A218320 -id:A218320 - cp's OEIS frontend

A291928 Positions of records in A218320.

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 240, 288, 360, 480, 576, 720, 1080, 1440, 2160, 2520, 2880, 3360, 3600, 4320, 5040, 7200, 7560, 8640, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 60480, 75600, 90720, 100800, 110880, 120960

Offset: 1

Michael De Vlieger, Sep 06 2017

nonn

Distinct from A033833; first term not in A033833 is a(24) = 2520. There appear to be increasingly many terms a(n) not in A033833 as n increases.
A291834(13) = 192 is the smallest term not in a(n).
Subsequence of A025487.

David A. Corneth, Table of n, a(n) for n = 1..160

Cf. A002182, A007416, A025487, A033833, A218320.

f[n_, i_, t_] := f[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i], Sum[f[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; With[{s = Array[f[#, #, 4] &, 10^5]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Sep 06 2017, after Alois P. Heinz at A218320 *)

A291927 Records transform of A218320.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 15, 16, 20, 25, 27, 33, 36, 46, 50, 53, 77, 86, 118, 145, 158, 173, 174, 184, 224, 270, 282, 304, 330, 422, 522, 625, 656, 820, 881, 899, 1030, 1218, 1276, 1416, 1529, 1590, 1722, 2012, 2106, 2161, 2369, 2478, 2994, 3132, 3361, 3484

Offset: 1

Michael De Vlieger, Sep 06 2017

nonn

			A218320(n) for 1 <= n <= 24 is {1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7}; the records are {1, 2, 3, 4, 5, 7}, thus these are the first 6 terms of this sequence.

Cf. A033833, A218320.

f[n_, i_, t_] := f[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i], Sum[f[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Union@ FoldList[Max, Array[f[#, #, 4] &, 10^5]] (* Michael De Vlieger, Sep 06 2017, after Alois P. Heinz at A218320 *)

A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 11, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

Offset: 1

Amiram Eldar, Jun 16 2022

nonn

First differs from A305254 at n = 40, from A001055 and A252665 at n = 36, from A218320 at n = 32 and from A317791, A318559 and A326334 at n = 30.

			a(2) = 1 since numbers with 2 divisors are primes, i.e., numbers k with the single value Omega(k) = 1.
a(4) = 2 since numbers with 4 divisors are either of the following 2 forms: p1 * p2 with p1 and p2 being distinct primes, or of the form p^3 with p prime.
a(8) = 3 since numbers with 8 divisors are either of the following 3 forms: p1 * p2 * p3 with p1, p2 and p3 being distinct primes, p1 * p2^3, or p1^7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Row lengths of A355029.
Cf. A000005, A001222, A118914, A162247, A355027.
Cf. A001055, A218320, A305254, A317791, A318559, A326334.

Table[Length[Union[Total[#-1]& /@ f[n]]], {n, 1, 100}] (* using the function f by T. D. Noe at A162247 *)

a(n) <= A001055(n).
a(p) = 1 for p prime.
a(A355031(n)) = n.

A218431 Cyclic quadrilateral numbers: numbers m = abc*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.

Original entry on oeis.org

2304, 36864, 57600, 186624, 230400, 451584, 589824, 630000, 806400, 921600, 1440000, 2073600, 2822400, 2985984, 3686400, 4665600, 5531904, 6969600, 7225344, 8960000, 9437184, 10080000, 12672000, 12902400, 14745600, 15116544, 16257024, 18662400, 19079424

Offset: 1

Michel Lagneau, Oct 28 2012

A cyclic quadrilateral is a quadrilateral for which a circle can be circumscribed so that it touches each polygon vertex.
A cyclic quadrilateral number m is an integer with at least one decomposition m = a*b*c*d such that the area of the quadrilateral of sides (a,b,c,d) and the circumradius are integers. Because this property is not always unique, we introduce the notion of "cyclic quadrilateral order" for each cyclic quadrilateral number m, denoted by CQO(m). For example, CQO(2304) = 1 because the decomposition 2304 = 8*8*6*6 is unique with the quadrilateral (8,8,6,6) whose area A is given by Brahmagupta's formula: A = sqrt((s - a)*(s - b)*(s - c)*(s - d)) where the semiperimeter is s = (a+b+c+d)/2 and the circumradius R (the radius of the circumcircle) is given by R = sqrt((ab+cd)*(ac+bd)*(ad+bc))/(4A) => A = sqrt((14-8)*(14-8)*(14-6)*(14-6)) = 48, and R = 5, but CQO(2822400) = 2 because 2822400 = 24*24*70*70 = 40*40*42*42 and the area of the quadrilateral (24,24,70,70) equals 1680 with R = 37 and the area of the quadrilateral (40,40,42,42) also equals 1680 with R = 29.
The number of ways to write m = a*b*c*d with 1 <= a <= b <= c <= d <= m is given by A218320, thus: CQO(m) <= A218320(m).
If m is in this sequence, so is m*k^4 for any k > 0. Thus this sequence is infinite.
In view of the preceding comment, one might call "primitive" the terms m of the sequence for which there is no k > 1 such that m/k^4 is again a term of the sequence. These terms are 2304, 57600, 230400, 451584, 630000, ...

			2304 is in the sequence because 2304 = 8*8*6*6 and we obtain:
s = (8+8+6+6)/2 = 14;
A = sqrt((14-8)*(14-8)*(14-6)*(14-6)) = 48;
R = sqrt((8*8 + 6*6)*(8*6 + 8*6)*(8*6 + 8*6))/(4*48) = 5.

E. Gürel, Solution to Problem 1472, Maximal Area of Quadrilaterals, Math. Mag. 69 (1996), 149.
Eric W. Weisstein, MathWorld: Cyclic Quadrilateral.

Cf. A210250.

nn=200; lst={}; Do[s=(a+b+c+d)/2; If[IntegerQ[s], area2=(s-a)*(s-b)*(s-c)*(s-d); If[0

Typos in comment fixed by Zak Seidov and M. F. Hasler, Sep 21 2013, Sep 21 2013

A252665 Number of ways to write n as n = abcde with 1 <= a <= b <= c <= d <= e <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 10, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2

Offset: 1

Michel Lagneau, Dec 20 2014

nonn

Starts the same as, but is different from A218320 where a(n) = A218320(n) for n = 1..31. First values of n such that a(n) differs from A218320(n) are 32, 48, 64, 72, 80, ... .

Also starts the same as A001055, but differs from it for n = 64, ...

			a(12) = 4 because we can write 12 = 1*1*1*1*12 = 1*1*1*2*6 = 1*1*1*3*4 = 1*1*2*2*3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Records and first positions of records

Cf. A001055, A034836, A218320.

with(numtheory):
b:= proc(n, i, t) option remember;
      `if`(n=1, 1, `if`(t=1, `if`(n<=i, 1, 0),
       add(b(n/d, d, t-1), d=select(x->x<=i, divisors(n)))))
    end:
a:= proc(n) local l, m;
      l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);
      m:= mul(ithprime(i)^l[i][2], i=1..nops(l));
      b(m, m, 5)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 31 2017

Table[c=0; Do[If[i<=j<=k<=l<=m && i*j*k*l*m==n, c++], {i, t=Divisors[n]}, {j, t}, {k, t}, {l, t}, {m, t}]; c, {n, 90}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i], Sum[b[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Parallelize@ Array[b[#, #, 5] &@ Apply[Times, Power @@@ Sort[FactorInteger[#], #1[[2]] > #2[[2]] &]] &, 120] (* Michael De Vlieger, Aug 31 2017, after Jean-François Alcover at A218320 *)

A321566 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} 1/(1 - x^(i_1i_2i_3*i_4)).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 32, 44, 67, 91, 139, 186, 269, 362, 518, 687, 960, 1267, 1747, 2294, 3106, 4052, 5449, 7063, 9365, 12092, 15914, 20422, 26639, 34029, 44090, 56075, 72108, 91303, 116802, 147264, 187210, 235182, 297562, 372346, 468777, 584553, 732803, 910744

Offset: 0

Seiichi Manyama, Nov 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Product_{1 <= i_1 <= i_2 <= ... <= i_b} 1/(1 - x^(i_1 * i_2 * ... * i_b)): A000041 (b=1), A182269 (b=2), A321360 (b=3), this sequence (b=4).

Cf. A066739, A218320, A321567.

Euler transform of A218320.

G.f.: Product_{k>0} 1/(1 - x^k)^A218320(k).

A321567 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 + x^(i_1i_2i_3*i_4)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 16, 21, 27, 38, 49, 63, 84, 109, 138, 180, 228, 289, 369, 463, 578, 732, 911, 1128, 1407, 1741, 2140, 2646, 3243, 3967, 4861, 5924, 7196, 8767, 10616, 12827, 15516, 18707, 22486, 27054, 32440, 38835, 46488, 55502, 66136, 78836, 93727, 111265

Offset: 0

Seiichi Manyama, Nov 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{1 <= i_1 <= i_2 <= ... <= i_b} (1 + x^(i_1 * i_2 * ... * i_b)): A000009 (b=1), A211856 (b=2), A321359 (b=3), this sequence (b=4).

Cf. A066806, A218320, A321566.

G.f.: Product_{k>0} (1 + x^k)^A218320(k).

A291833 Records transform of A252665.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 18, 21, 28, 30, 37, 43, 51, 53, 59, 66, 92, 103, 150, 188, 189, 235, 239, 312, 351, 396, 400, 493, 593, 628, 751, 947, 954, 1283, 1433, 1632, 2063, 2074, 2088, 2446, 2629, 3143, 3582, 3952, 4008, 4121, 4602, 5803, 6046, 6323, 6899

Offset: 1

Michael De Vlieger, Sep 03 2017

nonn

See a-file "Records and first positions of records in A252665" in that sequence for more information. - Michael De Vlieger, Sep 03 2017

			A252665(n) for 1 <= n <= 24 is {1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7}; the records are {1, 2, 3, 4, 5, 7}, thus these are the first 6 terms of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..76

Cf. A033833, A252665, A291834.

f[n_, i_, t_] := b[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i],
Sum[f[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Union@ FoldList[Max, Array[f[#, #, 5] &, 10^5]] (* Michael De Vlieger, Sep 03 2017, after Alois P. Heinz at A218320 *)

A319517 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 - x^(i_1i_2i_3*i_4)).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 0, 2, -1, 0, 3, -1, -2, -1, 1, -6, -1, 0, 0, 0, 7, -1, 1, -2, 4, 1, -2, 11, 1, -2, -10, 11, -11, 15, -16, -6, -7, -10, -1, 10, -5, -10, 12, -20, 19, -16, 24, -2, 28, -9, 41, -6, 15, 20, 4, -21, -15, -13, -14, 13, -73, 67, -30, -44, -19, 31, -30

Offset: 0

Seiichi Manyama, Nov 14 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A321566.
Product_{1 <= i_1 <= i_2 <= ... <= i_b} (1 - x^(i_1 * i_2 * ... * i_b)): A010815 (b=1), A321299 (b=2), A321361 (b=3), this sequence (b=4).
Cf. A218320, A321460, A321567.

G.f.: Product_{k>0} (1 - x^k)^A218320(k).

A321379 Number of ways to write n as n = abc*d with 1 < a <= b <= c <= d < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 4

Offset: 1

Seiichi Manyama, Nov 08 2018

nonn

This sequence is different from A101638.

If p is prime, a(p^k) = A026810(k). - Robert Israel, Nov 08 2018

			16 = 2*2*2*2. So a(16) = 1.
24 = 2*2*2*3. So a(24) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A026810, A122179, A218320.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for a from 2 to floor(N^(1/4)) do
  for b from a to floor((N/a)^(1/3)) do
    for c from b to floor((N/a/b)^(1/2)) do
      for d from c to N/(a*b*c) do
        V[a*b*c*d]:= V[a*b*c*d]+1
od od od od:
convert(V,list); # Robert Israel, Nov 08 2018

cp's OEIS Frontend

A291928 Positions of records in A218320.

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A291927 Records transform of A218320.

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A355030 a(n) is the number of possible values of the number of prime divisors (counted with multiplicity) of numbers with n divisors.

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A218431 Cyclic quadrilateral numbers: numbers m = a*b*c*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.

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A252665 Number of ways to write n as n = a*b*c*d*e with 1 <= a <= b <= c <= d <= e <= n.

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A321566 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} 1/(1 - x^(i_1*i_2*i_3*i_4)).

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A321567 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 + x^(i_1*i_2*i_3*i_4)).

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A291833 Records transform of A252665.

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A319517 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 - x^(i_1*i_2*i_3*i_4)).

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A218431 Cyclic quadrilateral numbers: numbers m = abc*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.

A252665 Number of ways to write n as n = abcde with 1 <= a <= b <= c <= d <= e <= n.

A321566 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} 1/(1 - x^(i_1i_2i_3*i_4)).

A321567 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 + x^(i_1i_2i_3*i_4)).

A319517 Expansion of Product_{1 <= i_1 <= i_2 <= i_3 <= i_4} (1 - x^(i_1i_2i_3*i_4)).

A321379 Number of ways to write n as n = abc*d with 1 < a <= b <= c <= d < n.