A169819 -id:A169819 - cp's OEIS frontend

A178538 Records in A169819.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 20, 23, 25, 26, 29, 32, 33, 37, 38, 43, 47, 49, 54, 58, 59, 66, 68, 71, 73, 76, 80, 88, 93, 96, 103, 104, 105, 106, 112, 113, 117, 126, 129, 130, 143, 147, 151, 161, 171, 176, 187, 192, 200, 205, 216

Offset: 1

Zak Seidov, May 29 2010

Cf. A169819, A177834, A178539.

mx=0;s={};Do[id=IntegerDigits[n];FLA=Flatten[Table[Partition[id,k,1],
{k,Length[id]}],1];fd=Union[FromDigits/@FLA];
dv=Length[Union[Flatten[Divisors/@fd]]];If[dv>mx,mx=dv;AppendTo[s2,{mx,n}]],
{n,200000}]; A178538 =First/@s

A178539 Where records occur in A169819.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 30, 48, 56, 60, 108, 120, 144, 180, 288, 396, 504, 720, 840, 1008, 1176, 1260, 1440, 1680, 1980, 2520, 3360, 3780, 6720, 7560, 9240, 10080, 11088, 11760, 13608, 15120, 15840, 19800, 22680, 25200, 26460, 27720, 31680

Offset: 1

Zak Seidov, May 29 2010

Next terms corresponding to terms in A178538: 32760,36960,49504,55440,64680,95040,98280,110880,123760,128520, 159840,163800,191520,196560.

Cf. A169819, A177834, A178538.

mx=0;s={};Do[id=IntegerDigits[n];FLA=Flatten[Table[Partition[id,k,1],
{k,Length[id]}],1];fd=Union[FromDigits/@FLA];
dv=Length[Union[Flatten[Divisors/@fd]]];If[dv>mx,mx=dv;AppendTo[s2,{mx,n}]],
{n,200000}]; A178539 = Last/@s

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.

Original entry on oeis.org

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

Martins Opmanis, May 14 2010

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)

			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.

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

Cf. A003418 (a weak upper bound), A169819, A169858, A178544.

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 *)

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

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

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

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A178538 Records in A169819.

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A178539 Where records occur in A169819.

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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.

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