A065491 -id:A065491 - cp's OEIS frontend

A065479 Decimal expansion of Product_{p prime >= 3} (1 - 1/(p^2-p-1)).

7, 1, 5, 4, 6, 8, 2, 3, 5, 9, 8, 9, 9, 5, 5, 8, 4, 5, 0, 9, 4, 7, 7, 4, 7, 0, 5, 7, 1, 1, 7, 2, 8, 0, 7, 7, 6, 7, 5, 9, 7, 6, 2, 4, 8, 9, 8, 3, 7, 6, 7, 7, 6, 7, 4, 2, 6, 7, 2, 4, 7, 6, 9, 4, 4, 2, 4, 9, 5, 3, 5, 5, 5, 5, 1, 9, 7, 5, 5, 8, 5, 6, 8, 3, 3, 1, 5, 5, 5, 4, 0, 9, 0, 9, 0, 1, 1, 3

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.715468235989955845094774705711728...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A065491, A078089.

$MaxExtraPrecision = 600; digits = 98; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n;LR = LinearRecurrence[{2, 2, -3, -2}, {0, 0, -2, -3}, terms + 10]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p^2-p-1), 1, 3) \\ Amiram Eldar, Mar 15 2021

A376789 Table read by antidiagonals: T(n,k) is the number of Lyndon words of length k on the alphabet {0,1} whose prefix is the bitwise complement of the binary expansion of n with n >= 1 and k >= 1.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 3, 1, 0, 0, 6, 1, 1, 0, 0, 9, 2, 2, 1, 0, 0, 18, 2, 4, 1, 0, 0, 0, 30, 4, 7, 1, 0, 1, 0, 0, 56, 5, 14, 1, 1, 1, 0, 0, 0, 99, 8, 25, 2, 1, 2, 1, 0, 0, 0, 186, 11, 48, 2, 2, 3, 2, 1, 0, 0, 0, 335, 18, 88, 3, 3, 6, 4, 1, 0, 0, 0, 0

Offset: 1

Peter Kagey, Oct 04 2024

T(n,k) = 0 if n is in A366195.
Row 1 is A059966.
Row 2 is A006206 for n > 1.
Row 3 is A065491 for n > 2.
Row 4 is A065417.
Row 6 is A349904.

			Table begins
n\k| 1  2  3  4  5  6   7   8   9  10   11   12
---+-------------------------------------------
 1 | 1, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335
 2 | 0, 1, 1, 1, 2, 2,  4,  5,  8, 11,  18,  25
 3 | 0, 0, 1, 2, 4, 7, 14, 25, 48, 88, 168, 310
 4 | 0, 0, 1, 1, 1, 1,  2,  2,  3,  4,   6,   7
 5 | 0, 0, 0, 0, 1, 1,  2,  3,  5,  7,  12,  18
 6 | 0, 0, 1, 1, 2, 3,  6, 10, 18, 31,  56,  96
 7 | 0, 0, 0, 1, 2, 4,  8, 15, 30, 57, 112, 214
 8 | 0, 0, 0, 1, 1, 1,  1,  1,  2,  2,   3,   3
 9 | 0, 0, 0, 0, 0, 0,  1,  1,  1,  2,   3,   4
10 | 0, 0, 0, 0, 1, 1,  2,  3,  5,  7,  12,  18
11 | 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,   0,   0
12 | 0, 0, 0, 1, 1, 2,  3,  5,  9, 15,  26,  43
T(6,5) = 2 because 6 is 110 in base 2, its bitwise complement is 001, and there are T(6,5) = 2 length-5 Lyndon words that begin with 001: 00101 and 00111.

Cf. A059966, A006206, A065491, A065417, A349904.

Cf. A365746.

cp's OEIS Frontend

A065479 Decimal expansion of Product_{p prime >= 3} (1 - 1/(p^2-p-1)).

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