A002182 -id:A002182 - 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

A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 6, 4, 6, 8, 2, 4, 6, 8, 12, 16, 20, 12, 14, 18, 12, 12, 12, 12, 12, 12, 12, 24, 8, 8, 8, 4, 16, 8, 16, 8, 16, 24, 16, 32, 6, 14, 30, 12, 18, 18, 24, 12, 18, 18, 24, 18, 36, 8, 14, 32, 28, 6, 24, 38, 12, 18, 36, 20, 24, 30, 40, 26, 10, 40, 20, 30, 18, 38, 26, 36, 36, 24, 24, 44, 50, 48, 14, 42

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

Among the first 10000 highly composite numbers, only in two cases a(n) < A112779(n). This happens on A002182(12) = 240 and A002182(18) = 2520. Note that A112779(n) gives the number of primorials needed when A002182(n) is expressed as a product [not as a sum] of primorials.

			For n=12, A002182(12) = 240, which is written as "11000" in primorial base (A049345) because 240 = 1*A002110(4) + 1*A002110(3) = 210+30, thus a(12) = 1+1 = 2. (Note that 240 = 30*2*2*2).
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 1+1 = 2. (Note that 2520 = 210*6*2).
For n=26, A002182(26) = 45360 = "1670000" in primorial base because 45360 = 1*A002110(6) + 6*A002110(5) + 7*A002110(4), thus a(26) = 1+6+7 = 14. (Note that 45360 = 210*6*6*6).

Cf. A002110, A002182, A276150, A324381.

Cf. also A108602, A112778, A112779, A324342.

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324382(n) = A276150(A002182(n));

a(n) = A276150(A002182(n)).

a(n) >= A324381(n).

A181806 Positive integers with more highly composite divisors (A002182) than any smaller positive integer.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 120, 240, 360, 720, 5040, 10080, 15120, 30240, 60480, 151200, 166320, 332640, 665280, 1663200, 1995840, 3326400, 8648640, 17297280, 21621600, 43243200, 86486400, 129729600, 259459200, 735134400

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

Numbers n such that A181801(n) > A181801(m) for all m < n. Also, numbers n such that row n of triangles A181802 and A181803 is longer than any previous row in either triangle.

Not a subsequence of A002182. The smallest positive integer which has a record number of highly composite divisors, but which is not highly composite itself, is 30240.

			12 has five divisors (namely, 1, 2, 4, 6 and 12) that are members of A002182. No positive integer smaller than 12 has more than three members of A002182 among its divisors; hence, 12 is a member of the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..153
A. Flammenkamp, List of the first 1200 highly composite numbers
Eric Weisstein's World of Mathematics, Highly composite number

A181807(n) = number of highly composite divisors of a(n) (i.e., A181801(a(n))).
Subsequence of A025487, A181804. Numbers A181804(n) such that A181805(n) increases to a record.
Includes all members of A136253.

a(20)-a(30) from Charles R Greathouse IV, Jan 14 2011

A301413 a(n) = A002182(n)/A002110(A108602(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 8, 2, 4, 6, 8, 12, 24, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 12, 216, 240, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 72, 1440, 120, 144, 216, 240, 288, 360, 480, 576

Offset: 1

Michael De Vlieger, Mar 30 2018

nonn

This sequence appears in Siano paper, page 5 of 12, as the "variable part" v. - Michael De Vlieger, Oct 11 2023

			Let m be a value in this sequence. The table below shows m*A002110(A108602(k)). Columns are A108602(k), rows are m whose products m*A002110(A108602(k)) appear in A002182 are in this sequence. Numbers in A002182 that also appear in A002201 are followed by (*).
        0  1   2    3     4       5       6 ...
      +------------------------------------
    1 | 1* 2*  6*
    2 |    4  12*  60*
    4 |       24  120*  840
    6 |       36  180  1260
    8 |       48  240  1680
   12 |           360* 2520*  27720
   24 |           720  5040*  55440* 720720*
   ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, On a graph of highly composite numbers
A. Flammenkamp, Highly composite numbers
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.

Cf. A002110, A002182, A108602, A301414.

(* Load b-file from A002182 *)
With[{s = Import["b002182.txt","Data"][[All,-1]]}, Array[#/Product[Prime@ i, {i, PrimeNu[#]}] &@ s[[#]] &, 62]]

a(n) = A002182(n)/A007947(A002182(n)).

A112781 Number of highly composite numbers (definition 1, A002182) < 10^n.

Original entry on oeis.org

4, 9, 15, 20, 29, 38, 47, 56, 66, 76, 86, 95, 106, 117, 125, 135, 146, 156, 167, 177, 186, 196, 209, 219, 231, 241, 254, 267, 280, 292, 305, 316, 330, 343, 356, 368, 381, 396, 409, 423, 436, 450, 463, 476, 491, 503, 517, 530, 547, 561, 577, 593, 608, 625, 640

Offset: 1

Ray Chandler, Nov 11 2005

nonn

			a(1) = 4 since there are four highly composite numbers < 10^1 {1,2,4,6}.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Highly Composite Number
A. Flammenkamp, First 1200 highly composite numbers

Cf. A002182, A002183, A108602, A112778, A112779, A112780.

Partial sums of A112780. - Lekraj Beedassy, Sep 02 2006

A166735 Superabundant numbers (A004394) that are not highly composite (A002182).

Original entry on oeis.org

1163962800, 4658179125600, 13974537376800, 144403552893600, 433210658680800, 10685862914126400, 21371725828252800, 32057588742379200, 37400520199442400, 64115177484758400, 1533421328177138400

Offset: 1

T. D. Noe, Oct 20 2009

nonn

Alaoglu and Erdos mention the first term in footnote 14.

Because the "shapes" of superabundant and highly composite numbers are different, there is a last superabundant number that is also highly composite. In factored form, that 154-digit number is N = A004394(1023) = A002182(2567) = 2^10 3^6 5^4 7^3 11^3 13^2 17^2 19^2 23^2 29 31 37...347. In other words, this sequence contains all superabundant numbers greater than N. - T. D. Noe, Oct 26 2009

T. D. Noe, Table of n, a(n) for n = 1..574
L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. Mentions this sequence.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 - 409.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.

Cf. A166981 (intersection of SA and HC numbers). - T. D. Noe, Oct 26 2009

Cf. A189228 (SA numbers that are not CA).

a(574+i) = A004394(1023+i) for i>0.

A181808 Numbers that set a record for number of even divisors: a(n) = 2*A002182(n).

Original entry on oeis.org

2, 4, 8, 12, 24, 48, 72, 96, 120, 240, 360, 480, 720, 1440, 1680, 2520, 3360, 5040, 10080, 15120, 20160, 30240, 40320, 50400, 55440, 90720, 100800, 110880, 166320, 221760, 332640, 443520, 554400, 665280, 997920, 1108800, 1330560, 1441440, 2162160, 2882880, 4324320

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

In other words, a positive integer n appears in the sequence iff more even numbers divide n than divide any positive integer smaller than n.

For all positive integer values (j,k) such that jk = n, the number of divisors of n that are multiples of j equals A000005(k). Therefore, n sets a record for the number of its divisors that are multiples of j iff k=n/j is highly composite (A002182). Cf. A181803, A181809, A181810.

			a(4)=12 has exactly four even divisors (2, 4, 6 and 12).  (Note that these are precisely the numbers that are twice a divisor of A002182(4)=6; see row 6 of A027750.)  No positive integer smaller than 12 has as many as four even divisors; hence, 12 is a member of the sequence.

Eric Weisstein's World of Mathematics, Highly composite number

Numbers n such that 2 appears in row n of A181803. See also A181809, A181810.
A002183(n) gives number of even divisors of a(n).
A053624 gives numbers that set records for number of odd divisors. No number sets records both for its number of odd divisors and its number of even divisors.

a(n)=2*A002182(n).

A054481 Highest common factor of successive highly composite numbers (1), A002182.

Original entry on oeis.org

1, 2, 2, 6, 12, 12, 12, 12, 60, 60, 60, 120, 360, 120, 420, 420, 840, 2520, 2520, 2520, 5040, 5040, 5040, 2520, 2520, 5040, 5040, 27720, 27720, 55440, 55440, 55440, 55440, 166320, 55440, 110880, 55440, 360360, 360360

Offset: 2

Henry Bottomley, Mar 31 2000

Not the same as the first differences of A002182. The latter are given by A262501, which differs from this sequence for the first time at n=25, where A262501(25) = 17640, while here the 25th term a(26) is 2520. The sequences differ next time at positions n = 52, 53, 54, 64, 67, 82, 83, 84, 85, 86, 87, 88, 90, 91, 96, 100, 106, ... (when one-based indexing as in A262501 is used). - Antti Karttunen, Sep 24 2015

			a(7)=12 because A002182(7)=36, A002182(6)=24 and GCD(36,24)=12.

Antti Karttunen, Table of n, a(n) for n = 2..1000 (computed from the b-file of A002182 provided by _T. D. Noe_)

Cf. A002182, A054482, A054483, A262501.

a(n) = GCD(A002182(n-1), A002182(n)) = A002182(n)/A054483(n) = A002182(n-1)/A054482(n).

Erroneous comment (wrong interpretation) removed by Antti Karttunen, Sep 25 2015

A141345 Distance from the n-th highly composite number, A002182(n), to the next prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 5, 1, 7, 1, 1, 7, 7, 13, 17, 13, 1, 11, 1, 11, 1, 1, 19, 13, 1, 11, 1, 17, 1, 29, 13, 13, 1, 1, 17, 13, 23, 17, 19, 17, 17, 19, 1, 19, 23, 37, 53, 1, 17, 29, 43, 29, 1, 19, 19, 1, 23, 23, 1, 41, 41, 1, 53, 29, 19, 19, 23, 23, 47, 29, 23, 37, 1, 59, 71, 41, 1, 29, 37

Offset: 1

T. D. Noe, Jun 26 2008

nonn

It appears that (1) every term is either 1 or a prime and (2) every prime greater than 3 appears. Note that a prime can occur only a finite number of times. Similar to Fortune's conjecture (A005235) and McEachen's conjecture (A117825).

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

T. D. Noe, Table of n, a(n) for n=1..10000

With[{s = Array[DivisorSigma[0, #] &, 10^6]}, Map[NextPrime[#] - # &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* or *)
Map[NextPrime[#] - # &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[1 ;; 80, -1]] ] (* Michael De Vlieger, Dec 11 2020 *)

A181802 Triangle read by rows: T(n,k) is k-th smallest divisor of n that is highly composite (A002182).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 24, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 36, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 6, 12, 24, 48

Offset: 1

Matthew Vandermast, Nov 27 2010

Row n contains A181801(n) numbers. T(n,k) * A180803(n, A181801(n)-k+1) = n.

Row n is identical to row (n+12) if n is not a multiple of 12.

			First rows read: 1; 1,2; 1; 1,2,4; 1; 1,2,6; 1; 1,2,4; 1; 1,2; 1; 1,2,4,6,12;...
8 has four divisors, of which three (1, 2 and 4) are members of A002182. Row 8 therefore reads 1, 2, 4.

Eric Weisstein's World of Mathematics, Highly composite number

See also A181804, A181805, A181806, A181807.

T(n,k) = n/(A180803(n, A181801(n)-k+1)).

cp's OEIS Frontend

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

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A324382 Minimal number of primorials that add to the n-th highly composite number: a(n) = A276150(A002182(n)).

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A181806 Positive integers with more highly composite divisors (A002182) than any smaller positive integer.

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A301413 a(n) = A002182(n)/A002110(A108602(n)).

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A112781 Number of highly composite numbers (definition 1, A002182) < 10^n.

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A166735 Superabundant numbers (A004394) that are not highly composite (A002182).

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A181808 Numbers that set a record for number of even divisors: a(n) = 2*A002182(n).

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A054481 Highest common factor of successive highly composite numbers (1), A002182.

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