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 1-3 of 3 results.

A383178 Numbers k such that omega(k) = 4 and p^omega(k) < k^(1/4) < lpf(k)^(omega(k)+1) for all primes p | k such that p > lpf(k), where lpf = A020639(k).

Original entry on oeis.org

81719, 268801, 565471, 626603, 631997, 657169, 700321, 799459, 838457, 893513, 916453, 1108927, 1212083, 1239389, 1271209, 1354681, 1366817, 1408637, 1420763, 1500313, 1527619, 1574359, 1602137, 1639877, 1700557, 1719871, 1751173, 1758203, 1775341, 1783511, 1843969
Offset: 1

Views

Author

Michael De Vlieger, May 09 2025

Keywords

Comments

Let primes p, q, r, s, p < q < r < s, divide k = a(n).
Then floor(log(k)/log(p)) = 4 and floor(log(k)/log(q)) = floor(log(k)/log(r)) = floor(log(k)/log(s)) = 3.
Let R(k) = row k of A162306 = {m <= k : rad(m) | k}. Then A010846(k) = c(k) is the number of terms in row k of A162306.
A010846(a(n)) = 51 for k such that p*r^3 < k.
A010846(a(n)) = 50 for k such that p*r^3 > k.

Examples

			Table of n, a(n), prime decomposition of a(n), and A010846(n) = c(n) for n = 1..12:
 n      a(n)    facs(a(n))  c(n)  p*r^3
--------------------------------------------------
 1     81719   11*17*19*23   51   11*19^3 = 75449
 2    268801   13*23*29*31   50   13*29^3 = 317057
 3    565471   17*29*31*37   51   17*31^3 = 506447
 4    626603   17*29*31*41   51   17*31^3 = 506447
 5    631997   19*29*31*37   51   19*31^3 = 566029
 6    657169   17*29*31*43   51   17*31^3 = 506447
 7    700321   19*29*31*41   51   19*31^3 = 566029
 8    799459   17*31*37*41   50   17*37^3 = 861101
 9    838457   17*31*37*43   50   17*37^3 = 861101
10    893513   19*31*37*41   50   19*37^3 = 962407
11    916453   17*31*37*47   51   17*37^3 = 861101
12   1108927   17*37*41*43   50   17*41^3 = 1171657
Let S(k) = row k of A162306 = {m <= k : rad(m) | k}.
Writing p^a*q^b*r^c*s^d instead as "abcd" (i.e., catenating prime power exponents), the following combinations are in S(a(n)). In brackets we show p*r^3, which is in S(a(n)) for n such that c(n) = 51, but not in S(a(n)) for n such that c(n) = 50.
0000 1000 2000 3000 4000    0010 1010 2010 3010    0020 1020 2020    0030 [1030]
0100 1100 2100 3100         0110 1110 2110         0120 1120
0200 1200 2200              0210 1210
0300 1300
.
0001 1001 2001 3001         0011 1011 2011         0021
0101 1101 2101              0111 1111
0201 1201
.
0002 1002 2002              0012
0102
.
0003

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..10000
  • Michael De Vlieger, Hasse diagram of R(268801) with logarithmic vertical scale. Gray represents the empty product, red represents primes, gold represents proper prime powers, green squarefree composites, and blue numbers that are neither squarefree nor prime powers.
  • Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 8X vertical exaggeration. The green bar at the bottom of the graph emphasizes the x axis that rides on the top edge of the bar.

Crossrefs

Cf. A010846, A020639, A162306, A383177.

Programs

  • Mathematica
    f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
  If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
  Union@ Reap[Do[
    While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
      If[Or[k == 1, Union[#2] == #1 - 1 & @@
        TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
        Sow[k]];
      j = 1; w[[-j]]++];
      If[j == i, Break[], j++; w[[-j]]++;
        w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
f[4, 2000000]

A383179 Numbers k such that omega(k) = 5 and p^omega(k) < k^(1/5) < lpf(k)^(omega(k)+1) for all primes p | k such that p > lpf(k), where lpf = A020639(k).

Original entry on oeis.org

101007559, 112442377, 145352341, 370621421, 392748073, 396181519, 403811399, 496492847, 510478561, 530733733, 540954893, 545683979, 552435703, 578262127, 580407131, 585416939, 590534717, 594163571, 620435209, 625790521, 633456391, 635140369, 643418423, 651300233
Offset: 1

Views

Author

Michael De Vlieger, May 09 2025

Keywords

Comments

A010846(a(n)) >= 176.

Examples

			Table of n, a(n), prime decomposition of a(n), and A010846(n) = c(n) for n = 1..12 and n = 209 (the smallest term with c(n) = 176):
  n         a(n)    facs(a(n))    c(a(n))
--------------------------------------
  1   101007559   23*41*43*47*53    180
  2   112442377   23*41*43*47*59    182
  3   145352341   23*43*47*53*59    179
  4   370621421   29*53*59*61*67    179
  5   392748073   29*53*59*61*71    180
  6   396181519   31*53*59*61*67    179
  7   403811399   29*53*59*61*73    181
  8   496492847   29*59*61*67*71    179
  9   510478561   29*59*61*67*73    179
 10   530733733   31*59*61*67*71    179
 11   540954893   29*59*61*71*73    179
 12   545683979   31*59*61*67*73    179
209  3433936673   41*83*97*101*103  176

Links

Crossrefs

Cf. A010846, A020639, A162306, A383177, A383178.

Programs

  • Mathematica
    f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
  If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
  Union@ Reap[Do[
    While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
      If[Or[k == 1, Union[#2] == #1 - 1 & @@
        TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
        Sow[k]];
      j = 1; w[[-j]]++];
      If[j == i, Break[], j++; w[[-j]]++;
        w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
f[5, 10^9, 5]

A384000 Smallest number k with n distinct prime factors such that A010846(k) = A024718(n) (a tight lower bound), or -1 if such k does not exist.

Original entry on oeis.org

1, 2, 6, 1001, 268801, 3433936673, 2603508937756211
Offset: 0

Views

Author

Michael De Vlieger, May 19 2025

Keywords

Comments

These numbers k have the smallest A010846(k) for a number with n distinct prime factors.
a(7) <= 1398483454696343742813089 = 1049 * 2819 * 3319 * 3433 * 3457 * 3463 * 3467.
a(8) <= 32829974457045619959776094471833047127947.

Examples

			Table of a(n), n = 0..6, showing prime decomposition and cardinality of row a(n) of A162306, c(n) = A010846(a(n)) = A024718(n).
n               a(n)   c(n)    prime factors of a(n)        a(n)
----------------------------------------------------------------------
0                  1     1     -
1                  2     2     2                            A000040(1)
2                  6     5     2,   3                       A138109(1)
3               1001    15     7,  11,  13                  A383177(1)
4             268801    50    13,  23,  29,  31             A383178(2)
5         3433936673   176    41,  83,  97, 101, 103        A383179(209)
6   2603508937756211   638   163, 373, 439, 457, 461, 463
Tables of terms m in r(a(n)) = row a(n) of A162306, writing instead only exponents i of prime power factors p^i | m for  each p | a(n), written in order of the prime base:
For n = 2, i.e., squarefree semiprime k in A138109 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-2 table:
  00  10  20
  01  11
Thus row 6 of A162306 has the following elements:
   1   2   4
   3   6
For n = 3, i.e., sphenic k in A383177 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-3 table:
  000 100 200 300     001 101 201     002
  010 110 210         011 111
  020 120
Thus row 1001 of A162306 has the following elements:
    1   7  49 343      13   91 637    169
   11  77 539         141 1001
  121 857

Crossrefs

Cf. A001700, A001221, A005117, A007947, A010846, A024718, A138109, A162306, A383177, A383178, A383179.
Showing 1-3 of 3 results.

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