A000959 -id:A000959 - cp's OEIS frontend

A034589 Lucky numbers N (A000959) such that Fibonacci iterations starting with (1, N) lead to a "nine digits anagram".

8955, 24405, 30013, 59325, 62025, 71493, 72123, 76885, 85461, 92077, 99165, 106185, 109717, 112251, 119077, 148773, 153007, 155077, 163771, 163803, 196797, 211369, 221137, 223365, 227119, 228271, 228631

Offset: 1

Patrick De Geest, Oct 15 1998

"Nine digits anagram" is a number whose digits are a permutation of {1, ..., 9}, also called restricted zeroless pandigital number. These are listed as the first 9! terms of A050289. - M. F. Hasler, Jan 10 2020

			Denote by F(1, N) the Fibonacci sequence x(k+1) = x(k) + x(k-1) starting with x(0) = 1, x(1) = N.
Then for N = 228631, F(1, N) = (1, 228631, 228632, 457263, 685895, 1143158, 1829053, 2972211, 4801264, 7773475, 12574739, 20348214, 32922953, 53271167, 86194120, 139465287, ...), where a nine-digits anagram has been reached.

M. F. Hasler, Table of n, a(n) for n = 1..275 (using A000959(1..2e5) from b-file), Jan 10 2020
Patrick De Geest, Nine Digits Digressions
Eric Weisstein's World of Mathematics, Lucky Numbers

Cf. A000959 (lucky numbers), A050289 (zeroless pandigital numbers).

Cf. A034587 (all starting values leading to 9-digit anagrams), A034588 (subset of primes), A034306 (subset of palindromes).

A034589_upto(N)=[n|n<-A000959_upto(N),is_A034587(n)] \\ or select(is_A034587, A959=from("b000959.txt")). - M. F. Hasler, Jan 10 2020

Intersection of A000959 (lucky numbers) and A034587. - M. F. Hasler, Jan 10 2020

Name, example & crossrefs edited, offset changed to 1 by M. F. Hasler, Jan 06 2020

A057589 Numbers n which are both lucky (A000959) and Fibonacci (A000045).

Original entry on oeis.org

1, 3, 13, 21, 1597, 6765, 75025

Offset: 1

Naohiro Nomoto, Oct 05 2000

Next term > 7*10^9. - Sascha Kurz, Mar 25 2002

a(8) >= 12586269025. - Kevin P. Thompson, Nov 24 2021

			a(6) = 6765 since it is the sixth number that is both lucky [A000959(797) = 6765] and Fibonacci [A000045(20) = 6765].

Cf. A000045, A000959.

A257733 Permutation of natural numbers: a(1) = 1, a(ludic(n)) = lucky(1+a(n-1)), a(nonludic(n)) = unlucky(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 3, 9, 2, 33, 5, 7, 14, 4, 45, 163, 8, 15, 11, 20, 6, 25, 59, 203, 12, 22, 17, 63, 28, 13, 10, 35, 78, 235, 251, 18, 30, 24, 83, 39, 19, 1093, 16, 47, 101, 31, 290, 67, 309, 26, 41, 43, 34, 107, 53, 27, 1283, 87, 23, 61, 128, 42, 354, 88, 376, 21, 36, 55, 57, 46, 137, 115, 70, 38, 1499, 321, 112, 32, 81, 161, 56, 1401, 430, 113, 454, 29, 48, 49

Offset: 1

Antti Karttunen, May 06 2015

Antti Karttunen, Table of n, a(n) for n = 1..8587
Index entries for sequences that are permutations of the natural numbers

Inverse: A257734.
Cf. A000959, A050505, A003309, A192607, A192490, A192512, A236863.
Related or similar permutations: A237427, A255422, A257726, A257731.
Cf. also A256486, A256487.
Differs from A257731 for the first time at n=19, where a(19) = 203, while A257731(19) = 63.

a(1) = 1; for n > 1: if A192490(n) = 1 [i.e., if n is ludic], then a(n) = A000959(1+a(A192512(n)-1)), otherwise a(n) = A050505(a(A236863(n))).
As a composition of other permutations:
a(n) = A257731(A255422(n)).
a(n) = A257726(A237427(n)).

A257734 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = ludic(1+a(n-1)), a(unlucky(n)) = nonludic(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and ludic = A003309, nonludic = A192607.

Original entry on oeis.org

1, 4, 2, 9, 6, 16, 7, 12, 3, 26, 14, 20, 25, 8, 13, 38, 22, 31, 36, 15, 61, 21, 54, 33, 17, 45, 51, 24, 81, 32, 41, 73, 5, 48, 27, 62, 119, 69, 35, 105, 46, 57, 47, 96, 10, 65, 39, 82, 83, 151, 115, 92, 50, 135, 63, 76, 64, 124, 18, 86, 55, 106, 23, 108, 189, 146, 43, 118, 193, 68, 169, 84, 91, 100, 149, 85, 156, 28, 179, 111, 74, 136, 34

Offset: 1

Antti Karttunen, May 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257733.
Cf. A000959, A050505, A003309, A109497, A145649, A192607.
Related or similar permutations: A237126, A255421, A257725, A257732.
Cf. also A256486, A256487.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = A003309(1+a(A109497(n)-1)), otherwise a(n) = A192607(a(n-A109497(n))).
As a composition of other permutations:
a(n) = A255421(A257732(n)).
a(n) = A237126(A257725(n)).

A350001 Iterated differences of lucky numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A000959(k), T(n,k) = T(n-1,k+1) - T(n-1,k) for n > 0.

Original entry on oeis.org

1, 3, 2, 7, 4, 2, 9, 2, -2, -4, 13, 4, 2, 4, 8, 15, 2, -2, -4, -8, -16, 21, 6, 4, 6, 10, 18, 34, 25, 4, -2, -6, -12, -22, -40, -74, 31, 6, 2, 4, 10, 22, 44, 84, 158, 33, 2, -4, -6, -10, -20, -42, -86, -170, -328, 37, 4, 2, 6, 12, 22, 42, 84, 170, 340, 668

Offset: 0

Pontus von Brömssen, Dec 08 2021

			Array begins:
  n\k|    1    2    3    4    5    6    7     8    9    10   11   12
  ---+--------------------------------------------------------------
   0 |    1    3    7    9   13   15   21    25   31    33   37   43
   1 |    2    4    2    4    2    6    4     6    2     4    6    6
   2 |    2   -2    2   -2    4   -2    2    -4    2     2    0   -4
   3 |   -4    4   -4    6   -6    4   -6     6    0    -2   -4   14
   4 |    8   -8   10  -12   10  -10   12    -6   -2    -2   18  -32
   5 |  -16   18  -22   22  -20   22  -18     4    0    20  -50   56
   6 |   34  -40   44  -42   42  -40   22    -4   20   -70  106  -82
   7 |  -74   84  -86   84  -82   62  -26    24  -90   176 -188  102
   8 |  158 -170  170 -166  144  -88   50  -114  266  -364  290 -100
   9 | -328  340 -336  310 -232  138 -164   380 -630   654 -390   50
  10 |  668 -676  646 -542  370 -302  544 -1010 1284 -1044  440   78

Winston de Greef, Table of n, a(n) for n = 0..11324 (150 antidiagonals)
Index entries for sequences related to lucky numbers

Cf. A000959 (row n = 0), A031883 (row n = 1), A123593 (column k = 1).

Cf. A254967 (absolute differences), A095195 (iterated differences of primes), A350004 (iterated differences of ludic numbers).

T(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*A000959(k+j).

A091431 Happy-go-Lucky numbers: numbers that are both Happy (A007770) and Lucky (A000959).

Original entry on oeis.org

1, 7, 13, 31, 49, 79, 129, 133, 193, 219, 319, 331, 367, 391, 409, 487, 655, 673, 739, 931, 937, 1009, 1029, 1039, 1093, 1209, 1233, 1251, 1275, 1281, 1285, 1303, 1309, 1323, 1339, 1533, 1575, 1587, 1599, 1663, 1771, 1857, 1933, 1959, 1995, 2019, 2115

Offset: 1

G. L. Honaker, Jr., Jan 07 2004

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
Lexico Dictionary.com, Happy-go-lucky

Cf. A007770, A000959, A091432, A091280.

Extended by Ray Chandler, Jan 18 2004

A137168 Lucky numbers (A000959) which are congruent to 1 mod 4.

Original entry on oeis.org

1, 9, 13, 21, 25, 33, 37, 49, 69, 73, 93, 105, 129, 133, 141, 169, 189, 193, 201, 205, 237, 241, 261, 273, 285, 289, 297, 321, 349, 357, 361, 385, 393, 409, 421, 429, 433, 477, 489, 517, 529, 537, 541, 553, 577, 601, 613, 621, 645, 673, 685, 693, 717, 729, 741, 745, 769

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

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

Intersection of A000959 and A016813.

A137170 Lucky numbers (A000959) which are congruent to 3 mod 4.

Original entry on oeis.org

3, 7, 15, 31, 43, 51, 63, 67, 75, 79, 87, 99, 111, 115, 127, 135, 151, 159, 163, 171, 195, 211, 219, 223, 231, 235, 259, 267, 283, 303, 307, 319, 327, 331, 339, 367, 391, 399, 415, 427, 451, 463, 475, 483, 487, 495, 511, 519, 535, 559, 579, 583, 591, 615, 619, 631, 639

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

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

Intersection of A000959 and A004767.

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.

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).

cp's OEIS Frontend

A034589 Lucky numbers N (A000959) such that Fibonacci iterations starting with (1, N) lead to a "nine digits anagram".

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A057589 Numbers n which are both lucky (A000959) and Fibonacci (A000045).

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A257733 Permutation of natural numbers: a(1) = 1, a(ludic(n)) = lucky(1+a(n-1)), a(nonludic(n)) = unlucky(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and lucky = A000959, unlucky = A050505.

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A257734 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = ludic(1+a(n-1)), a(unlucky(n)) = nonludic(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and ludic = A003309, nonludic = A192607.

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A350001 Iterated differences of lucky numbers. Array read by antidiagonals, n >= 0, k >= 1: T(0,k) = A000959(k), T(n,k) = T(n-1,k+1) - T(n-1,k) for n > 0.

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A091431 Happy-go-Lucky numbers: numbers that are both Happy (A007770) and Lucky (A000959).

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A137168 Lucky numbers (A000959) which are congruent to 1 mod 4.

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A137170 Lucky numbers (A000959) which are congruent to 3 mod 4.

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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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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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