John Bodeen - cp's OEIS frontend

User: John Bodeen

John Bodeen has authored 2 sequences.

A263856 Let S_n be the list of the first n primes written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n-th prime in S_n.

1, 2, 2, 4, 4, 3, 2, 6, 9, 5, 11, 4, 3, 11, 14, 6, 13, 9, 11, 17, 3, 20, 14, 5, 2, 9, 23, 20, 12, 4, 31, 17, 5, 23, 12, 32, 17, 22, 32, 15, 26, 14, 42, 2, 11, 37, 29, 46, 27, 14, 9, 48, 6, 40, 2, 43, 22, 51, 18, 12, 43, 17, 39, 56, 14, 32, 45, 6, 50

Offset: 1

John Bodeen, Oct 28 2015

a(A264647(n)) = n and a(m) != n for m < A264647(n). - Reinhard Zumkeller, Nov 19 2015

A264662(n,a(n)) = A000040(n): a(n) = index of prime(n) in n-th row of triangle A264662. - Reinhard Zumkeller, Nov 20 2015

			S_1 = [01], a(1) = 1;
S_2 = [01, 11], a(2) = 2;
S_3 = [01, 101, 11], a(3) = 2;
S_4 = [01, 101, 11, 111], a(4) = 4;
S_5 = [01, 101, 11, 1101, 111], a(5) = 4;
S_5 = [01, 101, 1011, 11, 1101, 111], a(6) = 3;
...

John Bodeen, Table of n, a(n) for n = 1..24771
John Bodeen, OCaml program to generate the sequence

A004676 is the sequence upon which the lexicographic ordering is based.
Cf. A264596.
Cf. A264647, A264662.

import Data.List (insertBy); import Data.Function (on)
import Data.List (elemIndex); import Data.Maybe (fromJust)
a263856 n = a263856_list !! (n-1)
a263856_list = f [] a004676_list where
   f bps (x:xs) = y : f bps' xs where
     y = fromJust (elemIndex x bps') + 1
     bps' = insertBy (compare `on` (reverse . show)) x bps
-- Reinhard Zumkeller, Nov 19 2015

s:= proc(n) s(n):= cat("", convert(ithprime(n), base, 2)[]) end:
a:= n-> ListTools[BinarySearch](sort([seq(s(i), i=1..n)]), s(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 19 2015

S[n_] := S[n] = SortBy[Prime[Range[n]], StringJoin @@ ToString /@ Reverse[IntegerDigits[#, 2]]&];
a[n_] := FirstPosition[S[n], Prime[n]][[1]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 22 2021 *)

from sympy import prime
def A263856(n):
    return 1+sorted(format(prime(i),'b')[::-1] for i in range(1,n+1)).index(format(prime(n),'b')[::-1]) # Chai Wah Wu, Nov 22 2015

A259356 Triangle T(n,k) read by rows: T(n,k) is the number of closed lambda-terms of size n with size 0 for the variables and k abstractions.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 2, 9, 3, 0, 5, 38, 35, 4, 0, 14, 181, 284, 95, 5, 0, 42, 938, 2225, 1320, 210, 6, 0, 132, 5210, 17816, 15810, 4596, 406, 7

Offset: 0

John Bodeen, Jun 24 2015

			In table format, the first few rows:
{0},
{0,1},
{0,1,2},
{0,2,9,3},
{0,5,38,35,4},
...
For n=3,k=2 we have the number of closed lambda terms of size three with exactly two abstractions, T(3,2,0) = 9:
\x.\y.x x
\x.\y.x y
\x.\y.y x
\x.\y.y y
(\x.x) (\y.y)
\x.(\y.y) x
\x.(\y.x) x
\x.x (\y.y)
\x.x (\y.x)

Cf. A220894 (row sums), A000108.

T(n,k) = T(n,k,0) where T(n,k,b) where n is size, k is number of abstractions, and b is number of free variables, T(0,0,b) = b, and T(n,k,b) = T(n-1,k-1,b+1) + Sum_{i=0..n-1} Sum_{j=0..k} T(i,j,b) * T(n-1-i,k-j,b).

T(n+1,1) = A000108(n).

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User: John Bodeen

A263856 Let S_n be the list of the first n primes written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n-th prime in S_n.

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