A341432 -id:A341432 - cp's OEIS frontend

A034709 Numbers divisible by their last digit.

1, 2, 3, 4, 5, 6, 7, 8, 9, 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, 111, 112, 115, 121, 122, 123, 124, 125, 126, 128, 131, 132

Offset: 1

Erich Friedman

Union of A017281, A017293, A139222, A139245, A017329, A139249, A139264, A139279 and A139280. - Reinhard Zumkeller, Jun 22 2008
The differences between consecutive terms repeat with period 1177 and the corresponding terms differ by 2520 = LCM(1,2,...,9). In other words, a(k*1177+i) = 2520*k + a(i). - Giovanni Resta, Aug 20 2015
The asymptotic density of this sequence is 1177/2520 = 0.467063... (see A341431 and A341432 for the values in other base representations). - Amiram Eldar, Nov 24 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010879, A034838, A007602.
Cf. A017281, A017293, A139222, A139245, A017329, A139249, A139264, A139279, A139280.
Cf. A341431, A341432.

import Data.Char (digitToInt)
a034709 n = a034709_list !! (n-1)
a034709_list =
   filter (\i -> i `mod` 10 > 0 && i `mod` (i `mod` 10) == 0) [1..]
-- Reinhard Zumkeller, Jun 19 2011

N:= 1000: # to get all terms <= N
sort([seq(seq(ilcm(10,d)*x+d, x=0..floor((N-d)/ilcm(10,d))), d=1..9)]); # Robert Israel, Aug 20 2015

dldQ[n_]:=Module[{idn=IntegerDigits[n],last1},last1=Last[idn]; last1!= 0&&Divisible[n,last1]]; Select[Range[150],dldQ]  (* Harvey P. Dale, Apr 25 2011 *)
Select[Range[150],Mod[#,10]!=0&&Divisible[#,Mod[#,10]]&] (* Harvey P. Dale, Aug 07 2022 *)

for(n=1,200,if(n%10,if(!(n%digits(n)[#Str(n)]),print1(n,", ")))) \\ Derek Orr, Sep 19 2014

A034709_list = [n for n in range(1, 1000) if n % 10 and not n % (n % 10)]
# Chai Wah Wu, Sep 18 2014

A341431 a(n) is the numerator of the asymptotic density of numbers divisible by their last digit in base n.

Original entry on oeis.org

1, 1, 7, 5, 37, 7, 421, 347, 1177, 671, 14939, 6617, 135451, 140311, 271681, 143327, 5096503, 751279, 91610357, 24080311, 9098461, 830139, 2188298491, 77709491, 925316723, 6609819823, 3567606143, 10876020307, 123417992791, 300151059037, 37903472946337, 32271030591223

Offset: 2

Amiram Eldar, Feb 11 2021

			The sequence of fractions begins with 1/2, 1/2, 7/12, 5/12, 37/60, 7/20, 421/840, 347/840, 1177/2520, 671/2520, 14939/27720, 6617/27720, 135451/360360, 140311/360360, ...
For n=2, the numbers divisible by their last binary digit are the odd numbers (A005408) whose density is 1/2, therefore a(2) = 1.
For n=3, the numbers divisible by their last digit in base 3 are the numbers that are congruent to {1, 2, 4} mod 6 (A047236) whose density is 1/2, therefore a(3) = 1.
For n=10, the numbers divisible by their last digit in base 10 are A034709 whose density is 1177/2520, therefore a(10) = 1177.

Amiram Eldar, Table of n, a(n) for n = 2..2300

Cf. A005408, A034709, A047236, A341432 (denominators).

a[n_] := Numerator[Sum[GCD[k, n]/k, {k, 1, n - 1}]/n]; Array[a, 32, 2]

a(n)/A341432(n) = (1/n) * Sum_{k=1..n-1} gcd(k, n)/k. [corrected by Amiram Eldar, Nov 16 2022]

A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

Original entry on oeis.org

2, 3, 15, 35, 231, 5005, 255255, 1616615, 10140585, 462120945, 6685349671, 1236789689135, 30425026352721, 311494317420715, 13367169186706335, 1253429172199617105, 33151040519900217915, 3909612711980232366109, 119065478046670712967865, 7970583287524270870963077

Offset: 1

Amiram Eldar, Jul 26 2022

See A356093 for details.

Amiram Eldar, Table of n, a(n) for n = 1..351

Cf. A002110, A356093 (numerators).

Similar sequences: A038111, A338560, A340819, A341432, A342451, A342480.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Denominator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 20}]]

a(n) = denominator((prime(n)-1)/factorback(primes(n))); \\ Michel Marcus, Jul 26 2022

from math import gcd
from sympy import primorial, prime
def A356094(n): return (p:=primorial(n))//gcd(p,prime(n)-1) # Chai Wah Wu, Jul 26 2022

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A034709 Numbers divisible by their last digit.

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A341431 a(n) is the numerator of the asymptotic density of numbers divisible by their last digit in base n.

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A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

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