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
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.
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(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.
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
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.
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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).
A359841
Integers Xd which are divisible by X, where d is the last decimal digit.
Original entry on oeis.org
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420
Offset: 1
- N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics, problem 15, pp. 11 and 102, Dover publications, Inc., New York, 1993.
- Index entries for linear recurrences with constant coefficients, signature (2,-1).
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Select[Range[10, 500], Divisible[#, Floor[#/10]] &] (* Amiram Eldar, Jan 15 2023 *)
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isok(k) = (k>9) && (k % (k \ 10) == 0); \\ Michel Marcus, Jan 20 2023
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def ok(n): return n > 9 and n%(n//10) == 0
print([k for k in range(421) if ok(k)]) # Michael S. Branicky, Jan 15 2023
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def A359841(n): return (10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99)[n-1] if n <= 32 else (n-23)*10 # Chai Wah Wu, Jan 20 2023
Showing 1-3 of 3 results.
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