Peter Shor - cp's OEIS frontend

A270780 Let p_i = the i-th prime. a(i) is the smallest n>1 such that p_i divides n!-1.

3, 5, 9, 11, 5, 17, 4, 10, 15, 35, 39, 41, 45, 15, 18, 42, 48, 35, 17, 77, 41, 21, 43, 99, 96, 53, 22, 111, 125, 129, 120, 69, 25, 75, 155, 161, 83, 171, 177, 179, 189, 90, 195, 81, 105, 111, 82, 227, 101, 28, 239, 125, 255, 261, 267, 135, 236, 279, 141, 291

Offset: 3

Peter Shor, Mar 22 2016

Since p divides (p-2)!-1, the i-th term a(i) cannot be much larger than i log i.

			For i=3, the third prime is 5, and 5 divides 3!-1.
The 7th prime is 17, and 17 divides 5!-1, so a(7)=5.

Alois P. Heinz, Table of n, a(n) for n = 3..5000

Cf. A000040, A002982, A270838.

a:= proc(n) local k, p; p:=ithprime(n);
      for k from 2 do if irem(k!, p)=1 then return k fi od
    end:
seq(a(n), n=3..100);  # Alois P. Heinz, Mar 23 2016

snpd[p_]:=Module[{n=2},While[!Divisible[n!-1,p],n++];n]; Table[snpd[p],{p,Prime[Range[3,70]]}] (* Harvey P. Dale, Jun 06 2017 *)

More terms from Alois P. Heinz, Mar 23 2016

A226220 Maximum number of entries of the multiplication table mod p realizable additively, where p is the n-th prime.

Original entry on oeis.org

3, 6, 12, 19, 37, 47

Offset: 1

Peter Shor, May 31 2013

The maximum number of pairs i,j such that ij=f(i)+g(j) mod p, maximized over all functions f, g (mod p), for p=2,3,5,7,11... Lower bounds are known for the next five terms (starting with p=19): 66, 79, 99, 135, 148

			For the second term a(2)=6, p=3, and one can take f(0)=g(0)=f(1)=g(1)=0, f(2)=g(2)=2.

Y.-C. Liang, C.-W. Lim and D.-L. Deng, Reexamination of a multisetting Bell inequality for qudits, Phys. Rev. A, 80 (2009), 052116.

A151908 Number of nonisomorphic cube tilings of dimension n which can be constructed using the recipe presented at the beginning of Section 3 of the Lagarias-Shor paper.

Original entry on oeis.org

1, 2, 3, 7, 22, 95

Offset: 2

Peter Shor, Jul 30 2009

A weak lower bound for a(8) is 404.

It appears that there is exactly one trivial tiling in each dimension. If so, and this tiling is excluded, we get a sequence which potentially matches two existing sequences in the OEIS.

J. C. Lagarias and P. W. Shor, Cube-tilings of R^n and nonlinear codes, preprint, 1993.
J. C. Lagarias and P. W. Shor, Cube-tilings of R^n and nonlinear codes, Discrete and Computational Geometry, Vol. 11, pp. 359-391, 1994.

A064509 Marks (in fathoms) on lead line used by ships on the Mississippi River.

Original entry on oeis.org

2, 3, 5, 7, 10, 13, 15, 17, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255

Offset: 1

Peter Shor, Oct 06 2001

Certain depths have (or had) a visual and tactile indicator at positions on the lead line. All depths with such attachments are "marks". All others are "deeps." A leadsman measuring 12 feet of water calls "by the mark two (or twain)." If the depth on the lead is 36 feet (6 fathoms) he would call "by the deep six!".

Samuel Clemens chose the nom de plume Mark Twain because, for a riverboat skipper on the Mississippi, when the water was 12 feet deep, it was safe sailing for those boats.

Bowditch, The American Practical Navigator, 1931 edition.
Postings to newsgroup rec.org.sca, circa Oct 22, 1994 by djheydt(AT)uclink.berkeley.edu (Dorothy J. Heydt), Jeff Suzuki (jeffs(AT)math.bu.EDU) and Hal Ravn.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Mark S. Harris, Period measures and cautions for recipes.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

LinearRecurrence[{2, -1}, {2, 3, 5, 7, 10, 13, 15, 17, 20, 25}, 100] (* Paolo Xausa, Jul 13 2025 *)

For n >= 9, a(n) = 5(n-5).

a(55)-a(56) corrected by Sean A. Irvine, Jul 14 2023

A001309 Order of real Clifford group L_n connected with Barnes-Wall lattices in dimension 2^n.

Original entry on oeis.org

2, 16, 2304, 5160960, 178362777600, 96253116206284800, 819651496316379542323200, 110857799304670627788849414144000, 238987988705420266773820308079698247680000

Offset: 0

N. J. A. Sloane, Peter Shor

A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
G. Nebe, E. M. Rains and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
Index entries for sequences related to Barnes-Wall lattices.
Index entries for sequences related to groups.

2^(2n+2) times order of Chevalley group D_n (2) (cf. A001308). Twice A014115. See also A014116, A003956 (for the complex group).

Cf. A100221.

2^(n^2+n+2) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1);

a[0] = 2; a[n_] := 2^(n^2+n+2) * (2^n-1) * Product[2^(2*i)-1, {i, 1, n-1}]; Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jul 16 2015, after Maple *)

from math import prod
def A001309(n): return 2 if n == 0 else ((1<Chai Wah Wu, Jun 20 2022

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 2^(n^2+n+2) * (2^n-1) * Product_{k=1..n-1} (2^(2*k)-1).
a(n) ~ c * 2^(2*n^2+n+2), where c = A100221. (End)

A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

Original entry on oeis.org

8, 192, 92160, 743178240, 97029351014400, 203286581427673497600, 6819500449352277792129024000, 3660967964237442812098963052691456000, 31446995505814020383166371418359014222725120000

Offset: 0

N. J. A. Sloane and Peter Shor

T. D. Noe, Table of n, a(n) for n = 0..20
Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, Genons, Double Covers and Fault-tolerant Clifford Gates, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Edwin Pednault, An alternative approach to optimal wire cutting without ancilla qubits, arXiv:2303.08287 [quant-ph], 2023.
Tefjol Pllaha, Olav Tirkkonen, and Robert Calderbank, Binary Subspace Chirps, arXiv:2102.12384 [cs.IT], 2021.
Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
Peter Selinger, Generators and relations for n-qubit Clifford operators, arXiv:1310.6813 [quant-ph], 2013; Log. Methods Comput. Sci. 11 (2:10) (2015), 1-17, doi:10.2168/LMCS-11(2:10)2015.
Index entries for sequences related to groups.

Cf. A001309, A014116, A014115, A027672, A100221.

Equals twice A027638.

List([0..10], n-> 2^((n+1)^2 +2)*Product([1..n], j-> 4^j -1) ); # G. C. Greubel, Sep 24 2019

[n eq 0 select 8 else 2^((n+1)^2+2)*(&*[4^j-1: j in [1..n]]): n in [0..10]]; // G. C. Greubel, Sep 24 2019

a(n):= 2^(n^2+2*n+3)*mul(4^j-1, j=1..n); seq(a(n), n=0..10); # modified by G. C. Greubel, Sep 24 2019

Table[2^(n^2+2n+3) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, Nov 03 2017 *)

vector(11, n, 2^(n^2 +2)*prod(j=1,n-1, 4^j-1) ) \\ G. C. Greubel, Sep 24 2019

from math import prod
def A003956(n): return prod((1<Chai Wah Wu, Jun 20 2022

[2^((n+1)^2 +2)*product(4^j -1 for j in (1..n)) for n in (0..10)] # G. C. Greubel, Sep 24 2019

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 2^(n^2+2*n+3) * Product_{k=1..n} (4^k-1).
a(n) ~ c * 2^(2*n^2+3*n+3), where c = A100221. (End)

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User: Peter Shor

A270780 Let p_i = the i-th prime. a(i) is the smallest n>1 such that p_i divides n!-1.

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A226220 Maximum number of entries of the multiplication table mod p realizable additively, where p is the n-th prime.

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A151908 Number of nonisomorphic cube tilings of dimension n which can be constructed using the recipe presented at the beginning of Section 3 of the Lagarias-Shor paper.

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A064509 Marks (in fathoms) on lead line used by ships on the Mississippi River.

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A001309 Order of real Clifford group L_n connected with Barnes-Wall lattices in dimension 2^n.

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A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

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