A279001 -id:A279001 - cp's OEIS frontend

A279000 Numbers of the form (11h+j)11^k-1 for h,k in N and j in {1,3,4,5,9}.

0, 2, 3, 4, 8, 10, 11, 13, 14, 15, 19, 22, 24, 25, 26, 30, 32, 33, 35, 36, 37, 41, 43, 44, 46, 47, 48, 52, 54, 55, 57, 58, 59, 63, 66, 68, 69, 70, 74, 77, 79, 80, 81, 85, 88, 90, 91, 92, 96, 98, 99, 101, 102, 103, 107, 110, 112, 113, 114, 118, 120, 121, 123, 124, 125, 129

Offset: 1

N. J. A. Sloane, Dec 07 2016

nonn

Created in a failed attempt to explain sequences J and K on page 10 of Fu and Han (2016). See A279194 and A279195. - N. J. A. Sloane, Dec 15 2016
Numbers n such that the least-significant non-0 digit of n+1 in base 11 is one of {1,3,4,5,9}. - R. J. Mathar, Dec 15 2016
It appears that the correct J and K have form more closely resembling the J and K of F_3(n): n is in J if either k is odd and j is in {2,6,7,8,10} or k is even and j is in {1,3,4,5,9}, and n is in K if either k is even and j is in {2,6,7,8,10} or k is odd and j is in {1,3,4,5,9}. - Charlie Neder, Mar 10 2019

Hao Fu and G.-N. Han, Computer assisted proof for Apwenian sequences related to Hankel determinants, arXiv preprint arXiv:1601.04370 [math.NT], 2016.

Complement of A279001.

Cf. A279194, A279195.

isA279000 := proc(n)
    local x,dgs11,i ;
    x := n+1 ;
    dgs11 := convert(x,base,11) ;
    for i from 1 to nops(dgs11) do
        if op(i,dgs11) in {1,3,4,5,9} then
            return true;
        elif op(i,dgs11) in {2,6,7,8,10} then
            return false;
        end if;
    end do:
    false ;
end proc:
for n from 0 to 200 do
    if isA279000(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 15 2016

okQ[n_] := MatchQ[IntegerDigits[n+1, 11], {_, 1 | 3 | 4 | 5 | 9, 0...}]; Select[Range[0, 200], okQ] (* Jean-François Alcover, Feb 25 2018, after R. J. Mathar *)

from sympy import integer_log
def A279000(n):
    def f(x): return n-1+sum(((m:=(x+1)//11**i)-2)//11+(m-6)//11+(m-7)//11+(m-8)//11+(m-10)//11+5 for i in range(integer_log(x+1,11)[0]+1))
    m, k = n-1, f(n-1)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 23 2025

Corrected by Lars Blomberg (10 added, 21 removed, 32 added...), Dec 15 2016

A279194 Numbers of the form {(11h+p)11^2k-1 | h,k in N and p in {1,3,4,5,9} } U {(11h+q)11^(2k+1)-1 | h,k in N and q in {2,6,7,8,10} }.

Original entry on oeis.org

0, 2, 3, 4, 8, 11, 13, 14, 15, 19, 21, 22, 24, 25, 26, 30, 33, 35, 36, 37, 41, 44, 46, 47, 48, 52, 55, 57, 58, 59, 63, 65, 66, 68, 69, 70, 74, 76, 77, 79, 80, 81, 85, 87, 88, 90, 91, 92, 96, 99, 101, 102, 103, 107, 109, 110, 112, 113, 114, 118, 120, 121, 123, 124, 125, 129

Offset: 1

N. J. A. Sloane, Dec 15 2016

nonn

The sequence J related to the Apwenian power series F_{11}(x).

Ray Chandler, Table of n, a(n) for n = 1..10000
Hao Fu, G.-N. Han, Computer assisted proof for Apwenian sequences related to Hankel determinants, arXiv preprint arXiv:1601.04370 [math.NT], 2016. See sequence "J" in Section 2.3.
Guo-Niu Han, The program Apwen.py

Cf. A279000, A279001, A279195.

isok[n_]:=Module[{ord=IntegerExponent[n+1,11],pq={{1,3,4,5,9},{2,6,7,8,10}}},MemberQ[pq[[Mod[ord,2]+1]],Mod[(n+1)/11^ord,11]]];Select[Range[0,131],isok] (* Ray Chandler, Dec 17 2016 *)

Definition (from p. 5, Definition 2.1 of the arXiv reference) provided by Arie Groeneveld, Dec 16 2016

More terms from Ray Chandler, Dec 17 2016

A279195 Numbers of the form {(11h+p)11^(2k+1)-1 | h,k in N and p in {1,3,4,5,9} } U {(11h+q)11^2k-1 | h,k in N and q in {2,6,7,8,10} }.

Original entry on oeis.org

1, 5, 6, 7, 9, 10, 12, 16, 17, 18, 20, 23, 27, 28, 29, 31, 32, 34, 38, 39, 40, 42, 43, 45, 49, 50, 51, 53, 54, 56, 60, 61, 62, 64, 67, 71, 72, 73, 75, 78, 82, 83, 84, 86, 89, 93, 94, 95, 97, 98, 100, 104, 105, 106, 108, 111, 115, 116, 117, 119, 122, 126, 127, 128, 130, 131

Offset: 1

N. J. A. Sloane, Dec 15 2016

nonn

The sequence K related to the Apwenian power series F_{11}(x).

Ray Chandler, Table of n, a(n) for n = 1..10000
Hao Fu, G.-N. Han, Computer assisted proof for Apwenian sequences related to Hankel determinants, arXiv preprint arXiv:1601.04370 [math.NT], 2016. See sequence "K" in Section 2.3.
Guo-Niu Han, The program Apwen.py

Cf. A279000, A279001, A279194.

isok[n_]:=Module[{ord=IntegerExponent[n+1,11],pq={{2,6,7,8,10},{1,3,4,5,9}}},MemberQ[pq[[Mod[ord,2]+1]],Mod[(n+1)/11^ord,11]]];Select[Range[0,131],isok] (* Ray Chandler, Dec 17 2016 *)

Definition (from p. 5, Definition 2.1 of the arXiv reference) provided by Arie Groeneveld, Dec 16 2016

More terms from Ray Chandler, Dec 17 2016

cp's OEIS Frontend

A279000 Numbers of the form (11h+j)11^k-1 for h,k in N and j in {1,3,4,5,9}.

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A279194 Numbers of the form {(11h+p)11^2k-1 | h,k in N and p in {1,3,4,5,9} } U {(11h+q)11^(2k+1)-1 | h,k in N and q in {2,6,7,8,10} }.

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A279195 Numbers of the form {(11h+p)11^(2k+1)-1 | h,k in N and p in {1,3,4,5,9} } U {(11h+q)11^2k-1 | h,k in N and q in {2,6,7,8,10} }.

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cp's OEIS Frontend

A279000 Numbers of the form (11*h+j)*11^k-1 for h,k in N and j in {1,3,4,5,9}.

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A279194 Numbers of the form {(11*h+p)*11^2k-1 | h,k in N and p in {1,3,4,5,9} } U {(11*h+q)*11^(2k+1)-1 | h,k in N and q in {2,6,7,8,10} }.

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Mathematica

Extensions

A279195 Numbers of the form {(11*h+p)*11^(2k+1)-1 | h,k in N and p in {1,3,4,5,9} } U {(11*h+q)*11^2k-1 | h,k in N and q in {2,6,7,8,10} }.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A279000 Numbers of the form (11h+j)11^k-1 for h,k in N and j in {1,3,4,5,9}.

A279194 Numbers of the form {(11h+p)11^2k-1 | h,k in N and p in {1,3,4,5,9} } U {(11h+q)11^(2k+1)-1 | h,k in N and q in {2,6,7,8,10} }.

A279195 Numbers of the form {(11h+p)11^(2k+1)-1 | h,k in N and p in {1,3,4,5,9} } U {(11h+q)11^2k-1 | h,k in N and q in {2,6,7,8,10} }.