A033619 -id:A033619 - cp's OEIS frontend

A057333 Numbers of n-digit primes that undulate.

4, 20, 74, 347, 1743, 8385, 44355, 229952, 1235489, 6629026, 37152645, 202017712, 1142393492, 6333190658

Offset: 1

Patrick De Geest, Sep 15 2000

'Undulate' means that the alternate digits are consistently greater than or less than the digits adjacent to them (e.g., 70769). Smoothly undulating palindromic primes (e.g., 95959) are a subset and included in the count.

C. A. Pickover, "Wonders of Numbers", Oxford New York 2001, Chapter 52, pp. 123-124, 316-317.

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
Eric Weisstein's World of Mathematics, Undulating Number.

Cf. A046075, A033619, A032758, A039944, A016073, A046076, A046077, A057332.

from sympy import isprime
def f(w,dir):
    if dir == 1:
        for s in w:
            for t in range(int(s[-1])+1,10):
                yield s+str(t)
    else:
        for s in w:
            for t in range(0,int(s[-1])):
                yield s+str(t)
def A057333(n):
    c = 0
    for d in '123456789':
        x = d
        for i in range(1,n):
            x = f(x,(-1)**i)
        c += sum(1 for p in x if isprime(int(p)))
        if n > 1:
            y = d
            for i in range(1,n):
                y = f(y,(-1)**(i+1))
            c += sum(1 for p in y if isprime(int(p)))
    return c # Chai Wah Wu, Apr 25 2021

Offset corrected and a(10)-a(11) from Donovan Johnson, Aug 08 2010
a(12) from Giovanni Resta, Feb 24 2013
a(2) corrected by Chai Wah Wu, Apr 25 2021
a(13)-a(14) from Chai Wah Wu, May 02 2021

A129120 Undulating Harshad numbers: numbers divisible by the sum of their own digits with decimal expansions in an abab...ab pattern.

Original entry on oeis.org

10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 171, 252, 414, 828, 1010, 1212, 1818, 2020, 2424, 3030, 3636, 4040, 4848, 5050, 5454, 6060, 7070, 7272, 8080, 9090, 10101, 13131, 20202, 23232, 26262, 30303, 39393

Offset: 1

Jason Earls, May 25 2007

This definition of "undulating" is more restrictive than the one used in A033619 and excludes single-digit numbers and numbers of the pattern aaaaaa. - R. J. Mathar, Jun 15 2007

Jason Earls, Red Zen, Lulu Press, NY, 2006, pp. 56-57. ISBN: 978-1-4303-2017-3.

Amiram Eldar, Table of n, a(n) for n = 1..2146

Cf. A005349, A033619.

undHarsQ[n_] := Module[{d = IntegerDigits[n]}, Divisible[n, Plus @@ d] &&  Length @ Union[d] > 1 && Length @ Union[d[[1 ;; -1 ;; 2]]] == 1 && Length @ Union[d[[2 ;; -1 ;; 2]]] == 1 ]; Select[Range[40000], undHarsQ] (* Amiram Eldar, Jan 27 2021 *)

More terms from R. J. Mathar, Jun 15 2007

A343763 Common prime factors of A007663(183)/1093 and A007663(490)/3511.

Original entry on oeis.org

3, 7, 79, 2731, 8191, 121369, 22366891

Offset: 1

Felix Fröhlich, Apr 28 2021

The sequence is complete (cf. Michon).
Are these primes necessarily prime factors of A007663(i)/p when p = prime(i) is a Wieferich prime (A001220)?
Note the following curious observation: 3 * 7 * 79 * 2731 * 8191 * 121369 * 22366891 = 100743818301219097892181. That number is a repdigit in bases 4 and 64 and undulating in bases 2, 8, 128, 2048 and 8192. Compare that to observations from John Blythe Dobson (cf. Dobson).

J. B. Dobson, A note on the two known Wieferich primes
Gerard P. Michon, Wieferich Primes 1093 and 3511

Cf. A001220, A007663, A010785, A033619, A172290, A242715.

fq(n) = (2^(n-1)-1)/n
my(x=fq(1093)/1093, y=fq(3511)/3511); forprime(p=1, , if(Mod(x, p)==0 && Mod(y, p)==0, print1(p, ", ")))

A344888 a(n) is the least base k >= 2 where n is an undulating number (i.e., with digits of the form abababab...).

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 3, 2, 3, 4, 2, 4, 4, 3, 4, 2, 3, 4, 5, 5, 3, 2, 5, 3, 5, 4, 3, 6, 6, 4, 3, 2, 6, 6, 4, 6, 5, 6, 4, 7, 3, 5, 2, 6, 7, 7, 4, 7, 7, 6, 3, 4, 5, 8, 8, 4, 8, 5, 8, 4, 3, 6, 5, 2, 7, 8, 9, 5, 4, 9, 3, 7, 5, 8, 6, 9, 9, 9, 5, 9, 3, 8, 9, 5, 10, 2, 6

Offset: 0

Rémy Sigrist, Jun 01 2021

			For n = 49:
- we have:
     b  49 in base b  Undulating?
     -  ------------  -----------
     2        110001  No
     3          1211  No
     4           301  No
     5           144  No
     6           121  Yes
- so a(49) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A000196, A000225, A000975, A033619.

is(n, base=10) = my (d=digits(n, base)); #d<=2 || d[1..#d-2]==d[3..#d]
a(n) = for (b=2, oo, if (is(n, b), return (b)))

def A344888(n):
    b, m = 2, n
    while True:
        m, x = divmod(m, b)
        m, y = divmod(m, b)
        while m > 0:
            m, z = divmod(m,b)
            if z != x:
                break
            if m > 0:
                m, z = divmod(m,b)
                if z != y:
                    break
            else:
                return b
        else:
            return b
        b += 1
        m = n # Chai Wah Wu, Jun 02 2021

a(n) <= A000196(n) + 1 for any n > 0.
a(n) <= 10 for any n in A033619.
a(n) = 2 iff n belongs to A000225 or to A000975.

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A057333 Numbers of n-digit primes that undulate.

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A129120 Undulating Harshad numbers: numbers divisible by the sum of their own digits with decimal expansions in an abab...ab pattern.

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A343763 Common prime factors of A007663(183)/1093 and A007663(490)/3511.

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A344888 a(n) is the least base k >= 2 where n is an undulating number (i.e., with digits of the form abababab...).

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