A177834 Opmanis's sequence: a(n) is the smallest integer k such that k or one of its nonzero substrings (regarded as an integer) is divisible by every integer in the range 1 through n.

1, 2, 6, 12, 45, 54, 56, 56, 245, 504, 1440, 1440, 5044, 5044, 10456, 10569, 11704, 11704, 11704, 13608, 13608, 13608, 26460, 26460, 198007, 258064, 264600, 264600, 475440, 475440, 1754608, 1754608, 2258064, 2258064, 2646004, 2646004, 2992520

Offset

1,2

Comments

Comment from N. J. A. Sloane, May 28 2010: (Start)

The factorizations of the initial terms are:

1, 2, 2*3, 2^2*3, 3^2*5, 2*3^3, 2^3*7, 2^3*7, 5*7^2, 2^3*3^2*7, 2^5*3^2*5, 2^5*3^2*5, 2^2*13*97, 2^2*13*97, 2^3*1307, 3*13*271, 2^3*7*11*19,

2^3*7*11*19, 2^3*7*11*19, 2^3*3^5*7, 2^3*3^5*7, 2^3*3^5*7, 2^2*3^3*5*7^2, 2^2*3^3*5*7^2, 23*8609, 2^4*127^2, 2^3*3^3*5^2*7^2, 2^3*3^3*5^2*7^2, 2^4*3*5*7*283,

2^4*3*5*7*283, 2^4*109663, 2^4*109663, 2^4*3^3*5227, 2^4*3^3*5227, 2^2*139*4759, 2^2*139*4759, 2^3*5*79*947, ...

The name "Opmanis's sequence" is due to N. J. A. Sloane, not the author. (End)

Links

Robert Gerbicz, Table of n, a(n) for n = 1..102

Example

a(8)=56 because 56 is divisible by 1,2,4,7,8; 5 is divisible by 5; 6 is divisible by 3 and 6. Therefore the set {1,2,3,4,5,6,7,8} is covered by the divisors. 56 is the smallest number with this property.

Mathematica

k = 1; lst = {}; mx = 0; f[n_] := Block[{a, d, id = IntegerDigits@ n}, a = Complement[ Union[ FromDigits /@ Flatten[ Table[ Partition[ id, k, 1], {k, Length@ id}], 1]], {0}]; d = Union[ Flatten[ Divisors /@ a]]; Complement[ Range@ 100, d][[1]] - 1]; While[k < 3000000, a = f@k; If[a > mx, Print[{a, k}]; AppendTo[lst, k]; mx = a]; k++ ] (* Zak Seidov & Robert G. Wilson v, May 30 2010 *)

Prog

(Python)
def substrings(n): # returns set of nonzero substrings of n
    s = str(n)
    ss = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1))
    return set(int(sij) for sij in ss) - {0}
def a(n, startk=1):
    k = startk
    while True:
        subsk = substrings(k)
        if all(any(kij%m == 0 for kij in subsk) for m in range(1, n+1)):
            return k
        k += 1
def afind():
    n, an = 1, 1
    while True:
        n, an = n+1, a(n, startk=an)
        print(an, end=", ")
afind() # Michael S. Branicky, Jan 22 2022

Crossrefs

Cf. A003418 (a weak upper bound), A169819, A169858.

Sequence in context: A332868 A261467 A180070 * A169858 A292132 A208147

Adjacent sequences: A177831 A177832 A177833 * A177835 A177836 A177837

Keyword

nonn,base,nice

Author

Martins Opmanis, May 14 2010

Extensions

Edited by N. J. A. Sloane, May 28 2010

a(1)-a(37) confirmed by Zak Seidov, May 28 2010

Status

approved