A141345 -id:A141345 - 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

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

Original entry on oeis.org

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

Antti Karttunen, Feb 26 2019

nonn

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

			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.

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

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

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 *)

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

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

A329894 Terms of A025487 from which the distance to the next larger prime is a composite number.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

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

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

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

Cf. A025487, A129912, A141345, A151800.

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

A272817 Distance from n-th highly composite number (cf. A002182) to nearest prime or square.

Original entry on oeis.org

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

Offset: 1

Vladimir Shevelev, May 07 2016

nonn

Conjecture: Terms are either 1 or prime, n>7. - Bill McEachen, Jun 11 2025

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

Cf. A002182, A005235, A117825, A141345.

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

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.

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

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

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A329894 Terms of A025487 from which the distance to the next larger prime is a composite number.

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A272817 Distance from n-th highly composite number (cf. A002182) to nearest prime or square.

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Extensions

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

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