A212165 -id:A212165 - cp's OEIS frontend

A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

1, 2, 4, 12, 24, 36, 48, 120, 240, 360, 720, 1680, 5040, 10080, 15120, 20160, 25200, 45360, 50400, 110880, 221760, 332640, 554400, 665280, 2882880, 8648640, 14414400, 17297280, 43243200, 294053760

Offset: 1

Matthew Vandermast, May 22 2012

Sequence can be used to find the largest highly composite number in subsequences of A212165 (of which there are several in the database).
Ramanujan showed that, in the canonical prime factorization of a highly composite number with largest prime factor prime(n), the largest exponent cannot exceed 2*log_2(prime(n+1)). (See formula 54 on page 15 of the Ramanujan paper.) This limit is less than n for all n >= 9 (and prime(n) >= 23).
1. Direct calculation verifies this for 9 <= n <= 11.
2. Nagura proved that, for any integer m >= 25, there is always a prime between m and 1.2*m.  Let n = 11, at which point prime(11) = 31 and log_2(prime(n+1)) = log 37/log 2 = 5.209453.... Since log 1.2/log 2 is only 0.263034..., it follows that n must increase by at least 3k before 2*log_2(prime(n+1)) can increase by 2k, for all values of k. Therefore, 2*log_2(prime(n+1)) can never catch up to prime(n) for n > 11.
665280 = 2^6*3^3*5*7*11 is the largest highly composite number whose prime factorization contains an exponent that is strictly greater than the number of positive exponents in that factorization (including the implied 1's).

			A002182(62) = 294053760 = 2^7*3^3*5*7*11*13*17 has 7 positive exponents in its prime factorization, including 5 implied 1's. The maximal exponent in its prime factorization is also 7. Therefore, 294053760 is a term of this sequence.

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

A. Flammenkamp, List of the first 1200 highly composite numbers
J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952), 177-181.
S. Ramanujan, Highly Composite Numbers (p. 15) (pages 11-12 introduce some of the notation in formula 54)

A212165 also includes all terms in A006939, A066120, A087980, A130091, A138534, A141586, A166475, A181555, A181813-A181814, A181818, A181823-A181825, A182763.

Other finite subsequences of A002182 include A072938, A095921, A106037, A136253, A162935, A166981, A168263.

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; a = 0; t = {}; Do[b = DivisorSigma[0, n]; If[b > a, a = b; If[okQ[n], AppendTo[t, n]]], {n, 10^6}]; t (* T. D. Noe, May 24 2012 *)

A212166 Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Matthew Vandermast, May 22 2012

			36 = 2^2*3^2 has 2 positive exponents in its prime factorization. The maximal exponent in its prime factorization is also 2. Therefore, 36 belongs to this sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Includes subsequences A000040, A006939, A138534, A181555, A181825.
See also A212164, A212165, A212167, A212168.
Cf. A001221, A050326, A051903, A188654 (complement), A225230.

import Data.List (elemIndices)
a212166 n = a212166_list !! (n-1)
a212166_list = map (+ 1) $ elemIndices 0 a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] == Length[f]]; Select[Range[424], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) == #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) = 0; A050326(a(n)) = 1. - Reinhard Zumkeller, May 03 2013

A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142, 143

Offset: 1

Matthew Vandermast, May 22 2012

nonn

A225230(a(n)) > 1; A050326(a(n)) > 1. - Reinhard Zumkeller, May 03 2013

Subsequence of A130092. - Ivan N. Ianakiev, Sep 17 2019

			10 = 2^1*5^1 has 2 distinct prime factors, hence 2 positive exponents in its prime factorization (although the 1s are often left implicit). 2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 belongs to the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages)

Complement of A212165. See also A212164, A212166-A212167.

Subsequence of A188654.

import Data.List (findIndices)
a212168 n = a212168_list !! (n-1)
a212168_list = map (+ 1) $ findIndices (> 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] < Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)
Select[Range[200],Max[FactorInteger[#][[All,2]]]Harvey P. Dale, Nov 21 2018 *)

is(n,f=factor(n))=my(e=f[,2]); #e && vecmax(e)<#e \\ Charles R Greathouse IV, Jan 09 2022

A212164 Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k)).

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Matthew Vandermast, May 22 2012

			40 = 2^3*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (namely, 3 and 1, although the 1 is often left implicit).   2 is less than the maximal exponent in 40's prime factorization, which is 3. Therefore, 40 belongs to the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212167.
See also A212165, A212166, A212168.
Cf. A001221, A050326, A051903, A225230.
Subsequence of A188654.

import Data.List (elemIndices)
a212164 n = a212164_list !! (n-1)
a212164_list = map (+ 1) $ findIndices (< 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] > Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); #e && vecmax(e) > #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) < 0; A050326(a(n)) = 0. - Reinhard Zumkeller, May 03 2013

A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Matthew Vandermast, May 22 2012

nonn

Union of A212166 and A212168. Includes numerous subsequences that are subsequences of neither A212166 nor A212168.

			40 = 2^3*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although the 1 is often left implicit).  2 is less than the maximal exponent in 40's prime factorization, which is 3. Therefore, 40 does not belong to the sequence. But 10 = 2^1*5^1 and 20 = 2^2*5^1 belong, since the maximal exponents in their prime factorizations are 1 and 2 respectively.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212164. See also A212165.
Subsequences (none of which are subsequences of A212166 or A212168) include A002110, A051451, A129912, A179983, A181826, A181827, A182862, A182863. Includes all members of A003418.
Cf. A001221, A050326, A051903, A188654, A225230.

import Data.List (findIndices)
a212167 n = a212167_list !! (n-1)
a212167_list = map (+ 1) $ findIndices (>= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

isA212167 := proc(n)
    simplify(A051903(n) <= A001221(n)) ;
end proc:
for n from 1 to 1000 do
    if isA212167(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 06 2021

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] <= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) <= #e; } \\ Amiram Eldar, Sep 09 2024

A225230(a(n)) >= 0; A050326(a(n)) > 0. - Reinhard Zumkeller, May 03 2013

A211991 Difference between the arithmetic derivative of n and the sum of proper divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, -1, 0, 5, 2, -1, 0, 0, 0, -1, -1, 17, 0, 0, 0, 2, -1, -1, 0, 8, 4, -1, 14, 4, 0, -11, 0, 49, -1, -1, -1, 5, 0, -1, -1, 18, 0, -13, 0, 8, 6, -1, 0, 36, 6, 2, -1, 10, 0, 15, -1, 28, -1, -1, 0, -16, 0, -1, 10, 129, -1, -17, 0, 14, -1, -15, 0, 33

Offset: 1

Omar E. Pol, Dec 18 2012

sign

Observations: at least the first 50 indices of nonnegative terms are also the first 50 terms of A212165. Also at least the first 28 indices of negative terms are also the first 28 terms of A212168, since A212168 is the complement of A212165.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A001065, A003415.

Cf. A013661, A136141.

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Table[dn[n] - (DivisorSigma[1, n] - n), {n, 100}] (* T. D. Noe, Dec 27 2012 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A211991(n) = (A003415(n) - (sigma(n)-n)); \\ Antti Karttunen, Mar 08 2018

a(n) = A003415(n) - A001065(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (A136141 - A013661 + 1) / 2 = 0.0641113... . - Amiram Eldar, Mar 17 2024

A225230 In the canonical prime factorization of n: (number of distinct primes) minus (largest prime exponent).

Original entry on oeis.org

0, 0, 0, -1, 0, 1, 0, -2, -1, 1, 0, 0, 0, 1, 1, -3, 0, 0, 0, 0, 1, 1, 0, -1, -1, 1, -2, 0, 0, 2, 0, -4, 1, 1, 1, 0, 0, 1, 1, -1, 0, 2, 0, 0, 0, 1, 0, -2, -1, 0, 1, 0, 0, -1, 1, -1, 1, 1, 0, 1, 0, 1, 0, -5, 1, 2, 0, 0, 1, 2, 0, -1, 0, 1, 0, 0, 1, 2, 0, -2, -3

Offset: 1

Reinhard Zumkeller, May 03 2013

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001221, A051903.

Cf. A188654, A212164, A212165, A212166, A212167, A212168.

Haskell
```
a225230 n = a001221 n - a051903 n
```

a[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, Length[e] - Max[e]]; Array[a, 100] (* Amiram Eldar, Sep 09 2024 *)

a(n) = if (n>1, my(f=factor(n)); #f~ - vecmax(f[,2]), 0); \\ Michel Marcus, Jan 26 2022

a(n) = A001221(n) - A051903(n).

a(A212164(n)) < 0; a(A212165(n)) <= 0; a(A212166(n)) = 0; a(A188654(n)) <> 0; a(A212167(n)) >= 0; a(A212168(n)) > 0.

A367685 Numbers divisible by their multiset multiplicity kernel.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107

Offset: 1

Gus Wiseman, Nov 30 2023

nonn

First differs from A344586 in lacking 120.
We define the multiset multiplicity kernel (MMK) of a positive integer n to be the product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n. For example, 90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so MMK(90) = 12. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.
First differs from A212165 at n=73: A212165(73)=120 is not a term of this. - Amiram Eldar, Dec 04 2023

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   23: {9}
   24: {1,1,1,2}

Includes all prime-powers A000961.
The only squarefree terms are the primes A008578.
Partitions of this type are counted by A367684.
A007947 gives squarefree kernel.
A027746 lists prime factors, length A001222, indices A112798.
A027748 lists distinct prime factors, length A001221, indices A304038.
A071625 counts distinct prime exponents.
A124010 gives multiset of multiplicities (prime signature), sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A367579 lists MMK, ranks A367580, sum A367581, max A367583.
Cf. A005117, A039956, A051904, A052409, A072774, A130091, A367582, A367584, A367586, A367682.

mmk[n_Integer]:= Product[Min[#]^Length[#]&[First/@Select[FactorInteger[n], Last[#]==k&]], {k,Union[Last/@FactorInteger[n]]}];
Select[Range[100], Divisible[#,mmk[#]]&]

A344586 Numbers k for which A003415(k) >= A001065(k), where A003415 gives the arithmetic derivative, and A001065 is the sum of proper divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 127, 128, 131

Offset: 1

Antti Karttunen, May 24 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001065, A003415.
Cf. A212127, A212128 (subsequences), A344585 (complement).
Positions of nonnegative terms in A211991.
Differs from A212165 for the first time at n=121, where a(121) = 220, while A212165(121) = 223.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA344586(n) = (A003415(n) >= (sigma(n)-n));

A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 6, 10, 11, 13, 17, 20, 30, 36, 44, 55, 70, 86, 98, 128, 156, 190, 235, 288, 351, 409, 499, 603, 722, 863, 1025, 1227, 1461, 1757, 2061, 2444, 2892, 3406, 3996, 4708, 5497, 6430, 7595, 8835, 10294, 12027, 13971, 16252, 18887, 21878

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

These are also integer partitions of n with the same number of distinct parts as maximal anti-runs of parts.

			The partition (5,3,2,1,1,1,1) has 4 runs ((5),(3),(2),(1,1,1,1)) and 4 anti-runs ((5,3,2,1),(1),(1),(1)) so is counted under a(14).
The a(1) = 1 through a(10) = 10 reversed partitions (A = 10):
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (A)
                 (112)  (113)  (114)  (115)  (116)    (117)    (118)
                        (122)         (133)  (224)    (144)    (226)
                                      (223)  (233)    (225)    (244)
                                             (11123)  (11124)  (334)
                                                      (11223)  (11125)
                                                               (11134)
                                                               (11224)
                                                               (11233)
                                                               (12223)

Christian Sievers, Table of n, a(n) for n = 0..1000

The RHS is counted by A116608, rank statistic A297155.
The LHS is counted by A133121, rank statistic A046660.
For related inequalities see A212165, A212168, A361204.
For subsets instead of partitions see A217615, A385572, A385575.
These partitions are ranked by A385576.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number maximal runs, for partitions A384881, strict A116674.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A268193 counts partitions by maximal anti-runs, strict A384905, subsets A384893.
A355394 counts partitions with neighbors, complement A356236.
Cf. A000071, A003114, A008284, A010027, A047966, A210034, A325324, A325325, A356606, A384882, A384885.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union[#]]==Length[Split[#,#2!=#1&]]&]],{n,0,30}]

lista(n)=Vec(polcoef((prod(i=1,n,1+x^i/(t*(1-t*x^i))+O(x*x^n))-1)*t+1,0,t)) \\ Christian Sievers, Jul 18 2025

For a partition p, let s(p) be its sum, e(p) the number of equal adjacent pairs, and d(p) the number of distinct adjacent pairs. Then Sum_{p partition} x^s(p) * t^(e(p)-d(p)) = (Product_{i>=1} (1 + x^i/(t*(1-t*x^i))) - 1) * t + 1, so a(n) is the coefficient of x^n*t^0 of this expression. - Christian Sievers, Jul 18 2025

cp's OEIS Frontend

A212169 List of highly composite numbers (A002182) with an exponent in its prime factorization that is at least as great as the number of positive exponents; intersection of A002182 and A212165.

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A212166 Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)).

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A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

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A212164 Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k)).

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A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).

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A211991 Difference between the arithmetic derivative of n and the sum of proper divisors of n.

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A225230 In the canonical prime factorization of n: (number of distinct primes) minus (largest prime exponent).

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