A141345 -id:A141345

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A117825

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

+10
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.

KEYWORD

nonn,look

AUTHOR

Bill McEachen, May 01 2006

EXTENSIONS

More terms from Don Reble, May 02 2006

STATUS

approved

A324385

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,6

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

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 26 2019

STATUS

approved

A329894

Terms of A025487 from which the distance to the next larger prime is a composite number.

+10
2

512, 16384, 373248, 393216, 524288, 1119744, 4194304, 4718592, 5971968, 8388608, 10077696, 10616832, 17915904, 21233664, 31104000, 33554432, 35831808, 42467328, 47775744, 56623104, 67108864, 150994944, 159252480, 286654464, 322486272, 362797056, 679477248, 859963392, 1528823808, 2176782336, 2890137600, 4294967296, 5804752896, 8748000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

From the first 795641 terms of A025487 (terms that are in range 1 .. 2^101) only 4238 (~ 0.5 %) are included in this sequence.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..4238

EXAMPLE

As A151800(512) = 521, with 521 - 512 = 9 (a composite number), 512 is included in this sequence.

PROG

(PARI)

isc(n) = ((n > 1)&&!isprime(n));

for(n=1, 2000, if(isc(nextprime(1+A025487(n))-A025487(n)), print1(A025487(n), ", ")));

CROSSREFS

Cf. A025487, A129912, A141345, A151800.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 24 2019

STATUS

approved

A272817

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

+10
1

0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 1, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,24

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..1000 (based on A002182 b-file)

CROSSREFS

Cf. A002182, A005235, A117825, A141345.

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, May 07 2016

EXTENSIONS

a(25)-a(77) from Giovanni Resta, May 07 2016

STATUS

approved

A339385

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

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.

LINKS

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

Cf. A005250, A141345, A324385.

KEYWORD

nonn

AUTHOR

A.H.M. Smeets, Dec 02 2020

STATUS

approved

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