A350002 -id:A350002 - cp's OEIS frontend

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.

37, 87, 31, 87, 87, 87, 72979, 17781, 1263, 31

Offset: 2

Pontus von Brömssen, Dec 08 2021

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.

			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).

Index entries for sequences related to lucky numbers

Cf. A330362 (row n=2), A350002 (column m=0).

Cf. A000959, A349644 (counterpart for primes), A350001, A350007 (counterpart for ludic numbers).

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

Pontus von Brömssen, Dec 08 2021

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).

			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).

First column of A350007.

Cf. A003309, A349643, A350002, A350004.

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

cp's OEIS Frontend

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.

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