A177834 -id:A177834 - cp's OEIS frontend

A169819 a(n) = total number of distinct divisors of n and all of its substrings.

1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 3, 5, 4, 7, 3, 8, 4, 7, 5, 4, 4, 8, 4, 6, 6, 7, 5, 9, 3, 7, 4, 6, 5, 9, 4, 7, 5, 9, 4, 9, 5, 6, 8, 7, 5, 10, 7, 7, 5, 7, 4, 10, 4, 11, 6, 7, 5, 13, 5, 6, 8, 9, 7, 8, 6, 9, 7, 9, 3, 13, 4, 6, 7, 9, 4, 11, 5, 11, 8, 6, 6, 13, 7, 8, 8, 8, 7, 13, 6, 8, 5, 7, 6, 13, 5, 10, 6

Offset: 1

Zak Seidov, May 28 2010

Note that we are counting 0 when it occurs as a digit of n, but are not counting any other integers as divisors of 0. (If we did, there would be infinitely many of them; every integer divides 0.) [From Franklin T. Adams-Watters, May 29 2010]

			a(56) = 11 because divisors of 56 are d1= {1, 2, 4, 7, 8, 14, 28, 56}; 56 has two substrings 5,6; divisors of 5 are d2= {1, 5}, and divisors of 6 are d3= {1, 2, 3,6} ; union of d1,d2,d3 gives 11 distinct divisors of 56 and all of its substrings: {1, 2, 3, 4, 5, 6, 7, 8, 14, 28, 56}.

Cf. A177834

Table[id = IntegerDigits[n]; FLA = Flatten[Table[Partition[id, k, 1], {k, Length[id]}], 1]; fd = Union[FromDigits /@ FLA]; dv = Length[Union[Flatten[Divisors /@ fd]]], {n, 200}]

A169858 Smallest integer k such that k or one of its left substrings (or prefixes, regarded as an integer) is divisible by any integer from {1,2,...,n}.

Original entry on oeis.org

1, 2, 6, 12, 45, 60, 245, 245, 504, 504, 5049, 5049, 10296, 11760, 11760, 11760, 56160, 56160, 198016, 198016, 1008159, 1323008, 2340849, 6240366, 13442580, 13442580, 37536408, 37536408, 75432065, 75432065, 180092645, 319800096, 319800096, 800640126, 2201169600, 2201169600, 3780487275, 5250966084, 5250966084, 6832425609, 36960308625, 36960308625, 62244072512, 62244072512, 62244072512, 62244072512, 372960042489, 372960042489

Offset: 1

Hugo van der Sanden, Jun 01 2010

			a(5) = 45 as the left substrings of 45 are {4, 45} and for every d in {1,2,...,n} = {1, 2, 3, 4, 5} there is a left substring of 45 such that d | 45. That is: 1 | 4, 2 | 4, 3 | 45, 4 | 4, 5 | 45. - _David A. Corneth_, Jun 09 2023

Hugo van der Sanden A169858: C source code to calculate terms.

Cf. A177834, A175516, A003418.

from itertools import count, islice
def agen(): # generator of terms
    n = 1
    for k in count(1):
        s = str(k)
        prefixes = [int(s[:i+1]) for i in range(len(s))]
        if all(any(ki%m == 0 for ki in prefixes) for m in range(1, n+1)):
            yield k; n += 1
            while any(ki%n == 0 for ki in prefixes):
                yield k; n += 1
print(list(islice(agen(), 20))) # Michael S. Branicky, Jun 09 2023

a(n) = min m: forall d in {1..n}: exists k in {0..log_10(m)}: d | floor(m / 10^k).

a(n) <= A003418(n). - Michael S. Branicky, Jun 09 2023

Corrected and extended by Hugo van der Sanden, Jun 04 2010 (errors reported by Zak Seidov).

A178538 Records in A169819.

Original entry on oeis.org

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

A178544 a(n) are such that n or one of its substrings is divisible by every integer from 1 to a(n).

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 3, 2, 3, 4, 2, 3, 3, 2, 3, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 2, 4, 4, 2, 5, 4, 2, 4, 4, 2, 1, 2, 1, 6, 1, 8, 1, 2, 1, 6, 3, 3, 3, 4, 3, 3, 3, 4, 3, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 4, 2, 4, 4, 2, 4, 4, 2, 4, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 2, 3, 3, 2, 3

Offset: 1

Zak Seidov, May 29 2010

Positions of records 1,2,6,12,45,54,56,... are terms in A177834 (with repetitions).

			a(56)=8 because divisors of 56 are d1={1,2,4,7,8,14,28,56},
divisors of 5 are d2={1, 5}, divisors of 6 are d3={1,2,3,6},
and union of d1, d2, d3 gives 8 subsequent integers 1..8 (14,28,.. not counted).

A177834.

Table[id=IntegerDigits[n]; FLA=Flatten[Table[Partition[id, k, 1],
{k,Length[id]}], 1]; fd = Complement[Union[FromDigits /@ FLA], {0}];
dv=Union[Flatten[Divisors /@ fd]]; Complement[Range[100], dv][[1]]-1, {n,10^3}]

cp's OEIS Frontend

A169819 a(n) = total number of distinct divisors of n and all of its substrings.

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A169858 Smallest integer k such that k or one of its left substrings (or prefixes, regarded as an integer) is divisible by any integer from {1,2,...,n}.

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A178538 Records in A169819.

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A178539 Where records occur in A169819.

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A178544 a(n) are such that n or one of its substrings is divisible by every integer from 1 to a(n).

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