Join~Map[# - NextPrime[#, -1] &@ FirstPosition[s, #][[1]] &, Drop[Union@ FoldList[Max, s], 2]]] (* or *)
{0}~Join~Map[# - NextPrime[#, -1] &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[3 ;; 97, -1]] ] (* Michael De Vlieger, Dec 11 2020 *)

PARI

A324385(n) = (A002182(n)-precprime(A002182(n)));

Formula

a(n) = A002182(n) - A007917(A002182(n)).