A087980 -id:A087980 - cp's OEIS frontend

A363250 Numbers in A363063 arranged in lexicographic order according to ordered prime signature (i.e., multiplicities of prime power factors p^k, written in order of p).

1, 2, 4, 12, 8, 24, 16, 48, 144, 720, 32, 96, 288, 1440, 864, 4320, 21600, 151200, 64, 192, 576, 2880, 1728, 8640, 43200, 302400, 128, 384, 1152, 5760, 3456, 17280, 86400, 604800, 10368, 51840, 259200, 1814400, 256, 768, 2304, 11520, 6912, 34560, 172800, 1209600, 20736, 103680, 518400, 3628800, 62208

Offset: 0

Michael De Vlieger, May 23 2023

The sequence is also readable as an irregular triangle by rows in which row n lists the terms divisible by 2^k but not by 2^(k+1).
Numbers m in A363063 are products of prime powers p(j)^S(j), j = 1..N, where p(j) is the j-th prime, such that p(j+1)^S(j+1) < p(j)^S(j). As consequence of definition of A363063, S(j) > S(j+1), hence multiplicities S(j) are distinct. Consequently, A363063 is a subset of A025487; m is a product of primorials. A025487 in turn is a subset of A055932.
These qualities enable us to write an algorithm that increments S(j) or drops the last term in S until we can increment S(j) to attain a solution. This algorithm generates terms in lexicographic order as described in the Name. The same qualities enable expression of m = Product p(j)^S(j) instead as Sum 2^(S(j)-1), a strictly increasing sequence.

			Table of n, a(n), and multiplicities S(j) written such that Product p(j)^S(j) = a(n). a(n) = A000079(i) is shown in the penultimate column, while a(n) = A347284(k) appears in the last column.
   n      a(n) multiplicities  i    k
  -----------------------------------
   0:       1                  0    0
   1:       2           1      1    1
   2:       4         2        2
   3:      12         2 1           2
   4:       8       3          3
   5:      24       3   1           3
   6:      16     4            4
   7:      48     4     1
   8:     144     4   2
   9:     720     4   2 1           4
  10:      32   5              5
  11:      96   5       1
  12:     288   5     2
  13:    1440   5     2 1
  14:     864   5   3
  15:    4320   5   3   1
  16:   21600   5   3 2
  17:  151200   5   3 2 1           5
  ...
Sequence read as an irregular triangle T(n, k):
  n\k   1    2    3     4     5     6      7       8
  ---------------------------------------------------
  0:    1
  1:    2
  2:    4   12
  3:    8   24
  4:   16   48  144   720
  5:   32   96  288  1440   864  4320  21600  151200
  6:   64  192  576  2880  1728  8640  43200  302400
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..14158 (rows i = 0..30, flattened)
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n,e), n = 1..3526, 12X vertical exaggeration

Cf. A000079, A025487, A055932, A067255, A087980, A124010, A347284, A363063.

Subsequence of A362227.

nn = 12;
 f[x_] := Times @@ MapIndexed[Prime[First[#2]]^#1 &, x];
 {1}~Join~Reap[Do[s = {i}; Sow[2^i]; Set[k, 1];
     Do[
      If[Prime[k]^s[[-1]] > Prime[k + 1],
       AppendTo[s, 1]; k++; Sow[f[s]],
       If[Length[s] == 1, Break[],
        If[Prime[k - 1]^(s[[-2]]) > Prime[k]^(s[[-1]] + 1),
         s[[-1]]++; Sow[f[s]],
         While[And[k > 1,
           Prime[k - 1]^(s[[-2]]) < Prime[k]^(s[[-1]] + 1)], k--;
          s = s[[1 ;; k]]]; If[k == 1, Break[], s[[-1]]++; Sow[f[s]] ]
          ] ] ], {j, Infinity}], {i, nn}]][[-1, -1]]

from sympy import nextprime,oo
from itertools import islice
primes = [2] # global list of first primes
def f(pi, ppmax):
    # Generate numbers with nonincreasing prime-powers <= ppmax, starting at the (pi+1)-st prime.
    if len(primes) <= pi: primes.append(nextprime(primes[-1]))
    p0 = primes[pi]
    if ppmax < p0:
        yield 1
        return
    pp = 1
    while pp <= ppmax:
        for x in f(pi+1, pp):
            yield pp*x
        pp *= p0
def A363250_list(nterms):
    return list(islice(f(0,oo),nterms)) # Pontus von Brömssen, May 25 2023

Seen as an irregular triangle, the first term in row i is 2^i, and the last term in row i is A347284(i).

Edited by Michael De Vlieger/_Peter Munn_, May 27 2025

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.

Original entry on oeis.org

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 *)

A343014 Number with a record number of divisors whose prime factorizations contain no repeated exponents.

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 48, 72, 96, 144, 288, 432, 576, 720, 864, 1152, 1440, 2160, 2880, 4320, 5760, 8640, 12960, 17280, 25920, 34560, 43200, 51840, 69120, 77760, 86400, 103680, 129600, 155520, 172800, 207360, 259200, 345600, 388800, 518400, 777600, 907200, 1036800

Offset: 1

Amiram Eldar, Apr 02 2021

nonn

Indices of records of A181796.
Since A181796(n) depends only on the prime signature of n, this sequence is a subsequence of A025487.
The corresponding record values are 1, 2, 3, 4, 5, 7, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, ... (see the link for more values).
From David A. Corneth, Apr 04 2021: (Start)
Subsequence of A087980 and of A181824.
Let G_m be the gcd of terms k with omega(k) = m. So G_1 <= 2, G_2 <= 12, G_3 <= 720, G_4 <= 907200.
Do we have G_m | G_(m + 1)? (End)

			A181796 begins with 1, 2, 2, 3, 2, 3, 2, 4, .... The record values, 1, 2, 3 and 4 occur at 1, 2, 4 and 8, which are the first 4 terms of this sequence.

David A. Corneth, Table of n, a(n) for n = 1..700 (first 500 terms from Amiram Eldar)
Amiram Eldar, Table of n, a(n), A181796(a(n)) for n = 1..500

Cf. A025487, A087980, A130091, A181796, A181824.

q[n_] := UnsameQ @@ FactorInteger[n][[;; , 2]]; s[n_] := DivisorSum[n, 1 &, q[#] &]; sm = 0; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

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A363250 Numbers in A363063 arranged in lexicographic order according to ordered prime signature (i.e., multiplicities of prime power factors p^k, written in order of p).

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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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A343014 Number with a record number of divisors whose prime factorizations contain no repeated exponents.

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Mathematica