A031883 -id:A031883 - cp's OEIS frontend

A031884 Smaller of a pair of consecutive lucky numbers with a gap of 2n.

1, 3, 15, 79, 141, 51, 787, 495, 171, 937, 903, 745, 2851, 1611, 1057, 3811, 5769, 4893, 8719, 10683, 9841, 24073, 9567, 28245, 25687, 3507, 26943, 35743, 44805, 51979, 64723, 23205, 50779, 51475, 264075, 155833, 238057, 178755, 143311, 400591, 223095, 181581, 466813

Offset: 1

Patrick De Geest

nonn

a(n) is the first occurrence of a difference of 2n between A000959(k+1) and A000959(k). - Robert G. Wilson v, May 12 2006
a(136) > 10^9. - Donovan Johnson, Dec 07 2011
Unknown terms a(166), a(176), a(178), a(182), and a(185) through a(209) are all greater than 10^10. - Kevin P. Thompson, Nov 24 2021

			a(4) = 79 since the lucky numbers A000959(20) = 79 and A000959(21) = 87 are the first consecutive pair with difference 2*4 = 8.

Kevin P. Thompson, Table of n, a(n) for n = 1..165 (terms 1..135 from Donovan Johnson)
Kevin P. Thompson, Table of n, a(n) for n = 1..210 with unknown terms

Cf. A000959, A031885, A031883, A031158, A031886, A031888, A031890, A031892, A031894, A031896, A031898, A031900, A031902.

lst = Range[1, 10^6, 2]; i = 2; While[ i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++ ]; f[n_] := Block[{k = 1}, While[t[[k + 1]] - t[[k]] != 2n, k++ ]; t[[k]]]; Array[f, 41] (* Robert G. Wilson v, May 12 2006 *)

More terms from Robert G. Wilson v, May 12 2006

A184827 a(n) = largest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 5, 5, 11, 9, 17, 19, 29, 29, 31, 37, 47, 39, 59, 65, 65, 71, 71, 71, 81, 87, 93, 99, 107, 103, 125, 125, 131, 129, 131, 143, 155, 157, 167, 153, 185, 191, 189, 197, 199, 203, 215, 215, 227, 233, 233, 223, 257, 255, 261, 263

Offset: 1

Rémi Eismann, Jan 23 2011

From the definition, a(n) = A000959(n) - A031883(n) if A000959(n) - A031883(n) > A031883(n), 0 otherwise where A000959 are the lucky numbers and A031883 are the gaps between lucky numbers.

			For n = 1 we have A000959(1) = 1, A000959(2) = 3; there is no k such that 3 - 1 = 2 = (1 mod k), hence a(1) = 0.
For n = 3 we have A000959(3) = 7, A000959(4) = 9; 5 is the largest k such that 9 - 7 = 2 = (7 mod k), hence a(3) = 5; a(3) = 7 -2 = 5.
For n = 24 we have A000959(24) = 105, A000959(25) = 111; 99 is the largest k such that 111 - 105 = 6 = (105 mod k), hence a(24) = 99; a(24) = 105 - 6 = 99.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A000959, A031883, A130889, A184828, A117078, A117563, A001223, A118534.

A184828 a(n) = A184827(n)/A130889(n) unless A130889(n) = 0 in which case a(n) = 0.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 13, 13, 1, 1, 1, 9, 3, 3, 11, 1, 1, 25, 25, 1, 3, 1, 13, 31, 1, 1, 3, 37, 1, 27, 1, 1, 7, 43, 5, 1, 1, 1, 1, 1, 17, 29, 1, 1, 1, 1, 3, 23, 5, 1, 45, 19, 19, 7, 31, 1, 5, 1, 1, 1, 43, 1, 31, 1, 5, 85, 85, 5, 1, 11, 43, 3

Offset: 1

Rémi Eismann, Jan 23 2011

nonn

a(n) is the "level" of lucky numbers.
The decomposition of lucky numbers into weight * level + gap is A000959(n) = A130889(n) * a(n) + A031883(n) if a(n) > 0.
A184827(n) = A000959(n) - A031883(n) if A000959(n) - A031883(n) > A031883(n), 0 otherwise.

			For n = 1 we have A130889(1) = 0, hence a(1) = 0.
For n = 3 we have A184752(3)/A130889(3)= 5 / 5 = 1; hence a(3) = 1.
For n = 24 we have A184752(24)/A130889(24)= 99 / 9 = 11; hence a(24) = 11.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A000959, A031883, A130889, A184827, A117078, A117563, A001223, A118534.

A260722 Difference between n-th odd Ludic and n-th Lucky number: a(1) = 0; for n > 1: a(n) = A003309(n+1) - A000959(n).

Original entry on oeis.org

0, 0, -2, -2, -2, -2, -4, -2, -6, -4, 0, -2, -6, -4, -10, -6, -2, -2, 2, 4, 2, -2, -2, 2, 4, 4, -6, -2, -2, 8, 8, 6, 2, 10, 6, 8, -8, 0, 14, 10, 16, 12, 8, 10, 4, 4, 10, 16, 6, 16, 16, 14, 18, 22, 24, 32, 28, 30, 22, 32, 32, 30, 38, 34, 32, 36, 40, 30, 28, 28, 32, 24, 22, 24, 36, 38, 42, 30, 30, 22, 26, 26, 30, 38, 40, 30, 36, 46, 48, 46, 56, 54, 54, 54, 40, 46

Offset: 1

Antti Karttunen, Aug 06 2015

sign

Equally: for n >= 2, the difference between (n+1)-th Ludic and n-th Lucky number.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000959, A003309, A031883, A260721 (same terms divided by two), A260723, A256486, A256487.

Cf. also permutations A260435, A260436, A260741, A260742.

(define (A260722 n) (if (= 1 n) 0 (- (A003309 (+ 1 n)) (A000959 n))))

a(1) = 0; for n > 1: a(n) = A003309(n+1) - A000959(n).
Other identities. For all n >= 2:
a(n) = A256486(n) + A260723(n).
a(n) = A256486(n+1) + A031883(n).

A057700 The next new gap between successive lucky numbers.

Original entry on oeis.org

2, 4, 6, 12, 8, 10, 18, 16, 24, 14, 22, 20, 30, 28, 26, 52, 32, 36, 34, 38, 46, 42, 40, 64, 44, 50, 54, 48, 56, 58, 66, 68, 60, 62, 100, 78, 72, 94, 76, 84, 96, 82, 74, 108, 70, 90, 80, 92, 86, 112, 144, 132, 88, 120, 106, 122, 140, 102, 104, 114, 110, 98, 158

Offset: 0

Naohiro Nomoto, Oct 23 2000

Cf. A000959, A014320, A031883.

More terms from Sascha Kurz, Mar 24 2002

a(51) onward corrected by Sean A. Irvine, Jun 23 2022

cp's OEIS Frontend

A031884 Smaller of a pair of consecutive lucky numbers with a gap of 2n.

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A184827 a(n) = largest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists.

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A184828 a(n) = A184827(n)/A130889(n) unless A130889(n) = 0 in which case a(n) = 0.

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A260722 Difference between n-th odd Ludic and n-th Lucky number: a(1) = 0; for n > 1: a(n) = A003309(n+1) - A000959(n).

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A057700 The next new gap between successive lucky numbers.

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