cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A033273 Number of nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 9, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 9, 2, 2, 2, 6, 1, 9, 2, 4, 2, 2, 2, 10, 1, 4, 4, 7, 1, 5, 1, 6
Offset: 1

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000005, A001221, A010051, A027750.
Cf. A055079, A059992, A180040.
Cf. A001620, A077761.

Programs

Formula

a(n) = A000005(n) - A001221(n).
a((n)) = n and a(m) <> n for m < A055079(n). - Reinhard Zumkeller, Dec 16 2013
G.f.: Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Jan 17 2017
Dirichlet g.f.: zeta(s)*(zeta(s)-primezeta(s)). - Benedict W. J. Irwin, Jul 11 2018
Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 1 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Nov 27 2022

Extensions

More terms from Reinhard Zumkeller, Sep 02 2003
Corrected error in offset. - Jaroslav Krizek, May 04 2009
Extended by Ray Chandler, Aug 07 2010

A180040 Record values in A033273.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 13, 15, 17, 21, 27, 28, 29, 32, 33, 36, 37, 44, 45, 56, 60, 68, 76, 80, 86, 91, 92, 96, 104, 115, 116, 123, 139, 140, 155, 163, 175, 187, 195, 211, 219, 234, 235, 250, 251, 282, 283, 314, 330, 331, 354, 378, 394, 426, 442, 474, 498, 506, 570, 594
Offset: 1

Views

Author

Ray Chandler, Aug 07 2010

Keywords

Links

Crossrefs

Cf. A033273, A059992.

Programs

  • PARI
    lista(nn) = {my(m=0, nb); for (n=1, nn, nb = sumdiv(n, d, !isprime(d)); if (nb > m, m = nb; print1(nb, ", ")););} \\ Michel Marcus, Jul 16 2019

Formula

a(n) = A033273(A059992(n)).

A368523 Positive integers in decreasing order of tau(k)/k, where tau(k) = A000005(k).

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 12, 5, 10, 9, 18, 24, 16, 20, 7, 14, 15, 30, 36, 28, 48, 40, 60, 21, 42, 32, 11, 22, 72, 13, 26, 27, 54, 56, 84, 44, 45, 90, 120, 80, 96, 33, 66, 25, 50, 17, 34, 52, 35, 70, 108, 64, 19, 38, 144, 39, 78, 180, 63, 126, 168, 88, 132, 100, 112
Offset: 1

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Author

Keith J. Bauer, Dec 28 2023

Keywords

Comments

Because tau(k)/k is bounded by 2/sqrt(k), this sequence is well-defined.
In the case of ties, terms are sorted from least to greatest.
Let c be the j-th distinct value of tau(a(n))/a(n). Terms of this sequence for which tau(a(n))/a(n) >= c are the proper divisors of A059992(j + 1) for 1 <= j <= 4. True for j = 0 if the 0th value of c is taken to be infinity. Pattern breaks for j > 4.

Examples

			tau(1)/1 = tau(2)/2 = 1
tau(4)/4 = 3/4
tau(3)/3 = tau(6)/6 = 2/3
tau(8)/8 = tau(12)/12 = 1/2
tau(5)/5 = tau(10)/10 = 2/5
tau(9)/9 = tau(18)/18 = tau(24)/24 = 1/3

Crossrefs

Cf. A000005, A090387, A090395, A323383, A354768.

Programs

  • Lua
    length = 100
result = {}
for n = 1, length do
  local div_count = 0
  local root_n = math.sqrt(n)
  for d = 1, root_n do
    if n % d == 0 then
      div_count = div_count + 2
    end
  end
  if (root_n == math.floor(root_n)) then
    div_count = div_count - 1
  end
  result[n] = {n, div_count / n}
end
function compare(a, b)
  if a[2] ~= b[2] then
    return a[2] > b[2]
  else
    return a[1] < b[1]
  end
end
table.sort(result, compare)
i = 1
bound = 2 / math.sqrt(length)
while result[i][2] >= bound do
  io.write(result[i][1] .. ',')
  i = i + 1
end
  • Mathematica
    nmax = 100; s = Sort[Table[{k, DivisorSigma[0, k]/k}, {k, 1, nmax^2}], #1[[2]] >= #2[[2]] &]; Table[s[[j, 1]], {j, 1, nmax}] (* Vaclav Kotesovec, Jan 04 2024 *)

A376687 Numbers that set records in in A376281.

Original entry on oeis.org

24, 96, 120, 240, 360, 480, 840, 1080, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 27720, 30240, 55440, 60480, 83160, 110880, 151200, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 831600, 997920, 1330560, 1441440, 1663200, 2162160, 2882880
Offset: 1

Views

Author

Michael De Vlieger, Jan 08 2025

Keywords

Comments

Proper subset of the intersection A025487 and A379336.
There are three kinds of pairs (d, k/d), d | k, such that gcd(d, k/d) does not equal 1, d, or k/d:
Type A: rad(d) does not divide k/d, and rad(k/d) does not divide d (see A379752), where rad = A007947.
Type B: the squarefree kernel of one divisor divides the other but the reverse is not true (see A379772).
Type C: rad(d) = rad(k/d), i.e., d, k/d, and k are coreful (see A379552).
Conjecture: Numbers k that set records in A376281 do not have type C divisor pairs, i.e., those that are coreful but neither divides the other. This, since type C requires k to be powerful and divisible by cubes of 2 distinct primes (i.e., in A376936). Therefore the record is achieved only through large numbers of type A and B.
Since type A divisor pairs are common for composite k in A375055, this sequence is resembles A379752.
Since d and k/d are both composite, this sequence resembles A059992.
This sequence, to a lesser extent A379752, and a greater extent A059992, contains many highly composite numbers. (See plot of S(n) = union of this sequence and A002182 below, and corresponding graphs in respective other sequences.)

Examples

			Let b(n) = A376281(n).
Table showing exponents of prime power factors of a(n) for n = 1..20.
Example: a(5) = 360 = 2^3 * 3^2 * 5, hence we write "3.2.1".
   n    a(n)  Exp.   b(a(n))
  ----------------------------------
   1     24 *   3.1        1   4*6
   2     96     5.1        2   6*16 = 8*12
   3    120 **  3.1.1      3   4*30 = 6*20 = 10*12
   4    240 *   4.1.1      4   6*40 = 8*30 = 10*24 = 12*20
   5    360 **  3.2.1      5   4*90 = 10*36 = 12*30 = 15*24 = 18*20
   6    480     5.1.1      6   6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24
   7    840 *   3.1.1.1    7
   8   1080     3.3.1      9
   9   1680 *   4.1.1.1   10
  10   2160     4.3.1     11
  11   2520 **  3.2.1.1   13
  12   3360     5.1.1.1   14
*  = a(n) is highly composite (in A002182),
** = a(n) is superior highly composite (in both A002182 and A002201).

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..241
  • Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.
  • Michael De Vlieger, List of (d, k/d), d < k/d, k = a(n), n = 1..24, such that gcd(d, k/d) is not in {1, d, k/d}, showing type A in light gray, type B in either blue or gold, and type C (if it occurs) in black.
  • Michael De Vlieger, Prime power decomposition of a(n), n = 1..241.

Crossrefs

Cf. A002182, A002201, A007947, A025487, A059992, A379336, A376281, A377525, A379752.

Programs

  • Mathematica
    (* Load function f at A025487 *)
r = 0; s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not@*SquareFreeQ];
nn = Length[s]; Print[nn]
Reap[Monitor[
  Do[k = s[[i]];
    If[# > r, r = #; Sow[k]] &@
      Count[Transpose@{#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
        _?(And[1 < GCD @@ {##}, Mod[#1, #2] != 0,
           Mod[#2, #1] != 0] & @@ # &)], {i, nn}], i] ][[-1, 1]]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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