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

Author

Bill McEachen, May 01 2006

Keywords

• nonn
• look

Comments

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)

Examples

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

Links

• 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).

Crossrefs

Cf. A002100, A002182, A007917, A129912, A141345, A151800, A324385.

Sequences tied to conjecture a): A228943, A228945.

Cf. also A005235, A060270.

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

More terms from Don Reble, May 02 2006