A355301 Normal undulating numbers where "undulating" means that the alternate digits go up and down (or down and up) and "normal" means that the absolute differences between two adjacent digits may differ.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 130, 131, 132, 140, 141, 142, 143, 150

Offset

1,3

Comments

This definition comes from Patrick De Geest's link.

Other definitions for undulating are present in the OEIS (e.g., A033619, A046075).

When the absolute differences between two adjacent digits are always equal (e.g., 85858), these numbers are called smoothly undulating numbers and form a subsequence (A046075).

The definition includes the trivial 1- and 2-digit undulating numbers.

Subsequence of A043096 where the first different term is A043096(103) = 123 while a(103) = 130.

This sequence first differs from A010784 at a(92) = 101, A010784(92) = 102.

The sequence differs from A160542 (which contains 100). - R. J. Mathar, Aug 05 2022

Links

Table of n, a(n) for n=1..110.

Patrick De Geest, Smoothly Undulating Palindromic Primes, World of Numbers.

Example

111 is not a term here, but A033619(102) = 111.
a(93) = 102, but 102 is not a term of A046075.
Some terms: 5276, 918230, 1053837, 263915847, 3636363636363636.
Are not terms: 1331, 594571652, 824327182.

Maple

isA355301 := proc(n)
    local dgs, i, back, forw ;
    dgs := convert(n, base, 10) ;
    if nops(dgs) < 2 then
        return true;
    end if;
    for i from 2 to nops(dgs)-1 do
        back := op(i, dgs) -op(i-1, dgs) ;
        forw := op(i+1, dgs) -op(i, dgs) ;
        if back*forw >= 0 then
            return false;
        end if ;
    end do:
    back := op(-1, dgs) -op(-2, dgs) ;
    if back = 0 then
        return false;
    end if ;
    return true ;
end proc:
A355301 := proc(n)
    option remember ;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA355301(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A355301(n), n=1..110) ; # R. J. Mathar, Aug 05 2022

Mathematica

q[n_] := AllTrue[(s = Sign[Differences[IntegerDigits[n]]]), # != 0 &] && AllTrue[Differences[s], # != 0 &]; Select[Range[0, 100], q] (* Amiram Eldar, Jun 28 2022 *)

Prog

(PARI) isok(m) = if (m<10, return(1)); my(d=digits(m), dd = vector(#d-1, k, sign(d[k+1]-d[k]))); if (#select(x->(x==0), dd), return(0)); my(pdd = vector(#dd-1, k, dd[k+1]*dd[k])); #select(x->(x>0), pdd) == 0; \\ Michel Marcus, Jun 30 2022

Crossrefs

Cf. A033619, A043096, A046075.

Cf. A059168 (subsequence of primes).

Differs from A010784, A241157, A241158.

Cf. A355302, A355303, A355304.

Sequence in context: A241158 A241157 A043096 * A010784 A052081 A031995

Adjacent sequences: A355298 A355299 A355300 * A355302 A355303 A355304

Keyword

nonn,base

Author

Bernard Schott, Jun 27 2022

Status

approved