A341431 -id:A341431

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A034709

Numbers divisible by their last digit.

+10
35

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(Haskell)

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

(Python)

A034709_list = [n for n in range(1, 1000) if n % 10 and not n % (n % 10)]

# Chai Wah Wu, Sep 18 2014

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

CROSSREFS

Cf. A010879, A034838, A007602.

Cf. A017281, A017293, A139222, A139245, A017329, A139249, A139264, A139279, A139280.

Cf. A341431, A341432.

KEYWORD

nonn,base,easy,nice

AUTHOR

Erich Friedman

STATUS

approved

A341432

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

+10
4

2, 2, 12, 12, 60, 20, 840, 840, 2520, 2520, 27720, 27720, 360360, 360360, 720720, 720720, 12252240, 4084080, 232792560, 77597520, 33256080, 5173168, 5354228880, 356948592, 3824449200, 26771144400, 11473347600, 80313433200, 332727080400, 2329089562800, 144403552893600

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

a(n) divides A003418(n), and a(n) = A003418(n) for n = 1, 2, 4, 6, 8, 10, 12, ...

LINKS

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

FORMULA

A341431(n)/a(n) = (1/n) * Sum_{k=1..n-1} gcd(k, n)/k. [corrected by Amiram Eldar, Nov 16 2022]

a(prime(n)) = A185399(n), for n > 1.

EXAMPLE

For n=2, the numbers divisible by their last binary digit are the odd numbers (A005408) whose density is 1/2, therefore a(2) = 2.

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) = 2.

For n=10, the numbers divisible by their last digit in base 10 are A034709 whose density is 1177/2520, therefore a(10) = 2520.

MATHEMATICA

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

CROSSREFS

Cf. A003418, A005408, A034709, A047236, A185399, A341431 (numerators).

KEYWORD

nonn,base,frac,easy

AUTHOR

Amiram Eldar, Feb 11 2021

STATUS

approved

A356093

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

+10
2

1, 1, 2, 1, 1, 2, 8, 3, 1, 2, 1, 6, 4, 1, 1, 2, 1, 2, 1, 1, 12, 1, 1, 4, 16, 10, 1, 1, 18, 8, 3, 1, 4, 1, 2, 5, 2, 27, 1, 2, 1, 6, 1, 32, 14, 3, 1, 1, 1, 2, 4, 1, 8, 25, 128, 1, 2, 9, 2, 4, 1, 2, 3, 1, 4, 2, 1, 8, 1, 2, 16, 1, 1, 2, 9, 1, 2, 6, 40, 4, 1, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

f(n) = a(n)/A356094(n) is the asymptotic density of numbers k such that prime(n) = A053669(k) is the smallest prime not dividing k.

The asymptotic mean of A053669 is 2.9200509773... (A249270) which is the weighted mean of the primes with f(n) as weights. The corresponding asymptotic standard deviation, which can be evaluated from the second raw moment Sum_{n>=1} f(n) * prime(n)^2, is 2.8013781465... .

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = 1 iff prime(n) is in A039787.

Let f(n) = a(n)/A356094(n):

f(n) = A006093(n)/A002110(n).

Sum_{n>=1} f(n) = 1.

Sum_{n>=1} f(n) * prime(n) = A249270.

EXAMPLE

Fractions begin with 1/2, 1/3, 2/15, 1/35, 1/231, 2/5005, 8/255255, 3/1616615, 1/10140585, 2/462120945, ...

MATHEMATICA

primorial[n_] := Product[Prime[i], {i, 1, n}]; Numerator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 100}]]

PROG

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

(Python)

from math import gcd

from sympy import prime, primorial

def A356093(n): return (p:=prime(n)-1)//gcd(p, primorial(n)) # Chai Wah Wu, Jul 26 2022

CROSSREFS

Cf. A002110, A006093, A039787, A053669, A249270, A356094 (denominators).

Similar sequences: A038110, A338559, A340818, A341431, A342450, A342479.

KEYWORD

nonn,frac

AUTHOR

Amiram Eldar, Jul 26 2022

STATUS

approved

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