A098952 -id:A098952 - cp's OEIS frontend

A144688 "Magic" numbers: all numbers from 0 to 9 are magic; a number >= 10 is magic if it is divisible by the number of its digits and the number obtained by deleting the final digit is also magic.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 105, 108, 120, 123, 126, 129, 141, 144, 147, 162, 165, 168, 180

Offset: 1

N. J. A. Sloane, based on email from Roberto Bosch Cabrera, Feb 02 2009

Roberto Bosch Cabrera finds that there are exactly 20457 terms. (Total corrected by Zak Seidov, Feb 08 2009.)
The 20457th and largest term is the 25-digit number 3608528850368400786036725. - Zak Seidov, Feb 08 2009
a(n) is also the number such that every k-digit substring ( k <= n ) taken from the left, is divisible by k. - Gaurav Kumar, Aug 28 2009
A probabilistic estimate for the number of terms with k digits for the corresponding sequence in base b is b^k/k!, giving an estimate of e^b total terms. For this sequence, the estimate is approximately 22026, compared to the actual value of 20457. - Franklin T. Adams-Watters, Jul 18 2012
Numbers such that their first digit is divisible by 1, their first two digits are divisible by 2, and so on. - Charles R Greathouse IV, May 21 2013
These numbers are also called polydivisible numbers, because so many of their digits are divisible. - Martin Renner, Mar 05 2016
The unique zeroless pandigital (A050289) term, also called penholodigital, is a(7286) = 381654729 (see Penguin reference); so, the unique pandigital term (A050278) is a(9778) = 3816547290. - Bernard Schott, Feb 07 2022

			102 has three digits, 102 is divisible by 3, and 10 is also magic, so 102 is a member.

Robert Bosch, Tale of a Problem Solver, Arista Publishing, Miami FL, 2016.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 381654729, page 185.

Zak Seidov, The full table of n, a(n) for n=1..20457
James Grime and Brady Haran, Why 381,654,729 is awesome, Numberphile video (2013).
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Wikipedia, Polydivisible number

A subsequence of A098952.
Cf. A082399, A051883, A143671, A214437.
Cf. A050278, A050289.

P1:={seq(i,i=1..9)}:
for i from 2 to 25 do
  P||i:={}:
  for n from 1 to nops(P||(i-1)) do
    for j from 0 to 9 do
      if P||(i-1)[n]*10+j mod i = 0 then P||i:={op(P||i),P||(i-1)[n]*10+j}: fi:
    od:
  od:
od:
`union`({0},seq(P||i,i=1..25)); # Martin Renner, Mar 05 2016

divQ[n_]:=Divisible[n,IntegerLength[n]];
lessQ[n_]:=FromDigits[Most[IntegerDigits[n]]];
pdQ[n_]:=If[Or[n<10,And[divQ[n],divQ[lessQ[n]]]],True];
Select[Range[0,180],pdQ[#]&] (* Ivan N. Ianakiev, Aug 23 2016 *)

def agen(): # generator of terms
    yield 0
    magic, biggermagic, digits = list(range(1, 10)), [], 2
    while len(magic) > 0:
        yield from magic
        for i in magic:
            for d in range(10):
                t = 10*i + d
                if t%digits == 0:
                    biggermagic.append(t)
        magic, biggermagic, digits = biggermagic, [], digits+1
print([an for an in agen()][:70]) # Michael S. Branicky, Feb 07 2022

A171491 Natural numbers not divisible by their number of decimal digits.

Original entry on oeis.org

11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 101, 103, 104, 106, 107, 109, 110, 112, 113, 115, 116, 118, 119, 121, 122, 124, 125

Offset: 1

Jaroslav Krizek, Dec 10 2009

Complement of A098952.

Asymptotic density 1. - Charles R Greathouse IV, Nov 13 2010

			100 has three digits and is not divisible by 3, therefore 100 is in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A098952.

[n: n in [1..140] | not IsIntegral(n/#Intseq(n))]; // Bruno Berselli, Feb 09 2016

Select[Range[150],!Divisible[#,IntegerLength[#]]&]  (* Harvey P. Dale, Apr 24 2011 *)

def ok(n): return n%len(str(n))
print([k for k in range(142) if ok(k)]) # Michael S. Branicky, Feb 07 2022

A177869 Integers divisible by their number of digits in binary.

Original entry on oeis.org

1, 2, 6, 8, 12, 20, 25, 30, 36, 42, 48, 54, 60, 70, 77, 84, 91, 98, 105, 112, 119, 126, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 261, 270, 279, 288, 297, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 405

Offset: 1

Grant Garcia, Dec 13 2010

			105 is 1101001 in base 2 (length of 7); 105 / 7 is 15.

Daniel Arribas, Table of n, a(n) for n = 1..1000

Base 10 equivalent is A098952.

base_weight b g n | n == 0 = 0 | otherwise = (base_weight b g (n `div` b)) + (g $ n `mod` b)
interesting b g = filter f [1..] where f n = n `mod` (base_weight b g n) == 0
bin_interesting g = interesting 2 g
weights l n | (n >=0) && ((length l) > fromInteger n) = l !! fromInteger n | otherwise = 0
cnst = weights [1, 1]
let sequence = bin_interesting cnst -- Victor S. Miller, Oct 17 2011

Select[Range[410], IntegerQ[#/Length[IntegerDigits[#, 2]]] &] (* Alonso del Arte, Dec 13 2010 *)

for(d=1, 9, forstep(n=(2^(d-1)+d-1)\d*d, 2^d-1, d, print1(n", "))) \\ Charles R Greathouse IV, Oct 17 2011

import math
for n in range(1, 1000):
    if not n % int(math.log(n, 2) + 1): print(n) # Garcia

A117218 Squares divisible by their number of digits.

Original entry on oeis.org

0, 1, 4, 9, 16, 36, 64, 144, 225, 324, 441, 576, 729, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 6084, 6400, 6724, 7056, 7396, 7744, 8100, 8464, 8836, 9216, 9604, 10000

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 21 2006

			225 is in the sequence because it is a square divisible by 3.

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A000290 and A098952.

Join[{0},Select[Range[200]^2,Divisible[#,IntegerLength[#]]&]] (* Harvey P. Dale, May 06 2015 *)

lista(nn) = {for (i=0, nn, if (issquare(i) && (i % #Str(i) == 0) , print1(i, ", ")););} \\ Michel Marcus, Jun 01 2013

3364 inserted by Michel Marcus, Jun 01 2013

Corrected and extended by Harvey P. Dale, May 06 2015

A117219 Cubes divisible by their number of digits.

Original entry on oeis.org

0, 1, 8, 64, 216, 729, 1000, 1728, 2744, 4096, 5832, 8000, 15625, 27000, 42875, 64000, 91125, 110592, 157464, 216000, 287496, 373248, 474552, 592704, 729000, 884736, 1157625, 1404928, 1685159, 2000376, 2352637, 2744000, 3176523, 3652264

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 21 2006

			2744 is in the sequence because it is a cube divisible by 4.

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A000578 and A098952.

Join[{0},Select[Range[200]^3,Divisible[#,IntegerLength[#]]&]] (* Harvey P. Dale, Jan 28 2015 *)

lista(nn) = {for (i=0, nn, if (ispower(i, 3) && (i % #Str(i) == 0) , print1(i, ", ")););} \\ Michel Marcus, May 31 2013

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A172432 Duplicate of A098952.

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A144688 "Magic" numbers: all numbers from 0 to 9 are magic; a number >= 10 is magic if it is divisible by the number of its digits and the number obtained by deleting the final digit is also magic.

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A171491 Natural numbers not divisible by their number of decimal digits.

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Magma

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A177869 Integers divisible by their number of digits in binary.

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Haskell

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A117218 Squares divisible by their number of digits.

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A117219 Cubes divisible by their number of digits.

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Mathematica

PARI