A350006 -id:A350006 - cp's OEIS frontend

A350007 Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest ludic number L(k) such that all n-th differences of (L(k), ..., L(k+n+m)) are zero, where L is A003309; T(n,m) = 0 if no such number exists.

1, 71, 11, 6392047, 41, 41

Offset: 2

Pontus von Brömssen, Dec 08 2021

Equivalently, T(n,m) 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+m.

T(n,m) = A003309(k), where k is the smallest positive integer such that A350004(n,k+j) = 0 for 0 <= j <= m.

			Array begins:
  n\m|     0        1        2        3        4        5
  ---+---------------------------------------------------
   2 |     1       71  6392047        ?        ?        ?
   3 |    11       41     1111  2176387 61077491 93320837
   4 |    41     1111   545977 27244691 93320837        ?
   5 |    47       91 27244691 93320837        ?        ?
   6 |    91    23309 93320837        ?        ?        ?
   7 |  1361  9899189        ?        ?        ?        ?
   8 |  4261    26233        ?        ?        ?        ?
   9 |   481  7110347        ?        ?        ?        ?
  10 | 46067 79241951        ?        ?        ?        ?
For n = 5 and m = 1, the first seven (n+m+1) consecutive ludic numbers for which all fifth (n-th) differences are 0 are (91, 97, 107, 115, 119, 121, 127), so T(5,1) = 91. The successive differences are (6, 10, 8, 4, 2, 6), (4, -2, -4, -2, 4), (-6, -2, 2, 6), (4, 4, 4), and (0, 0).

Cf. A350005 (row n = 2), A350006 (column m = 0).

Cf. A003309, A349644 (counterpart for primes), A350003 (counterpart for lucky numbers), A350004.

T(n,m) <= T(n-1,m+1).
T(n,m) <= T(n, m+1).
Sum_{j=0..n} (-1)^j*binomial(n,j)*A003309(k+i+j) = 0 for 0 <= i <= m, where A003309(k) = T(n,m).

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

Pontus von Brömssen, Dec 08 2021

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.

			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).

Index entries for sequences related to lucky numbers

First column of A350003.

Cf. A000959, A349643 (counterpart for primes), A350001, A350006 (counterpart for ludic numbers).

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

cp's OEIS Frontend

A350007 Array read by antidiagonals, n >= 2, m >= 0: T(n,m) is the smallest ludic number L(k) such that all n-th differences of (L(k), ..., L(k+n+m)) are zero, where L is A003309; T(n,m) = 0 if no such number exists.

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