cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A032589 Lucky numbers (A000959) ending with digit 9.

Original entry on oeis.org

9, 49, 69, 79, 99, 129, 159, 169, 189, 219, 259, 289, 319, 339, 349, 399, 409, 429, 489, 519, 529, 559, 579, 619, 639, 679, 699, 729, 739, 769, 819, 979, 1009, 1029, 1039, 1179, 1189, 1209, 1219, 1249, 1309, 1329, 1339, 1369, 1389, 1419, 1459
Offset: 1

Views

Author

Patrick De Geest, Apr 15 1998

Keywords

Comments

Because lucky numbers are all odd, these are also the lucky numbers that are congruent to 4 mod 5. - Nathaniel Johnston, Jun 24 2011

Links

Crossrefs

Cf. A000959.

A130594 Numbers which are both lucky (A000959) and tribonacci (A000073).

Original entry on oeis.org

1, 7, 13, 927, 1705, 10609
Offset: 1

Views

Author

Jonathan Vos Post, Jun 16 2007

Keywords

Comments

No other terms below 15902591. The next candidate is the odd tribonacci number 15902591. Is this also a lucky number? - Harvey P. Dale, Jul 12 2008
This is to tribonacci as A057589 is to the Fibonacci numbers.
a(7) >= 23837527729. - Kevin P. Thompson, Nov 24 2021

Examples

			a(6) = 10609 because it is lucky A000959(1182) and tribonacci A000073(18).

Crossrefs

Cf. A000073, A057589, A000959.

A137164 Lucky numbers (A000959) which are congruent to 0 mod 3.

Original entry on oeis.org

3, 9, 15, 21, 33, 51, 63, 69, 75, 87, 93, 99, 105, 111, 129, 135, 141, 159, 171, 189, 195, 201, 219, 231, 237, 261, 267, 273, 285, 297, 303, 321, 327, 339, 357, 393, 399, 429, 477, 483, 489, 495, 519, 537, 579, 591, 615, 621, 639, 645, 651, 693, 699, 717, 723, 729, 735
Offset: 1

Views

Author

N. J. A. Sloane, Mar 07 2008

Keywords

Links

Crossrefs

Intersection of A000959 and A008585.

A137185 Lucky numbers (A000959) which are congruent to 3 mod 7.

Original entry on oeis.org

3, 31, 73, 87, 115, 129, 171, 241, 283, 297, 339, 367, 409, 451, 535, 577, 591, 619, 717, 745, 787, 801, 885, 927, 997, 1011, 1039, 1053, 1095, 1123, 1179, 1249, 1263, 1291, 1389, 1417, 1459, 1473, 1501, 1543, 1585, 1599, 1641, 1711, 1767, 1809, 1879, 1893, 1921
Offset: 1

Views

Author

N. J. A. Sloane, Mar 07 2008

Keywords

Links

Crossrefs

Intersection of A000959 and A017017.

A257256 Difference between {the first unlucky number removed at the n-th stage of Lucky sieve} and {the n-th Lucky number}: a(n) = A219178(n) - A000959(n).

Original entry on oeis.org

1, 2, 12, 18, 32, 40, 64, 84, 108, 124, 138, 170, 206, 214, 274, 296, 318, 338, 348, 378, 426, 454, 498, 532, 564, 600, 662, 678, 710, 736, 766, 836, 874, 920, 944, 954, 1078, 1104, 1120, 1170, 1202, 1254, 1296, 1332, 1378, 1416, 1434, 1466, 1592, 1612, 1660, 1696, 1778, 1786, 1820, 1888, 1932, 1952, 2058
Offset: 1

Views

Author

Antti Karttunen, Apr 29 2015

Keywords

Links

Crossrefs

First column of A257255.
Cf. A000959, A219178.

Programs

Formula

a(n) = A219178(n) - A000959(n).

A260721 Half the difference between n-th odd Ludic and n-th Lucky number: a(1) = 0; a(n) = (A003309(n+1) - A000959(n))/2.

Original entry on oeis.org

0, 0, -1, -1, -1, -1, -2, -1, -3, -2, 0, -1, -3, -2, -5, -3, -1, -1, 1, 2, 1, -1, -1, 1, 2, 2, -3, -1, -1, 4, 4, 3, 1, 5, 3, 4, -4, 0, 7, 5, 8, 6, 4, 5, 2, 2, 5, 8, 3, 8, 8, 7, 9, 11, 12, 16, 14, 15, 11, 16, 16, 15, 19, 17, 16, 18, 20, 15, 14, 14, 16, 12, 11, 12, 18, 19, 21, 15, 15, 11, 13, 13, 15, 19, 20, 15, 18, 23, 24, 23, 28, 27, 27, 27, 20, 23
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2015

Keywords

Links

Crossrefs

Cf. A000959, A003309, A260722, A256488.

Programs

Formula

a(1) = 0; a(n) = (A003309(n+1) - A000959(n))/2.
a(n) = A260722(n)/2.

A350002 a(n) is the smallest lucky number L(k) such that the n-th difference of (L(k), ..., L(k+n)) is zero, where L is A000959; a(n) = 0 if no such number exists.

Original entry on oeis.org

37, 31, 87, 31, 517, 1797, 1797, 267, 483, 5649, 23815, 198223, 985921, 508401, 3720765, 1936245, 8302279, 16713091, 9857049, 16756749, 8904175
Offset: 2

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Comments

Equivalently, a(n) is the smallest lucky number L(k) such that there is a polynomial f of degree at most n-1 such that f(j) = L(j) for k <= j <= k+n.
a(n) = A000959(k), where k is the smallest positive integer such that A350001(n,k) = 0.

Examples

			The first six consecutive lucky numbers for which the fifth difference is 0 are (31, 33, 37, 43, 49, 51), so a(5) = 31. The successive differences are (2, 4, 6, 6, 2), (2, 2, 0, -4), (0, -2, -4), (-2, -2), and (0).

Links

Crossrefs

First column of A350003.
Cf. A000959, A349643 (counterpart for primes), A350001, A350006 (counterpart for ludic numbers).

Formula

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

A350003 Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest lucky number L(k) such that all n-th differences of (L(k), ..., L(k+n+m)) are zero, where L is A000959; T(n,m) = 0 if no such number exists.

Original entry on oeis.org

37, 87, 31, 87, 87, 87, 72979, 17781, 1263, 31
Offset: 2

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Comments

Equivalently, T(n,m) is the smallest lucky number L(k) such that there is a polynomial f of degree at most n-1 such that f(j) = L(j) for k <= j <= k+n+m.
T(n,m) = A000959(k), where k is the smallest positive integer such that A350001(n,k+j) = 0 for 0 <= j <= m.

Examples

			Array begins:
  n\m|        0       1        2        3
  ---+-----------------------------------
   2 |       37      87       87    72979
   3 |       31      87    17781   196089
   4 |       87    1263   196089 63955483
   5 |       31    3687 17622975        ?
   6 |      517  390015        ?        ?
   7 |     1797    1797        ?        ?
   8 |     1797 2432367        ?        ?
   9 |      267 9157647        ?        ?
  10 |      483 1683501        ?        ?
For n = 4 and m = 1, the first six (n+m+1) consecutive lucky numbers for which all fourth (n-th) differences are 0 are (1263, 1275, 1281, 1285, 1291, 1303), so T(4,1) = 1263. The successive differences are (12, 6, 4, 6, 12), (-6, -2, ,2, 6), (4, 4, 4), and (0, 0).

Links

Crossrefs

Cf. A330362 (row n=2), A350002 (column m=0).
Cf. A000959, A349644 (counterpart for primes), A350001, A350007 (counterpart for ludic numbers).

Formula

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

A049781 a(n) = (b(n) + 1)/2, where b = A000959 (the lucky numbers).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 22, 25, 26, 32, 34, 35, 37, 38, 40, 44, 47, 50, 53, 56, 58, 64, 65, 67, 68, 71, 76, 80, 82, 85, 86, 95, 97, 98, 101, 103, 106, 110, 112, 116, 118, 119, 121, 130, 131, 134, 137, 142, 143, 145, 149, 152
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000959.

A057609 Powers of a prime lucky number (A031157) but excluding lucky numbers (A000959).

Original entry on oeis.org

27, 81, 243, 343, 1849, 2197, 2401, 4489, 5329, 6241, 6561, 16129, 16807, 19683, 22801, 26569, 28561, 37249, 44521, 49729, 58081, 59049, 79507, 80089, 94249, 109561, 117649, 134689, 177147, 177241, 187489, 214369, 237169, 361201, 371293, 375769, 383161, 389017
Offset: 1

Views

Author

Naohiro Nomoto, Oct 09 2000

Keywords

Comments

Up to 10^7, terms are 3^3, 3^4, 3^5, 3^8, 3^9, 3^10, 3^11, 3^12, 3^13, 7^3, 7^4, 7^5, 7^6, 13^3, 13^4, 13^5, 13^6, 31^4, 43^2, 43^3, 43^4, 67^2, ..., . - Robert G. Wilson v, May 12 2006

Examples

			In the first 23 terms of A000959, {1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99}, 3 is a prime lucky number (A031157), and 3^2 is also a lucky number, but 3^3=27 and 3^4=81 are not lucky numbers, so they are terms of this sequence.

Links

Crossrefs

Cf. A031157, A000959.

Programs

  • Mathematica
    lst = Range[1, 2*10^6, 2]; i = 2; While[i <= (len = Length[lst]) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++ ]; m = Last@ lst; Complement[ Reap[ Do[ If[x^2 > m, Break[]]; If[PrimeQ[x], y = x^2; While[y <= m, Sow@ y; y *= x]], {x, lst}]] [[2, 1]], lst] (* Robert G. Wilson v, May 12 2006, corrected by Giovanni Resta, May 10 2020 *)

Extensions

More terms from Robert G. Wilson v, May 12 2006
Data corrected and extended by Giovanni Resta, May 10 2020
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