A292684 -id:A292684 - cp's OEIS frontend

A292683 Numbers divisible by themselves with first digit removed (A217657), excluding multiples of 10.

11, 12, 15, 21, 22, 24, 25, 31, 32, 33, 35, 36, 41, 42, 44, 45, 48, 51, 52, 55, 61, 62, 63, 64, 65, 66, 71, 72, 75, 77, 81, 82, 84, 85, 88, 91, 92, 93, 95, 96, 99, 101, 102, 104, 105, 125, 201, 202, 204, 205, 208, 225, 301, 302, 303, 304, 305, 306, 312, 315, 325, 375, 401, 402, 404, 405, 408, 416, 425, 501

Offset: 1

M. F. Hasler, Oct 17 2017

Obviously, any term multiplied by 10 would again be a term, so we exclude trailing zeros.
This sequence cannot contain single-digit numbers (which would yield 0 with the initial digit removed), in contrast to A178158 (numbers divisible by every suffix of n) where the condition is vacuously satisfied for single-digit numbers.
416 is the first term in the present sequence which is not in A178158.
See A292684 and A292685 for the (number of) multiples of N = a(n) which have the same property and yield the same ratio N/A217657(N).

			12 is in the sequence because it is divisible by 2.
416 is in the sequence because it is divisible by 16, 416 = 4*4*25 + 16.

Harvey P. Dale, Table of n, a(n) for n = 1..712 (All terms up to 10^7.)

Cf. A217657, A178158, A034709.

Cf. A292684, A292685.

fQ[n_] := Mod[n, 10] > 0 && Mod[n, n - Quotient[n, 10^Floor@ Log10@ n] 10^Floor@ Log10@ n] == 0; Select[ Range[11, 501], fQ] (* Robert G. Wilson v, Oct 18 2017 *)
Select[Range[10,550],Mod[#,10]!=0&&Mod[#,FromDigits[Rest[IntegerDigits[#]]]]==0&] (* Harvey P. Dale, Sep 15 2024 *)

select( is(n)=n%10&&(m=n%10^logint(n,10))&&!(n%m), [0..500])

A292685 Irregular table where row n lists the positive integers k not divisible by 10 such that f(kN) = f(N) for N = A292683(n) and f(x) = x / (x without its first digit: A217657(x)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 15, 25, 35, 45, 1, 2, 3, 4, 5, 15, 25, 35, 45, 1, 2, 5, 15, 1, 5, 15, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 5, 15, 25, 75, 125, 175, 225, 1, 2, 5, 15, 25, 75, 125, 175, 225, 1, 2, 5, 15, 25, 75, 125, 175, 225

Offset: 1

M. F. Hasler, Oct 18 2017

The numbers N listed in A292683 are such that N is divisible by A217657(N) = N with its initial digit removed. Most of these numbers have several multiples k*N which again have this property, but furthermore, such that the ratio k*N / A217657(k*N) is always the same. The corresponding k-values are listed here.

It is not rare that there are 9 such k-values, although the set of these is usually different from { 1, ..., 9 }. Is there any N for which there more than 10 such k-values?

			The table starts as follows:
    n | N=A292683(n) | N/A217657(N) | A292685(n,k=1..A292684(n))
    1 |      11      |      11      | 1, 2, 3, 4, 5, 6, 7, 8, 9
    2 |      12      |       6      | 1, 2, 3, 4
    3 |      15      |       3      | 1
    4 |      21      |      21      | 1, 2, 3, 4, 5, 15, 25, 35, 45
    5 |      24      |       6      | 1, 2, 5, 15
      |    (...)     |    (...)     | (...)
   68 |     416      |      26      | 1, 2, 5, 15, 25, 75, 125, 175, 225
           (...)
For A292683(2) = 12, we have k = 1, 2, 3, 4 satisfying 12*k / A217657(12*k) = 6, e.g., 12*4 = 48, 48 / 8 = 6 (= 12 / 2).
There are other k such that 12*k is divisible by A217657(12*k), e.g., k = 6, 7, 8, 17, ... (=> 12*k = 72, 84, 96, 204: all divisible by their last digit), but which yield ratios (here 36, 21, 16, 51) different from 6.
For n = 4, we have, e.g., 21*15 = 315, 315 / 15 = 21 (= 21 / 1), or 21*45 = 945, 945 / 45 = 21. Here too, e.g., 21*24 = 504 is divisible by 04, but 504 / 4 = 126, not 21.

Cf. A292683, A292684 (gives the row lengths), A217657, A000030.

A292685_row(n, N=A292683(n), r=N/A217657(N), a=[1])={for(k=2, oo, if(k%10,A217657(k*N)*r==k*N&&a=concat(a,k), k<10*a[#a]||break)); a} \\ Instead of the 1st arg. n, one can directly give N (= A292683(n) by default) as 2nd arg. It is not checked whether N is in A292683 (else the resulting vector should be empty).

cp's OEIS Frontend

A292683 Numbers divisible by themselves with first digit removed (A217657), excluding multiples of 10.

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