A109375 -id:A109375 - cp's OEIS frontend

A093411 Divide n by the largest factorial that divides it and repeat until an odd number is reached, which will be the result; a(0) = 0.

0, 1, 1, 3, 1, 5, 1, 7, 1, 9, 5, 11, 1, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 1, 25, 13, 27, 7, 29, 5, 31, 1, 33, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 45, 23, 47, 1, 49, 25, 51, 13, 53, 9, 55, 7, 57, 29, 59, 5, 61, 31, 63, 1, 65, 11, 67, 17, 69, 35, 71, 3, 73, 37, 75, 19, 77, 13, 79

Offset: 0

Amarnath Murthy, Mar 30 2004

a(n) is odd for all positive n>0; a(n) = n iff n is odd.

			a(18) = 3, 18/6 = 3. though 18/2 = 9.

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial numbers

For bisection see A109375, for positions of ones, A344181.
Cf. A000142, A076934, A329379.
Cf. also A328478.

f[n_] := Block[{m = n, k = n}, While[k > 1, While[ IntegerQ[m/k! ], m = m/k! ]; k-- ]; m]; Table[ f[n], {n, 0, 79}] (* Robert G. Wilson v *)

A076934(n) = for(k=2, oo , if(n%k, return(n), n /= k));
A093411(n) = if(!n,n, if(n%2, n, A093411(A076934(n)))); \\ Antti Karttunen, May 18 2021

From Antti Karttunen, May 18 2021: (Start)
a(0) = 0, a(2n+1) = 2n+1, a(2n) = a(A076934(2n)).
a(n) = n / A329379(n).
(End)

Edited, corrected and extended by Robert G. Wilson v, Aug 25 2005

Definition further clarified by Antti Karttunen, May 18 2021

A264906 a(n) is the denominator of the 2nd term of the power series which is the loop length in a regular n-gon. (See comment.)

Original entry on oeis.org

25, 36, 49, 64, 81, 100, 121, 72, 169, 196, 225, 256, 289, 324, 361, 100, 441, 484, 529, 576, 625, 676, 729, 392, 841, 900, 961, 1024, 1089, 1156, 1225, 324, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 968, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 676, 2809

Offset: 5

Kival Ngaokrajang, Nov 28 2015

Inspired by A262343. Given a regular n-gon whose sides are of unit length, draw around each vertex V a circular arc connecting vertex V's two next-to-nearest neighbors. Connect the n arcs thus drawn into a single closed curve if n is odd, or a pair of identical (but rotated by 1/n of a turn) closed curves if n mod 4 = 2, or four identical (but rotated by 1/n of a turn) closed curves if n mod 4 = 0. (See illustration in Links.)

The values of the loop length L(n) appear to form a power series. Conjectures: the coefficient of the first term is 2*A060819; the numerator and denominator of the coefficient of the 2nd term are -1*A000265 and a(n), respectively; and the numerator of the coefficient of the 3rd term is A109375.

			L(5) = 2*Pi - 1/25*Pi^3 + 1/7500*Pi^5 - 1/5625000*Pi^7 + 1/7875000000*Pi^9 - ...
L(6) = 2*Pi - 1/36*Pi^3 + 1/15552*Pi^5 - 1/16796160*Pi^7 + 1/33861058560*Pi^9 - ...
L(7) = 6*Pi - 3/49*Pi^3 + 1/9604*Pi^5 - 1/14117880*Pi^7 + 1/38739462720*Pi^9 - ...
L(8) = 2*Pi - 1/64*Pi^3 + 1/49152*Pi^5 - 1/94371840*Pi^7 + 1/338228674560*Pi^9 - ...
L(9) = 10*Pi - 5/81*Pi^3 + 5/78732*Pi^5 - 1/38263752*Pi^7 + 1/173564379072*Pi^9 - ...
L(10) = 6*Pi - 3/100*Pi^3 + 1/40000*Pi^5 - 1/120000000*Pi^7 + 1/672000000000*Pi^9 - ...
...
Let T(n) be the total of the loop lengths, i.e., T(n) = L(n) if n is odd, 2*L(n) if n mod 4 = 2, and 4*L(n) if n mod 4 = 0. Multiplying each of the above series expansions for L(n) by the appropriate multiplier (i.e., 1, 2, or 4) to get T(n) gives expansions for L(5)..L(10) that agree with the general form
T(n) = 2*(n-4) * Sum_{k>=0} (-1)^k * Pi^(2k+1) / ((2k)! * n^(2k)) for n=5..10.

Kival Ngaokrajang, Illustration of loop length L(n) for n = 5..12

Cf. A000265, A060819, A109375, A262343.

[n^2 div Gcd((n-4) div Gcd(n-4,4),n): n in [5..60]]; // Vincenzo Librandi, Nov 29 2015

Table[n^2/GCD[(n - 4)/GCD[n - 4, 4], n], {n, 5, 46}] (* Michael De Vlieger, Nov 28 2015 *)

{for(n = 5, 100, k = 1; if (Mod(n,4)==0, k = 4); if (Mod(n,4)==2, k = 2); arc = 2*cos(x/n)*x*(1-4/n); loop = n*arc/k; print(loop))} \\ L(n)

{for(n = 5, 100, a = n^2/gcd((n-4)/gcd(n-4,4), n); print1(a, ", "))} \\ a(n)

a(n) = n^2/gcd((n-4)/gcd(n-4,4),n); for n >= 5.

More terms from Vincenzo Librandi, Nov 29 2015

cp's OEIS Frontend

A093411 Divide n by the largest factorial that divides it and repeat until an odd number is reached, which will be the result; a(0) = 0.

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A264906 a(n) is the denominator of the 2nd term of the power series which is the loop length in a regular n-gon. (See comment.)

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Mathematica

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