A034709 -id:A034709 - cp's OEIS frontend

A007602 Numbers that are divisible by the product of their digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 36, 111, 112, 115, 128, 132, 135, 144, 175, 212, 216, 224, 312, 315, 384, 432, 612, 624, 672, 735, 816, 1111, 1112, 1113, 1115, 1116, 1131, 1176, 1184, 1197, 1212, 1296, 1311, 1332, 1344, 1416, 1575, 1715, 2112, 2144

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

These are called Zuckerman numbers to base 10. [So-named by J. J. Tattersall, after Herbert S. Zuckerman. - Charles R Greathouse IV, Jun 06 2017] - Howard Berman (howard_berman(AT)hotmail.com), Nov 09 2008
This sequence is a subsequence of A180484; the first member of A180484 that is not a member of A007602 is 1114. - D. S. McNeil, Sep 09 2010
Complement of A188643; A188642(a(n)) = 1; A038186 is a subsequence; A168046(a(n)) = 1: subsequence of A052382. - Reinhard Zumkeller, Apr 07 2011
The terms of n digits in the sequence, for n from 1 to 14, are 9, 5, 20, 40, 117, 285, 747, 1951, 5229, 13493, 35009, 91792, 239791, 628412, 1643144, 4314987. Empirically, the counts seem to grow as 0.858*2.62326^n. - Giovanni Resta, Jun 25 2017
De Koninck and Luca showed that the number of Zuckerman numbers below x is at least x^0.122 but at most x^0.863. - Tomohiro Yamada, Nov 17 2017
The quotients obtained when Zuckerman numbers are divided by the product of their digits are in A288069. - Bernard Schott, Mar 28 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters (2005), 2nd Edition, p. 86 (see problems 44-45).

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below)
Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.
Qizheng He and Carlo Sanna, Counting numbers that are divisible by the product of their digits, arXiv:2403.14812 [math.NT], 2024.
Giovanni Resta, Zuckerman numbers, Numbers Aplenty.
Index entries for Colombian or self numbers and related sequences

Cf. A034709, A001103, A005349, A288069.
Cf. A286590 (for factorial-base analog).
Subsequence of A002796, A034838, and A055471.

import Data.List (elemIndices)
a007602 n = a007602_list !! (n-1)
a007602_list = map succ $ elemIndices 1 $ map a188642 [1..]
-- Reinhard Zumkeller, Apr 07 2011

[ n: n in [1..2144] | not IsZero(&*Intseq(n)) and IsZero(n mod &*Intseq(n)) ];  // Bruno Berselli, May 28 2011

filter:= proc(n)
local p;
p:= convert(convert(n,base,10),`*`);
p <> 0 and n mod p = 0
end proc;
select(filter, [$1..10000]); # Robert Israel, Aug 24 2014

zuckerQ[n_] := Module[{d = IntegerDigits[n], prod}, prod = Times @@ d; prod > 0 && Mod[n, prod] == 0]; Select[Range[5000], zuckerQ] (* Alonso del Arte, Aug 04 2004 *)

for(n=1,10^5,d=digits(n);p=prod(i=1,#d,d[i]);if(p&&n%p==0,print1(n,", "))) \\ Derek Orr, Aug 25 2014

from operator import mul
from functools import reduce
A007602 = [n for n in range(1,10**5) if not (str(n).count('0') or n % reduce(mul, (int(d) for d in str(n))))] # Chai Wah Wu, Aug 25 2014

A017281 a(n) = 10*n + 1.

Original entry on oeis.org

1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 211, 221, 231, 241, 251, 261, 271, 281, 291, 301, 311, 321, 331, 341, 351, 361, 371, 381, 391, 401, 411, 421, 431, 441, 451, 461, 471, 481, 491, 501, 511, 521, 531

Offset: 0

N. J. A. Sloane

Equals [1, 2, 3, ...] convolved with [1, 9, 0, 0, 0, ...]. - Gary W. Adamson, May 30 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1] = -1, and A[i,j]=0 otherwise. Then, for n>=2, a(n-1) = -coeff(charpoly(A,x),x^(n-1)). - Milan Janjic, Feb 21 2010
Positive integers with last decimal digit = 1. - Wesley Ivan Hurt, Jun 17 2015
Also the number of (not necessarily maximal) cliques in the 2n-crossed prism graph. - Eric W. Weisstein, Nov 29 2017
From Martin Renner, May 28 2024: (Start)
Also number of squares in a grid cross with equally long arms and a width of two points (cf. A017113), e.g. for n = 2 there are nine squares of size 1 unit of area, four of size 2, two of size 5, four of size 8 and two of size 13, thus a total of 21 squares.
    · ·           · ·           · ·           · ·           * ·
    · ·           · ·           * ·           * ·           · ·
* * · · · ·   · · * · · ·   · · · · * ·   · · · · · ·   · · · · · *
* * · · · ·   · * · * · ·   · * · · · ·   * · · · * ·   * · · · · ·
    · ·           * ·           · *           · ·           · ·
    · ·           · ·           · ·           * ·           · *
The possible areas of the squares are given by ceiling(k^2/2) for 1 <= k <= 2*n+1, cf. A000982. In general, there are 4*n + 1 squares with one unit area to be found in the cross, cf. A016813, for n > 0 always four squares of even area and two squares of odd area > 1. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Crossed Prism Graph
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093645 (column 1).
Subsequence of A034709, together with A017293, A017329, A139222, A139245, A139249, A139264, A139279 and A139280.
Cf. A048161, A154428.
Cf. A005408, A016813, A016921, A017533, A161700, A128470, A158057, A161705, A161709, A161714.
Cf. A030430 (primes).
Cf. A272914, first comment. [Bruno Berselli, May 26 2016]

List([0..60], n-> 10*n+1 ); # G. C. Greubel, Sep 18 2019

a017281 = (+ 1) . (* 10)
a017281_list = [1,11..]  -- Reinhard Zumkeller, Apr 16 2012

[10*n+1 : n in [0..60]]; // Zaki Khandaker, May 16 2015

A017281:=n->10*n + 1; seq(A017281(n), n=0..80); # Wesley Ivan Hurt, Jan 29 2014

f[n_] := FromDigits[IntegerDigits[n^2, n + 1]]; Array[f, 60] (* Robert G. Wilson v, Apr 14 2009 *)
Range[1, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
(* From Eric W. Weisstein, Nov 29 2017: (Start) *)
Table[10n+1, {n, 0, 60}]
10*Range[0, 60] + 1
LinearRecurrence[{2, -1}, {11, 21}, {0, 60}]
CoefficientList[Series[(1+9x)/(1-x)^2, {x, 0, 60}], x] (* End *)

Vec((1+9*x)/(1-x)^2 + O(x^80)) \\ Michel Marcus, Jun 17 2015

[10*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

G.f.: (1+9*x)/(1-x)^2.
a(n) = 20*n - a(n-1) - 8, with a(0)=1. - Vincenzo Librandi, Nov 20 2010
a(n) = 2*a(n-1) - a(n-2), for n > 2. - Wesley Ivan Hurt, Jun 17 2015
E.g.f.: (1 + 10*x)*exp(x). - G. C. Greubel, Sep 18 2019

A017329 a(n) = 10*n + 5.

Original entry on oeis.org

5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 165, 175, 185, 195, 205, 215, 225, 235, 245, 255, 265, 275, 285, 295, 305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 405, 415, 425, 435, 445, 455, 465, 475, 485, 495, 505, 515, 525, 535

Offset: 0

N. J. A. Sloane

Continued fraction expansion of tanh(1/5). - Benoit Cloitre, Dec 17 2002
n such that 5 divides the numerator of B(2n) where B(2n) = the 2n-th Bernoulli number. - Benoit Cloitre, Jan 01 2004
5 times odd numbers. - Omar E. Pol, May 02 2008
5th transversal numbers (or 5-transversal numbers): Numbers of the 5th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 5th column in the square array A057145. - Omar E. Pol, May 02 2008
Successive sums: 5, 20, 45, 80, 125, ... (see A033429). - Philippe Deléham, Dec 08 2011
3^a(n) + 1 is divisible by 61. - Vincenzo Librandi, Feb 05 2013
If the initial 5 is changed to 1, giving 1,15,25,35,45,..., these are values of m such that A323288(m)/m reaches a new record high value. - N. J. A. Sloane, Jan 23 2019

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000367, A002193, A016957, A057145, A094884, A139600, A139606, A182007.
Subsequence of A034709, together with A017281, A017293, A139222, A139245, A139249, A139264, A139279 and A139280.
Cf. A005408, A008587, A033429, A267541, A323288.

a017329 = (+ 5) . (* 10)
a017329_list = [5, 15 ..]  -- Reinhard Zumkeller, Apr 10 2015

[10*n + 5: n in [0..60]]; // Vincenzo Librandi, Feb 05 2013

Range[5, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{5,15},60] (* Harvey P. Dale, Nov 16 2019 *)

a(n)=10*n+5 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 5*A005408(n). - Omar E. Pol, Oct 19 2008
a(n) = 20*n - a(n-1) (with a(0)=5). - Vincenzo Librandi, Nov 19 2010
G.f.: 5*(x+1)/(x-1)^2. - Colin Barker, Nov 14 2012
a(n) = A057145(n+2,5). - R. J. Mathar, Jul 28 2016
E.g.f.: 5*exp(x)*(1 + 2*x). - Stefano Spezia, Feb 14 2020
Sum_{n>=0} (-1)^n/a(n) = Pi/20. - Amiram Eldar, Dec 12 2021
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(5-sqrt(5))/2 = sqrt(2)*sin(Pi/5) = A182007/A002193.
Product_{n>=0} (1 + (-1)^n/a(n)) = phi/sqrt(2) (A094884). (End)
a(n) = (n+3)^2 - (n-2)^2. - Alexander Yutkin, Mar 16 2025
From Elmo R. Oliveira, Apr 12 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008587(2*n+1). (End)

A034838 Numbers k that are divisible by every digit of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, 124, 126, 128, 132, 135, 144, 155, 162, 168, 175, 184, 212, 216, 222, 224, 244, 248, 264, 288, 312, 315, 324, 333, 336, 366, 384, 396, 412, 424, 432, 444, 448

Offset: 1

Erich Friedman

Subset of zeroless numbers A052382: Integers with at least one digit 0 (A011540) are excluded.
A128635(a(n)) = n.
Contains in particular all repdigits A010785 \ {0}. - M. F. Hasler, Jan 05 2020
The greatest term such that the digits are all different is the greatest Lynch-Bell number 9867312 = A115569(548) = A113028(10) [see Diophante link]. - Bernard Schott, Mar 18 2021
Named "nude numbers" by Katagiri (1982-83). - Amiram Eldar, Jun 26 2021

			36 is in the sequence because it is divisible by both 3 and 6.
48 is included because both 4 and 8 divide 48.
64 is not included because even though 4 divides 64, 6 does not.

Charles Ashbacher, Journal of Recreational Mathematics, Vol. 33 (2005), pp. 227. See problem number 2693.
Yoshinao Katagiri, Letter to the editor of the Journal of Recreational Mathematics, Vol. 15, No. 4 (1982-83).
Margaret J. Kenney and Stanley J. Bezuszka, Number Treasury 3: Investigations, Facts And Conjectures About More Than 100 Number Families, World Scientific, 2015, p. 175.
Thomas Koshy, Elementary Number Theory with Applications, Elsevier, 2007, p. 79.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Diophante, A1916. Le plus grand entier divisible par ses propres chiffres (in French).
Giovanni Resta, nude numbers, Numbersaplenty, 2013.
Roberto A. Ribas, The Nude Numbers, The Pentagon, Vol. 45, No. 1 (1985), pp. 18-31.
Voodooguru, Nude Numbers, Mathematical Meanderings, Oct 11 2020.
Eric Weisstein's World of Mathematics, Digit
Index entries for 10-automatic sequences.

Intersection of A002796 (numbers divisible by each nonzero digit) and A052382 (zeroless numbers), or A002796 \ A011540 (numbers with digit 0).
Subsequence of A034709 (divisible by last digit).
Contains A007602 (multiples of the product of their digits) and subset A059405 (n is the product of its digits raised to positive powers), A225299 (divisible by square of each digit), and A066484 (n and its rotations are divisible by each digit).
Cf. A113028, A346267 (number of terms with n digits), A087140 (complement).
Supersequence of A115569 (with all different digits).

a034838 n = a034838_list !! (n-1)
a034838_list = filter f a052382_list where
   f u = g u where
     g v = v == 0 || mod u d == 0 && g v' where (v',d) = divMod v 10
-- Reinhard Zumkeller, Jun 15 2012, Dec 21 2011

[n:n in [1..500]| not 0 in Intseq(n) and #[c:c in [1..#Intseq(n)]| n mod Intseq(n)[c] eq 0] eq #Intseq(n)] // Marius A. Burtea, Sep 12 2019

a:=proc(n) local nn,j,b,bb: nn:=convert(n,base,10): for j from 1 to nops(nn) do b[j]:=n/nn[j] od: bb:=[seq(b[j],j=1..nops(nn))]: if map(floor,bb)=bb then n else fi end: 1,2,3,4,5,6,7,8,9,seq(seq(seq(a(100*m+10*n+k),k=1..9),n=1..9),m=0..6); # Emeric Deutsch

divByEvryDigitQ[n_] := Block[{id = Union[IntegerDigits[n]]}, Union[ IntegerQ[ #] & /@ (n/id)] == {True}]; Select[ Range[ 487],  divByEvryDigitQ[#] &] (* Robert G. Wilson v, Jun 21 2005 *)
Select[Range[500],FreeQ[IntegerDigits[#],0]&&AllTrue[#/ IntegerDigits[ #], IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 31 2019 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8)); if(v[1]==0, return(0)); for(i=1, #v, if(n%v[i], return(0))); 1 \\ Charles R Greathouse IV, Apr 17 2012

is_A034838(n)=my(d=Set(digits(n)));d[1]&&!forstep(i=#d,1,-1,n%d[i]&&return) \\ M. F. Hasler, Jan 10 2016

A034838_list = []
for g in range(1,4):
    for n in product('123456789',repeat=g):
        s = ''.join(n)
        m = int(s)
        if not any(m % int(d) for d in s):
            A034838_list.append(m) # Chai Wah Wu, Sep 18 2014

for n in range(10**3):
    s = str(n)
    if '0' not in s:
        c = 0
        for i in s:
            if n%int(i):
                c += 1
                break
        if not c:
            print(n,end=', ') # Derek Orr, Sep 19 2014

# finite automaton accepting sequence (see comments in A346267)
from math import gcd
def lcm(a, b): return a * b // gcd(a, b)
def inF(q): return q[0]%q[1] == 0
def delta(q, c): return ((10*q[0]+c)%2520, lcm(q[1], c))
def ok(n):
    q = (0, 1)
    for c in map(int, str(n)):
        if c == 0: return False # computation dies
        else: q = delta(q, c)
    return inF(q)
print(list(filter(ok, range(450)))) # Michael S. Branicky, Jul 18 2021

A038772 Numbers not divisible by any of their digits.

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, 223, 227, 229, 233, 239, 247, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307, 308, 323, 329, 334, 337, 338

Offset: 1

Henry Bottomley, May 04 2000

A038769(a(n)) = 0; complement of A038770.
This is a regular language when written in decimal, though the minimal regular expression is probably thousands of characters long. - Charles R Greathouse IV, Aug 19 2011
Exponential density 0.954... = A104139. Asymptotically 8/35 * n^0.954... + O(n^0.903...) members up to n. - Charles R Greathouse IV, Jul 22 2012

			34 is divisible by neither 3 nor 4.
35 is excluded because 5 is a divisor of 35, but 37 is included because neither 3 nor 7 is a divisor of 37

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences

Cf. A327561 (counts), A038770 (complement).

Cf. also A034709, A034837, A038769.

import Data.Char (digitToInt)
a038772 n = a038772_list !! (n-1)
a038772_list = filter p [1..] where
   p n = all (> 0) $ map ((mod n) . digitToInt) $ filter (> '0') $ show n
-- Reinhard Zumkeller, Jun 19 2011

[k:k in [1..340]| forall{c:c in Set(Intseq(k)) diff {0}|k mod c ne 0}]; // Marius A. Burtea, Dec 22 2019

nddQ[n_]:=Module[{idn=DeleteCases[IntegerDigits[n],0]},And@@Table[ !Divisible[n, idn[[i]]],{i,Length[idn]}]]; Select[Range[350],nddQ] (* Harvey P. Dale, Nov 01 2011 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))), , 8)); for(i=if(v[1], 1, 2), #v, if(n%v[i]==0, return(0))); 1 \\ Charles R Greathouse IV, Jul 22 2011

def ok(n): return not any(n%int(d) == 0 for d in str(n) if d != '0')
print(list(filter(ok, range(1, 339)))) # Michael S. Branicky, May 20 2021

Edited by N. J. A. Sloane, Nov 17 2008 at the suggestion of R. J. Mathar

A017293 a(n) = 10*n + 2.

Original entry on oeis.org

2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122, 132, 142, 152, 162, 172, 182, 192, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 302, 312, 322, 332, 342, 352, 362, 372, 382, 392, 402, 412, 422, 432, 442, 452, 462, 472, 482, 492, 502, 512, 522, 532

Offset: 0

N. J. A. Sloane, Dec 11 1996

Number of 5 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1A008574; m=3: A016933; m=4: A022144; m=6: A017569. - Sergey Kitaev, Nov 13 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A139222, A139245, A017329, A139249, A139264, A139279 and A139280.

Cf. A008574, A008592, A016861, A016873, A016933, A017569, A022144.

[10*n+2: n in [0..50]]; // Vincenzo Librandi, May 04 2011

A017293:=n->10*n+2; seq(A017293(n), n=0..100); # Wesley Ivan Hurt, May 03 2014

Range[2, 1000, 10] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
CoefficientList[Series[(2 + 8 x) / (1 - x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Jul 23 2016 *)
10 Range[0,60]+2 (* or *) LinearRecurrence[{2,-1},{2,12},60] (* Harvey P. Dale, Jul 04 2019 *)

a(n) = 2*A016861(n) = A008592(n) + 2. - Wesley Ivan Hurt, May 03 2014
G.f.: 2*(1 + 4*x)/(1-x)^2. - Vincenzo Librandi, Jul 23 2016
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 2*exp(x)*(1 + 5*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = A016873(2*n). (End)

A034837 Numbers that are divisible by the first, i.e., the leftmost, digit.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 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, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

Erich Friedman

A 10-automatic sequence. - Charles R Greathouse IV, Jun 13 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A034709 (divisible by last digit).

import Data.Char (digitToInt)
a034837 n = a034837_list !! (n-1)
a034837_list = filter (\i -> i `mod` (a000030 i) == 0) [1..]
-- Reinhard Zumkeller, Jun 19 2011

Select[Range[150],Divisible[#,IntegerDigits[#][[1]]]&] (* Harvey P. Dale, Jul 11 2017 *)

for(n=1,1000,n%(Vecsmall(Str(n))[1]-48) || print1(n",")) \\ M. F. Hasler, Jun 19 2011

a(n)=for(k=1,1e9,k%(Vecsmall(Str(k))[1]-48) || n-- || return(k)) \\  M. F. Hasler, Jun 19 2011

def ok(n): return n and n%int(str(n)[0]) == 0
print([k for k in range(123) if ok(k)]) # Michael S. Branicky, Jan 15 2023

from itertools import count, islice
def agen(): # generator of terms
    yield from (i for e in count(0) for f in range(1, 10) for i in range(f*10**e, (f+1)*10**e, f))
print(list(islice(agen(), 64))) # Michael S. Branicky, Jan 15 2023

a(n) mod A000030(a(n)) = 0. - Reinhard Zumkeller, Sep 20 2003

Definition clarified by Harvey P. Dale, May 01 2023

A139279 a(n) = 40*n - 32.

Original entry on oeis.org

8, 48, 88, 128, 168, 208, 248, 288, 328, 368, 408, 448, 488, 528, 568, 608, 648, 688, 728, 768, 808, 848, 888, 928, 968, 1008, 1048, 1088, 1128, 1168, 1208, 1248, 1288, 1328, 1368, 1408, 1448, 1488, 1528, 1568, 1608, 1648, 1688, 1728, 1768, 1808, 1848

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 8 with unit digit equal to 8.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139249, A139264 and A139280.

[40*n-32: n in [1..50]]; // Vincenzo Librandi, Jun 19 2011

NestList[#+40&,8,50] (* Harvey P. Dale, Oct 02 2013 *)

a(n)=40*n-32 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 40.
From G. C. Greubel, Jul 18 2017: (Start)
G.f.: (72*x - 32)/(1-x)^2.
E.g.f.: (40*x - 32)*exp(x). (End)

More terms from Reinhard Zumkeller, Jun 22 2008

New definition from Paolo P. Lava, Sep 06 2010

A139280 a(n) = 90*n - 81.

Original entry on oeis.org

9, 99, 189, 279, 369, 459, 549, 639, 729, 819, 909, 999, 1089, 1179, 1269, 1359, 1449, 1539, 1629, 1719, 1809, 1899, 1989, 2079, 2169, 2259, 2349, 2439, 2529, 2619, 2709, 2799, 2889, 2979, 3069, 3159, 3249, 3339, 3429, 3519, 3609, 3699, 3789, 3879, 3969

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 9 with final digit 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139249, A139264 and A139279.

Cf. A017281, A139222.

[90*n-81: n in [1..50]]; // Vincenzo Librandi, Jun 19 2011

Table[90*n-81, {n, 0, 50}] (* G. C. Greubel, Jul 18 2017 *)

for(n=1, 1e2, a=90*n-81; print1(a, ", ")) \\ Felix Fröhlich, Jul 07 2014

a(n) = a(n-1) + 90.
From G. C. Greubel, Jul 18 2017: (Start)
G.f.: 9*(19*x-9)/(x-1)^2.
E.g.f.: 81 + 9*(10*x - 9)*exp(x). (End) [G.f. corrected by Georg Fischer, May 12 2019]; [E.g.f. corrected by Elmo R. Oliveira, Apr 04 2025]
From Elmo R. Oliveira, Apr 04 2025: (Start)
a(n) = 9*A017281(n-1) = 3*A139222(n).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

A139245 a(n) = 20*n - 16.

Original entry on oeis.org

4, 24, 44, 64, 84, 104, 124, 144, 164, 184, 204, 224, 244, 264, 284, 304, 324, 344, 364, 384, 404, 424, 444, 464, 484, 504, 524, 544, 564, 584, 604, 624, 644, 664, 684, 704, 724, 744, 764, 784, 804, 824, 844, 864, 884, 904, 924, 944, 964, 984, 1004, 1024, 1044

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 4 with the unit digit equal to 4.

Positive integers that are the product of two integers ending with 2 (see A017293). - Stefano Spezia, Jul 25 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A017329, A139249, A139264, A139279 and A139280.

Cf. A017293.

[20*n-16: n in [1..60]]; // Vincenzo Librandi, Jun 19 2011

A139245:=n->20*n-16; seq(A139245(n), n=1..100); # Wesley Ivan Hurt, Jan 17 2014

Range[4, 1500, 20] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

a(n)=20*n-16 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 20.
a(n) = 4*A016861(n-1). - Wesley Ivan Hurt, Jan 17 2014
From Stefano Spezia, Jul 25 2021: (Start)
O.g.f.: 4*x*(1 + 4*x)/(1 - x)^2.
E.g.f.: 16 + 4*exp(x)*(5*x - 4).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

cp's OEIS Frontend

A007602 Numbers that are divisible by the product of their digits.

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A017281 a(n) = 10*n + 1.

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A034838 Numbers k that are divisible by every digit of k.

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A038772 Numbers not divisible by any of their digits.

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