A292685 -id:A292685 - 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])

A292684 a(n) is the number of 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

9, 4, 1, 9, 9, 4, 3, 3, 3, 3, 1, 1, 9, 9, 9, 7, 4, 9, 9, 9, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 9, 9, 9, 9, 9, 9, 9

Offset: 1

M. F. Hasler, Oct 17 2017

Sequence A292683 lists the numbers n which are divisible by A217657(n), i.e., by n with its first digit removed.
We exclude k with trailing 0's (just like in A292683) because if k*N has the property, then 10*k*N trivially also has the property.
Is there any number for which there are more than 9 possible k-values?
All of the k-values are listed in the table A292685.

			For A292683(1) = 11, we have k = 1, ..., 9 satisfying 11*k / A217657(11*k) = 11.
For A292683(2) = 12, we have k = 1, 2, 3, 4 satisfying 12*k / A217657(12*k) = 6.
For A292683(3) = 15, we have only k = 1 satisfying 15*k / A217657(15*k) = 3.
For A292683(4) = 21, we have k = 1, 2, 3, 4, 5, 15, 25, 35 and 45 satisfying 21*k / A217657(21*k) = 2.

Cf. A292683, A292685, A217657, A000030.

(A217657(n)=n%10^logint(n,10)); A292684(n,N=A292683(n),r=N/A217657(N),a=[1])={for(k=2,oo,k%10||next;k>10*a[#a]&&break;A217657(k*N)*r==k*N&&a=concat(a,k));#a} \\ Instead of the 1st arg. n, one can directly give N (= A292683(n) by default) as 2nd arg. One could store only the last 'a' (and increase a counter) instead of storing all 'a's.

cp's OEIS Frontend

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

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