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A117825 Distance from n-th highly composite number (cf. A002182) to nearest prime. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

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)

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).

FORMULA

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

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

Sequences tied to conjecture a): A228943, A228945.

Cf. also A005235, A060270.

Sequence in context: A081229 A109010 A268354 * A010143 A101027 A129408

Adjacent sequences:  A117822 A117823 A117824 * A117826 A117827 A117828

KEYWORD

nonn,look

AUTHOR

Bill McEachen, May 01 2006

EXTENSIONS

More terms from Don Reble, May 02 2006

STATUS

approved

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Last modified February 26 11:15 EST 2021. Contains 341631 sequences. (Running on oeis4.)