A324385 -id:A324385 - cp's OEIS frontend

A117825 Distance from n-th highly composite number (cf. A002182) to nearest prime.

1, 0, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 11, 13, 1, 11, 1, 17, 1, 1, 13, 13, 1, 1, 17, 1, 17, 1, 1, 17, 17, 17, 1, 1, 19, 37, 37, 1, 17, 23, 1, 29, 1, 1, 19, 1, 19, 23, 1, 19, 31, 1, 19, 1, 1, 1, 1, 23, 1, 29, 23, 23, 1, 23, 71, 37, 1, 1, 31, 1, 23, 53, 1, 31

Offset: 1

Bill McEachen, May 01 2006

a) Conjecture: entries > 1 will always be prime. The entry will be larger than the largest prime factor of the highly composite number.
b) Will 1 always be the most common entry?
c) While a prime may always be located close to each highly composite number, is the converse false?
d) Is there always a prime between successive highly composite numbers?
From Antti Karttunen, Feb 26 2019: (Start)
The second sentence of point (a) follows as both gcd(n, A151799(n)) = 1 and gcd(A151800(n), n) = 1 for all n > 2 and the fact that the highly composite numbers are products of primorials, A002110 (with the least coprime prime > the largest prime factor). See also the conjectures and notes in A129912 and A141345. (End)

			a(5) = abs(12-11) = 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..19999
Bill McEachen, Alternate plot, Wikimedia Commons.
Graeme McRae, Highly Composite Numbers.
Wikipedia, Highly Composite Numbers.
Wikipedia, Divisor Function (sigma).

Cf. A002100, A002182, A007917, A129912, A141345, A151800, A324385.
Sequences tied to conjecture a): A228943, A228945.
Cf. also A005235, A060270.

With[{s = DivisorSigma[0, Range[Product[Prime@ i, {i, 8}]]]}, Map[If[PrimeQ@ #, 0, Min[# - NextPrime[#, -1], NextPrime[#] - #]] &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Mar 11 2019 *)

A141345(n) = (nextprime(1+A002182(n))-A002182(n));
A324385(n) = (A002182(n)-precprime(A002182(n)));
A117825(n) = if(1==n,1,min(A141345(n), A324385(n))); \\ Antti Karttunen, Feb 26 2019

a(1) = 1; for n > 1, a(n) = min(A141345(n), A324385(n)). - Antti Karttunen, Feb 26 2019

More terms from Don Reble, May 02 2006

A339385 a(n) = (smallest prime >= A002182(n)) - (largest prime <= A002182(n)).

Original entry on oeis.org

0, 2, 2, 2, 6, 6, 6, 2, 14, 2, 2, 8, 8, 14, 18, 24, 18, 12, 2, 12, 14, 12, 30, 32, 18, 24, 2, 40, 2, 30, 26, 30, 18, 14, 34, 14, 40, 18, 20, 40, 34, 36, 18, 20, 42, 120, 90, 24, 34, 52, 44, 72, 20, 20, 38, 44, 42, 54, 24, 60, 72, 20, 72, 30, 20, 20, 24, 70

Offset: 2

A.H.M. Smeets, Dec 02 2020

nonn

The prime gap size at the n-th highly composite number A002182(n), for n > 2.
The obtained arithmetic mean of the normalized gap size, i.e., a(n)/log(A002182(n)), for the terms 3..10000 is 3.030.
From Gauss's prime counting function approximation, the expected gap size should be approximately log(A002182), however, the observed values seem to be closer to log(A002182(n)^3).
The maximum merit (= a(n)/log(prevprime(A002182))) in the range 3..10000 is 12.96 and is obtained for n = 6911.

A.H.M. Smeets, Table of n, a(n) for n = 2..10000

Cf. A005250, A141345, A324385.

s = {}; dm = 1; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[s, NextPrime[n - 1] - NextPrime[n + 1, -1]]], {n, 2, 10^6}]; s (* Amiram Eldar, Dec 02 2020 *)
{0}~Join~Map[Subtract @@ NextPrime[#, {1, -1}] &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[3 ;; 10^3, -1]] ] (* Michael De Vlieger, Dec 10 2020 *)

lista(nn) = my(r=1); forstep(n=2, nn, 2, if(numdiv(n)>r, r=numdiv(n); print1(nextprime(n) - precprime(n), ", "))); \\ Michel Marcus, Dec 03 2020

a(n) = A324385(n)+A141345(n), for n > 1.

cp's OEIS Frontend

A117825 Distance from n-th highly composite number (cf. A002182) to nearest prime.

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