A034709 -id:A034709 - 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])

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]

A341432 a(n) is the denominator of the asymptotic density of numbers divisible by their last digit in base n.

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Feb 11 2021

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

			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.

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

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

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

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.

A225296 Numbers divisible by their first digit cubed (excluding those whose first digit is 1).

Original entry on oeis.org

24, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 324, 351, 378, 448, 500, 648, 2000, 2008, 2016, 2024, 2032, 2040, 2048, 2056, 2064, 2072, 2080, 2088, 2096, 2104, 2112, 2120, 2128, 2136, 2144, 2152, 2160, 2168, 2176, 2184, 2192, 2200, 2208

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Numbers where floor(n/10^floor(log(n)))^3 | n.

			448 is divisible by 4^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam Spiral of first 10000 terms.

Cf. A132359, A034709, A034837, A005349, A007602, A034838.

dfdcQ[n_]:=Module[{fidn=IntegerDigits[n][[1]]},fidn!=1&&Divisible[ n,fidn^3]]; Select[Range[2500],dfdcQ] (* Harvey P. Dale, Jan 13 2019 *)

x=0; y=rep(0,1000); len=0
firstdig<-function(x) as.numeric(substr(as.character(x),1,1))
isint<-function(x) x==as.integer(x)
while(len<1000) if((fd=firstdig((x=x+1)))>1) if(isint(x/fd^3)) y[(len=len+1)]=x

A225722 Numbers divisible by their last digit cubed, excluding those whose last digit is 1.

Original entry on oeis.org

32, 64, 72, 112, 125, 152, 192, 216, 232, 243, 272, 312, 352, 375, 384, 392, 432, 472, 512, 513, 552, 592, 625, 632, 672, 704, 712, 729, 752, 783, 792, 832, 872, 875, 912, 952, 992, 1024, 1032, 1053, 1072, 1112, 1125, 1152, 1192, 1232, 1272, 1296, 1312, 1323

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

a(n) ~ n. For 69 < n < 10000, the formula 26.61*n - 2.76 provides an estimate of a(n) to within 1%.

The asymptotic density of this sequence is 601296607/16003008000 = 0.037573... . Therefore, contrary to the above comment, a(n) ~ c*n where c = 16003008000/601296607 = 26.614166... . - Amiram Eldar, Aug 08 2023

			a(5) = 125 is an example because its last digit is 5, and 5^3 = 125, and 125 is divisible by 125.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Subsequence of A225297.

Cf. A221651, A225296, A132359, A034709, A034837, A005349, A007602, A034838.

dldcQ[n_]:=Module[{ld=Last[IntegerDigits[n]]},ld>1&&Divisible[n,ld^3]]; Select[Range[1500],dldcQ] (* Harvey P. Dale, Aug 15 2014 *)

which(sapply(1:1000,function(x) x%%10>1 & (v=x/(x%%10)^3)==as.integer(v) ))

A210582 Numbers whose first digit is the remainder of their division by the last digit (in base 10).

Original entry on oeis.org

13, 19, 23, 26, 29, 39, 46, 49, 59, 69, 79, 89, 103, 109, 127, 133, 163, 193, 197, 199, 203, 206, 209, 214, 218, 233, 234, 236, 247, 254, 258, 263, 266, 274, 293, 294, 296, 298, 299, 309, 367, 399, 406, 409, 417, 428, 436, 466, 468, 487, 496, 499, 509, 537, 599, 609, 638, 657, 678, 699, 709, 799, 809, 899

Offset: 1

Eric Angelini and M. F. Hasler, Mar 22 2012

This is a restricted or simplified version of the definition of modest numbers A054986.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Angelini, Not Modest, Mar 22 2012
E. Angelini, Not modest [Cached copy, with permission]

Cf. A054986, A178158.

A subsequence of A067251, disjoint with A034709.

a210582 n = a210582_list !! (n-1)
a210582_list = filter (\x -> mod x (a010879 x) == a000030 x) a067251_list
-- Reinhard Zumkeller, Mar 26 2012

[ n: n in [1..1002] | not IsZero(d[1]) and n mod d[1] eq d[#d] where d is Intseq(n) ];  // Bruno Berselli, Mar 26 2012

is_nm( x )=x%10 && x%(x%10)==x\10^(#Str(x)-1)
for(n=1,999,is_nm(n)&print1(n","))

a(n) mod A010879(a(n)) = A000030(a(n)). [Reinhard Zumkeller, Mar 26 2011]

Edited by M. F. Hasler, Jan 14 2014

A225299 Numbers divisible by the square of each digit.

Original entry on oeis.org

1, 11, 12, 36, 111, 112, 128, 144, 212, 216, 224, 333, 432, 448, 612, 1111, 1112, 1116, 1212, 1296, 1332, 1424, 2112, 2144, 2212, 2224, 2232, 2916, 3132, 3312, 3636, 4112, 4144, 4224, 4288, 4464, 6336, 6624, 8128, 8448, 9396, 11111, 11112, 11133, 11172, 11212

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Includes all repunits.

			a(7) 128 is divisible by 1^2, by 2^2, and by 8^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..5000

Cf. A132359, A034709, A034837, A005349, A007602, A034838.

d[n_]:=IntegerDigits[n]; t={}; Do[If[!MemberQ[d[n],0] && Union[Mod[n,d[n]^2]] == {0}, AppendTo[t,n]], {n,11220}]; t (* Jayanta Basu, May 15 2013 *)
Select[Range[12000],DigitCount[#,10,0]==0&&And@@Divisible[ #,IntegerDigits[ #]^2]&] (* Harvey P. Dale, Jul 16 2018 *)

isint<-function(x) x==as.integer(x)
sqalldig<-function(x) as.numeric(strsplit(as.character(x),"")[[1]])^2
divby<-function(x) ifelse(length(grep(0,x))>0,F,all(isint(x/sqalldig(x))))
which(sapply(1:1000,divby))

A277804 Numbers n such that first digit of n divides n, last digit of n divides n, number of divisors of n divides n and phi(n) divides n, where phi(n) is the Euler totient function.

Original entry on oeis.org

1, 2, 8, 12, 24, 36, 128, 288, 384, 864, 972, 1152, 1944, 3456, 6144, 6912, 13122, 18432, 26244, 31104, 62208, 69984, 209952, 279936, 294912, 497664, 839808, 884736, 1679616, 3538944, 4478976, 13436928, 22674816, 25165824, 31850496, 45349632

Offset: 1

Ilya Gutkovskiy, Nov 01 2016

Numbers n such that A000030(n)|n, A010879(n)|n, A000005(n)|n and A000010(n)|n.

Intersection of A007694, A034709, A033950 and A034837.

			a(5) = 24 because 24/2 = 12, 24/4 = 6, 24 has 8 divisors {1,2,3,4,6,8,12,24}, 24/8 = 3, phi(24) = 8 {1,5,7,11,13,17,19,23} and 24/8 = 3 (all are an integers).

Robert G. Wilson v, Table of n, a(n) for n = 1..43
Index to divisibility sequences

Cf. A000005, A000010, A000030, A007694, A034709, A033950, A034837, A010879.

Select[Range[50000000], Divisible[#1, First[IntegerDigits[#1]]] && Divisible[#1, Last[IntegerDigits[#1]]] && Divisible[#1, DivisorSigma[0, #1]] && Divisible[#1, EulerPhi[#1]] & ]

a(24) - a(36) added by G. C. Greubel, Nov 02 2016

A176659 Partial sums of A038770.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 277, 302, 328, 356, 386, 417, 449, 482, 517, 553, 592, 632, 673, 715, 759, 804, 852, 902, 953, 1005, 1060, 1120, 1181, 1243, 1306, 1370, 1435, 1501, 1571, 1642

Offset: 1

Jonathan Vos Post, Apr 23 2010

Partial sums of numbers divisible by at least one of their digits. Identical to triangular numbers A000217 until a(23), then differs, because 23 is the smallest natural number in the complement of A038770 (A038772, i.e., not divisible by at least one of its digits). Hence this partial sum is the triangular numbers minus the partial sums of A038772, properly offset. The subsequence of primes (of course 3 is the largest prime triangular number) in the partial sum begins: 3, 277, 449, 673, 953, 1181, 1571, 1789, 2027. What are the equivalents in bases other than 10?

			a(23) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 24 = 277 is prime.

Cf. A038770, A034709, A034837, A038769, A038772.

Accumulate[Select[Range[110],MemberQ[Divisible[#,Cases[ IntegerDigits[ #], Except[ 0]]], True]&]] (* Harvey P. Dale, May 11 2017 *)

a(n) = Sum_{i=1..n} A038770(i).

A188454 Numbers n whose decimal digits are distinct and no digit divides n.

Original entry on oeis.org

23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 207, 209, 239, 247, 249, 253, 257, 259, 263, 267, 269, 283, 289, 293, 307, 308, 329, 346, 347, 349, 356, 358, 359, 367, 370, 374

Offset: 1

Andy Edwards, Mar 31 2011

These may not contain 1 or have a 2 or 5 as the last digit. They include prime numbers not containing the digit 1 and composites with a smallest prime factor > 10 and obeying the other constraints (e.g. the largest case is 987654203 = 31*31859813).
The first even case is 34. The first consecutive pair is {37, 38}. {56,57,58,59} is a consecutive quadruple which is the maximal size for such a subset.
There are 202623 terms in this sequence. - Nathaniel Johnston, May 19 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A038772, A034709, A034837, A038769, A038770.

dddQ[n_]:=Module[{dcn=DigitCount[n]},Max[dcn]==1&&First[dcn]==0 && Union[ Divisible[n,Select[IntegerDigits[n],#!=0&]]]=={False}]; Select[Range[ 400],dddQ] (* Harvey P. Dale, May 01 2012 *)

cp's OEIS Frontend

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

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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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A341432 a(n) is the denominator of the asymptotic density of numbers divisible by their last digit in base n.

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A225296 Numbers divisible by their first digit cubed (excluding those whose first digit is 1).

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A225722 Numbers divisible by their last digit cubed, excluding those whose last digit is 1.

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A210582 Numbers whose first digit is the remainder of their division by the last digit (in base 10).

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A225299 Numbers divisible by the square of each digit.

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A277804 Numbers n such that first digit of n divides n, last digit of n divides n, number of divisors of n divides n and phi(n) divides n, where phi(n) is the Euler totient function.