A002182 -id:A002182 - cp's OEIS frontend

A181810 a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.

1, 2, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 1, 2, 3, 4, 1, 2, 3, 2, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 1, 2, 3, 2, 1, 4, 1, 2, 4, 1, 2, 2, 3, 1, 1, 6, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 1, 6, 3, 2, 1, 4, 1, 2, 1, 2, 2, 3, 1, 6, 3, 2, 4, 1, 1, 1, 1, 2, 2

Offset: 1

Matthew Vandermast, Nov 27 2010

nonn

Also, largest number k such that, for each integer j from 1 to k, more multiples of j appear among the divisors of A002182(n) than appear among the divisors of any smaller positive integer.

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).

			360 is a member of A002182, twice a member of A002182 (360/2 = 180), and three times a member of A002182 (360/3 = 120), but is not four times a member of A002182 (360/4 = 90 is not a member of A002182). Since A002182(13) = 360, a(13) = 3.
360 also sets records for the number of its divisors, the number of its divisors that are multiples of 2 (cf. A181808), and the number of its divisors that are multiples of 3, but not the number of its divisors that are multiples of 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Flammenkamp, List of the first 1200 highly composite numbers
Eric Weisstein's World of Mathematics, Highly composite number

a(n) equals the largest number k such that each number from 1 to k appears in row A002182(n) of A181803. a(n) also equals the largest number k such that each of the first k members of row A002182(n) of A056538 is highly composite.

A262501 First differences of A002182 (highly composite numbers, definition 1).

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, 17640, 5040, 5040, 27720, 27720, 55440, 55440, 55440, 55440, 166320, 55440, 110880, 55440, 360360, 360360, 720720, 720720, 720720, 720720, 2162160, 720720, 1441440, 2162160, 3603600, 2882880, 4324320, 10810800, 4324320, 6486480, 18018000

Offset: 1

Antti Karttunen, Sep 24 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..999 (computed from the b-file of A002182 provided by _T. D. Noe_.)

Cf. A002182, A261100.

Cf. also A054481 (gives the gcds of successive terms of A002182, from which this sequence differs for the first time at n=25).

(define (A262501 n) (- (A002182 (+ 1 n)) (A002182 n)))

a(n) = A002182(n+1) - A002182(n).

A321995 Indices of highly composite numbers A002182 which are between a twin prime pair.

Original entry on oeis.org

3, 4, 5, 9, 11, 12, 20, 28, 30, 84, 108, 118, 143, 149, 208, 330, 362, 1002, 2395, 3160, 10535

Offset: 1

M. F. Hasler, Jun 23 2019

The highly composite numbers are listed in A068507, but their growth is such that one cannot list the terms beyond A002182(362), corresponding to a(17), in the DATA section.
The term a(21) corresponds to A002182(10535) = A108951(52900585920). - Daniel Suteu, Jun 27 2019
a(22) > 779674, if it exists. - Amiram Eldar, Dec 03 2020

Cf. A068507, A002182, A002822.

select( x->ispseudoprime(x-1)&&ispseudoprime(x+1), A2182, 1) \\ assuming A2182 holds enough terms of A002182. - M. F. Hasler, Jun 23 2019

Intersection of A306587 and A306588. - Daniel Suteu, Jun 27 2019

a(21) from Daniel Suteu, Jun 27 2019 (obtained from A. Flammenkamp's data)

A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 4

Offset: 1

Antti Karttunen, Feb 26 2019

nonn

			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) = 2, as there are two nonzero digits.
For n=18, A002182(18) = 2520 = "110000" in primorial base because 2520 = 1*A002110(5) + 1*A002110(4) = 2310+210, thus a(18) = 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) = 3, as there are three nonzero digits.

Cf. A002100, A002182, A049345, A267263, A324382.

Cf. also A324341.

A267263(n) = { my(s=0); forprime(p=2, , if(n%p, s++, if(n==0, return(s))); n\=p); }; \\ From A267263
A324381(n) = A267263(A002182(n));

a(n) = A267263(A002182(n)).

a(n) <= A324382(n).

A324581 a(n) = A276086(A002182(n)).

Original entry on oeis.org

2, 3, 9, 5, 25, 625, 35, 875, 49, 2401, 117649, 77, 184877, 456533, 14641, 1771561, 214358881, 143, 20449, 2924207, 418161601, 8550986578849, 174859124550883201, 3575694237941010577249, 23298085122481, 1599034490244763, 32698656291015158587, 30466726698629, 39841104144361, 52099905419549, 89093921102069, 152355876914189, 260537564663909

Offset: 1

Antti Karttunen, Mar 09 2019

nonn

Note that gcd(a(n), A002182(n)) = A324198(A002182(n)) = 1 for all n because each term of A002182 is a product of primorial numbers (A002110).

Antti Karttunen, Table of n, a(n) for n = 1..670
Michael De Vlieger, Small chart of first terms.
Michael De Vlieger, 2400 x 3600 chart of first terms.
Index entries for sequences related to primorial base
Index entries for sequences related to primorial numbers

Subsequence of A324576.

Cf. A002110, A002182, A276086, A324198, A324381, A324382, A324582.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20], s = DivisorSigma[0, Range[10^5]], t}, t = Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]; Array[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[(*a002182[[#]]*)t[[#]], b] &, Length@ t]] (* Michael De Vlieger, Mar 18 2019 *)

\\ A002182 assumed to be precomputed
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324581(n) = A276086(A002182(n));

a(n) = A276086(A002182(n)).
a(n) = A324582(n)/A002182(n).
A001221(a(n)) = A324381(n).
A001222(a(n)) = A324382(n).

A330748 Index of the smallest element in A002182 that has exactly n prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 12, 14, 19, 21, 23, 32, 37, 47, 50, 62, 70, 80, 91, 99, 105, 109, 124, 140, 143, 159, 166, 182, 198, 217, 221, 240, 253, 276, 297, 304, 327, 352, 357, 381, 398, 424, 449, 475, 485, 512, 540, 570, 584, 617, 642, 676, 704, 738, 765, 770, 805, 841, 877, 913, 937, 949, 985, 1021, 1058, 1096, 1134, 1169

Offset: 0

Antti Karttunen, suggested by M. F. Hasler, Jan 10 2020

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..4257 (computed using A. Flammenkamp's 779674-term HCN dataset; terms 0..329 from _Antti Karttunen_)

Cf. A001222, A002182, A112778, A328521, A329902, A330743.

A330748(n) = { for(k=1,#v112778,if(v112778[k]==n,return(k))); -(1/0); };

v329902 = readvec("a329902.txt"); \\ File for the first 779674 terms of A329902
A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1])));
A330748list() = { my(m=Map(), lista=List([]), t); for(i=1, #v329902, t = A056239(v329902[i]); if(!mapisdefined(m,t), mapput(m,t,i))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v330748 = A330748list();
A330748(n) = v330748[1+n];
for(n=0,#v330748-1,write("b330748.txt", n, " ", A330748(n))); \\ Antti Karttunen, Jan 13 2020

a(n) = min{k: A112778(k)=n}.
A002182(a(n)) = A328521(n).
A329902(a(n)) = A330743(n).

A263096 Square roots of highly composite numbers, floored down: a(n) = A000196(A002182(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 6, 7, 10, 13, 15, 18, 26, 28, 35, 40, 50, 70, 86, 100, 122, 141, 158, 166, 212, 224, 235, 288, 332, 407, 470, 526, 576, 706, 744, 815, 848, 1039, 1200, 1470, 1697, 1898, 2079, 2546, 2684, 2940, 3287, 3796, 4158, 4649, 5694, 6062, 6575, 7826, 8573, 10500, 11068, 12125, 13556, 15653, 17147, 19172, 23480, 26426, 27113, 33206, 37373, 45772, 46961, 48248, 52853, 59092, 68233, 74746, 83568, 102350

Offset: 1

Antti Karttunen, Oct 10 2015

nonn

a(n) = number of strictly positive squares <= A002182(n).

Antti Karttunen, Table of n, a(n) for n = 1..1000 (based on the b-file of A002182 provided by _T. D. Noe_.)

Cf. A000196, A002182, A263097, A263098.

(define (A263096 n) (A000196 (A002182 n)))

a(n) = A000196(A002182(n)).

A272605 a(1) = 1, for n>=1 a(n) is the largest prime factor of A002182(n).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 13, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 19, 17, 17, 19, 19, 17, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 19, 23, 19

Offset: 1

Joerg Arndt, Nov 01 2016

nonn

For n>=1, the largest prime factor of the n-th highly composite number.

			The first highly composite numbers with their prime factorizations:
n:  A002182(n) = [factorization]
1:  1  = []
2:  2   = [2]
3:  4   = [2^2]
4:  6   = [2 * 3]
5:  12   = [2^2 * 3]
6:  24   = [2^3 * 3]
7:  36   = [2^2 * 3^2]
8:  48   = [2^4 * 3]
9:  60   = [2^2 * 3 * 5]
10:  120   = [2^3 * 3 * 5]
11:  180   = [2^2 * 3^2 * 5]
12:  240   = [2^4 * 3 * 5]
13:  360   = [2^3 * 3^2 * 5]
14:  720   = [2^4 * 3^2 * 5]
15:  840   = [2^3 * 3 * 5 * 7]
16:  1260   = [2^2 * 3^2 * 5 * 7]
17:  1680   = [2^4 * 3 * 5 * 7]
18:  2520   = [2^3 * 3^2 * 5 * 7]
19:  5040   = [2^4 * 3^2 * 5 * 7]
20:  7560   = [2^3 * 3^3 * 5 * 7]
21:  10080   = [2^5 * 3^2 * 5 * 7]
22:  15120   = [2^4 * 3^3 * 5 * 7]
23:  20160   = [2^6 * 3^2 * 5 * 7]
24:  25200   = [2^4 * 3^2 * 5^2 * 7]
25:  27720   = [2^3 * 3^2 * 5 * 7 * 11]
26:  45360   = [2^4 * 3^4 * 5 * 7]
27:  50400   = [2^5 * 3^2 * 5^2 * 7]
28:  55440   = [2^4 * 3^2 * 5 * 7 * 11]
29:  83160   = [2^3 * 3^3 * 5 * 7 * 11]
30:  110880   = [2^5 * 3^2 * 5 * 7 * 11]

Joerg Arndt, Table of n, a(n) for n = 1..19999

A308530 The number of largely composite numbers (A067128) having the same number of divisors as the n-th highly composite number A002182(n).

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 1, 1, 6, 2, 1, 2, 9, 1, 2, 2, 2, 8, 1, 2, 3, 2, 1, 1, 9, 1, 1, 3, 2, 5, 2, 2, 2, 8, 1, 2, 1, 6, 3, 6, 2, 2, 1, 6, 1, 1, 1, 5, 1, 3, 8, 1, 2, 5, 3, 10, 1, 2, 2, 4, 2, 3, 8, 2, 1, 6, 4, 11, 1, 1, 2, 2, 4, 2, 2, 6, 2, 4, 4, 8, 1, 5, 2, 1, 3, 4

Offset: 1

Amiram Eldar, Jun 06 2019

nonn

			a(4) = 3 since there are 3 largely composite numbers with the same number of divisors as the 4th highly composite number 6: 6, 8, and 10.

Michael De Vlieger, Table of n, a(n) for n = 1..783

Cf. A000005, A002182, A067128.

s = {}; dm = 1; c = 0; Do[d = DivisorSigma[0, n]; If[d == dm, c++]; If[d > dm, dm = d; AppendTo[s, c]; c = 1], {n, 1, 10^8}]; s

A308913 Highly composite numbers (A002182) that are not superabundant numbers (A004394).

Original entry on oeis.org

7560, 20160, 45360, 50400, 83160, 221760, 498960, 1081080, 2882880, 6486480, 14414400, 17297280, 32432400, 43243200, 110270160, 245044800, 294053760, 551350800, 2095133040, 2205403200, 4655851200, 5587021440, 10475665200, 64250746560, 73329656400, 97772875200

Offset: 1

Amiram Eldar, Jun 30 2019

nonn

Pillai noted in 1941 that 7560 is the first term of this sequence. He also asked for the opposite sequence and wondered whether its first term (A166735(1) = 1163962800) is within the reach of modern computation.

Since the sequence of superabundant numbers that are also highly composite, A166981, is finite, this sequence contains all the highly composite numbers above A002182(2567) = A004394(1023).

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
S. Sivasankaranarayana Pillai, Highly Composite Numbers of the t th Order, J. Indian Math. Soc., Vol. 8 (1944), pp. 61-74.

Cf. A002182, A004394, A166735, A166981, A181309.

seq = {}; dm = 0; sm = 0; Do[d = DivisorSigma[0, n]; s = DivisorSigma[1, n]; If[d > dm, dm = d]; If[s > s, sm = s, AppendTo[seq, n]], {n, 1, 3000000}]; seq

a(2118+i) = A002182(2567+i) for i > 0.

cp's OEIS Frontend

A181810 a(n) = largest number k such that A002182(n)/j is highly composite for each integer j from 1 to k.

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A262501 First differences of A002182 (highly composite numbers, definition 1).

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A321995 Indices of highly composite numbers A002182 which are between a twin prime pair.

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A324381 Number of nonzero digits when the n-th highly composite number is written in primorial base: a(n) = A267263(A002182(n)).

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A324581 a(n) = A276086(A002182(n)).

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A330748 Index of the smallest element in A002182 that has exactly n prime factors counted with multiplicity.

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A263096 Square roots of highly composite numbers, floored down: a(n) = A000196(A002182(n)).

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A272605 a(1) = 1, for n>=1 a(n) is the largest prime factor of A002182(n).

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A308530 The number of largely composite numbers (A067128) having the same number of divisors as the n-th highly composite number A002182(n).