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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 02:30 EST 2022. Contains 358572 sequences. (Running on oeis4.)