A085301 -id:A085301 - cp's OEIS frontend

A085302 a(n) is the partial sum of A085301(j) from j=1 to n; a(n)-1 shows the number of factorials below n-th primorial.

2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 88, 89, 90, 92, 93

Offset: 1

Labos Elemer, Jun 26 2003

			For n = 26: a(26) = 34 since there are 33 factorials below the 26th primorial.

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

Cf. A085301, A002110, A000142.

Variant of A073071 and A048964.

fn[n_] := Module[{k = 1, r = n}, While[r >= 1, k++; r /= k]; k - 1]; prim[n_] := Times @@ Prime[Range[n]]; a[1] = 2; a[n_] := fn[prim[n]] + 1; Array[a, 100] (* Amiram Eldar, Feb 09 2025 *)

A085303 Positions of 2 in A085301.

Original entry on oeis.org

1, 2, 7, 10, 14, 17, 20, 23, 27, 30, 33, 37, 40, 43, 47, 50, 53, 57, 60, 63, 67, 70, 73, 77, 80, 84, 87, 90, 94, 97, 101, 104, 108, 111, 114, 118, 121, 125, 128, 132, 135, 139, 142, 146, 149, 153, 156, 160, 164, 167, 171, 174, 178, 181, 185, 188, 192, 196, 199, 203

Offset: 1

Labos Elemer, Jun 26 2003

nonn

Numbers k such that A085301(k) = 2, i.e., between primorial(k-1) and primorial(k) there are two distinct factorial numbers.

			10 is a term since between the 9th and the 10th primorials there are two factorials: 12! and 13!.
14 is a term since between the 13th and the 14th primorials there are two factorials: 17! and 18!.
584 is a term since between the 583rd and the 584th primorials there are two factorials: 745! and 746!.

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

Cf. A085301, A002110, A000142.

f[n_] := f[n] = Module[{k = 1, r = Times @@ Prime[Range[n]]}, While[r >= 1, k++; r /= k]; k - 1]; q[n_] := f[n] - f[n - 1] == 2; q[1] = q[2] = True; Select[Range[210], q] (* Amiram Eldar, Feb 18 2025 *)

f(n) = {my(k = 1); while(n >= 1, k++; n /= k); k-1;}
list(lim) = {my(c = 1, f1 = 1, r = 1, k = 0); print1("1, 2, "); forprime(p = 2, lim, k++; r* = p; f2 = f(r); if(f2 == f1 + 2, print1(k, ", ")); f1 = f2);} \\ Amiram Eldar, Feb 18 2025

Solutions x to A085301(x) = 2.

A106035 The "Octanacci" sequence: Trajectory of 1 under the morphism 1->{1,2,1}, 2->{1}.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Roger L. Bagula, May 05 2005

nonn

Silver mean chain substitution sequence: characteristic polynomial = -x^2+2*x+1.
A space-filling lattice is given by: bb = aa /. 1 -> {-0.4142135623730951, 2.414213562373095} /. 2 -> {1,-0.414213562373095`} /. 3 -> 0; ListPlot[FoldList[Plus, {0, 0}, bb], PlotRange -> All, PlotJoined -> False, Axes -> False];
The sequence is S_oo where S_0 = 2, S_1 = 1; S_{n+2} = S_{n+1} S_n S_{n+1}. Used to construct the "labyrinth" tiling. - N. J. A. Sloane, Mar 13 2019

G. C. Greubel, Table of n, a(n) for n = 0..1392
M. Baake and R. V. Moody, Self-Similar Measures for Quasicrystals, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, vol. 13, AMS, Providence, RI (2000), pp. 1-42; arXiv:math/0008063 [math.MG], 2000.
Clément Sire, Rémy Mosseri, and Jean-François Sadoc, Geometric study of a 2D tiling related to the octagonal quasiperiodic tiling, Journal de Physique 50.24 (1989): 3463-3476. See Eq. 2; HAL Id : jpa-00211156.
Index entries for sequences that are fixed points of mappings

Cf. A085301, A106036.

See A324772 for version over {0,1}.

f(1):= (1, 2, 1): f(2):= (1): A:= [1]:
for i from 1 to 6 do A:= map(f, A) od:
A; # - N. J. A. Sloane, Mar 13 2019

s[1] = {1, 2, 1}; s[2] = {1}; s[3] = {}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] aa = p[6]
Nest[Function[l, Flatten[l/.{1->{1, 2, 1}, 2->{1}}]], {1}, 6] (* Vincenzo Librandi, Mar 14 2019 *)
SubstitutionSystem[{1->{1,2,1},2->{1}},{1},{6}]//Flatten (* Harvey P. Dale, Nov 20 2021 *)

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A085302 a(n) is the partial sum of A085301(j) from j=1 to n; a(n)-1 shows the number of factorials below n-th primorial.

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A085303 Positions of 2 in A085301.

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A106035 The "Octanacci" sequence: Trajectory of 1 under the morphism 1->{1,2,1}, 2->{1}.

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