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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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

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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).

A350006 a(n) is the smallest ludic number L(k) such that the n-th difference of (L(k), ..., L(k+n)) is zero, where L is A003309; a(n) = 0 if no such number exists.

Original entry on oeis.org

1, 11, 41, 47, 91, 1361, 4261, 481, 46067, 5027, 31499, 888893, 126205, 36191, 7676353, 26794127, 206527, 2560375, 7716073
Offset: 2

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Comments

Equivalently, a(n) is the smallest ludic 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) = A003309(k), where k is the smallest positive integer such that A350004(n,k) = 0.
a(21) > 10^8 (unless a(21) = 0).

Examples

			The first six consecutive ludic numbers for which the fifth difference is 0 are (47, 53, 61, 67, 71, 77), so a(5) = 47. The successive differences are (6, 8, 6, 4, 6), (2, -2, -2, 2), (-4, 0, 4), (4, 4), and (0).

Crossrefs

First column of A350007.
Cf. A003309, A349643, A350002, A350004.

Formula

Sum_{j=0..n} (-1)^j*binomial(n,j)*A003309(k+j) = 0, where A003309(k) = a(n).

A350005 a(n) is the smallest number that starts an arithmetic progression of n consecutive ludic numbers (A003309), or 0 if no such number exists.

Original entry on oeis.org

1, 1, 1, 71, 6392047
Offset: 1

Views

Author

Pontus von Brömssen, Dec 08 2021

Keywords

Comments

a(n) is the smallest ludic number A003309(k), such that A260723(k) = A260723(k+1) = ... = A260723(k+n-2).
a(6) > 10^8 (unless a(6) = 0).

Examples

			The first arithmetic progression of 3 consecutive ludic numbers is (1, 2, 3), so a(3) = 1.
The first arithmetic progression of 4 consecutive ludic numbers is (71, 77, 83, 89), so a(4) = 71.
The first arithmetic progression of 5 consecutive ludic numbers is (6392047, 6392077, 6392107, 6392137, 6392167), so a(5) = 6392047.

Crossrefs

From n = 3, first row of A350007.
Cf. A003309, A260723.
Counterparts for other sequences than ludic numbers: A006560 (primes), A228433 (abundant numbers), A231623 (deficient numbers), A276821 (Sophie Germain primes), A330362 (lucky numbers).
Showing 1-3 of 3 results.

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