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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.

%I A144688 #62 Apr 08 2025 10:01:51
%S A144688 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,
%T A144688 42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,
%U A144688 88,90,92,94,96,98,102,105,108,120,123,126,129,141,144,147,162,165,168,180
%N 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.
%C A144688 Roberto Bosch Cabrera finds that there are exactly 20457 terms. (Total corrected by _Zak Seidov_, Feb 08 2009.)
%C A144688 The 20457th and largest term is the 25-digit number 3608528850368400786036725. - _Zak Seidov_, Feb 08 2009
%C A144688 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
%C A144688 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
%C A144688 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
%C A144688 These numbers are also called polydivisible numbers, because so many of their digits are divisible. - _Martin Renner_, Mar 05 2016
%C A144688 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
%D A144688 Robert Bosch, Tale of a Problem Solver, Arista Publishing, Miami FL, 2016.
%D A144688 David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised Edition), Penguin Books, 1997, entry 381654729, page 185.
%H A144688 Zak Seidov, <a href="/A144688/b144688.txt">The full table of n, a(n) for n=1..20457</a>
%H A144688 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=gaVMrqzb91w">Why 381,654,729 is awesome</a>, Numberphile video (2013).
%H A144688 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_22">On Some Special Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
%H A144688 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polydivisible_number">Polydivisible number</a>
%e A144688 102 has three digits, 102 is divisible by 3, and 10 is also magic, so 102 is a member.
%p A144688 P1:={seq(i,i=1..9)}:
%p A144688 for i from 2 to 25 do
%p A144688   P||i:={}:
%p A144688   for n from 1 to nops(P||(i-1)) do
%p A144688     for j from 0 to 9 do
%p A144688       if P||(i-1)[n]*10+j mod i = 0 then P||i:={op(P||i),P||(i-1)[n]*10+j}: fi:
%p A144688     od:
%p A144688   od:
%p A144688 od:
%p A144688 `union`({0},seq(P||i,i=1..25)); # _Martin Renner_, Mar 05 2016
%t A144688 divQ[n_]:=Divisible[n,IntegerLength[n]];
%t A144688 lessQ[n_]:=FromDigits[Most[IntegerDigits[n]]];
%t A144688 pdQ[n_]:=If[Or[n<10,And[divQ[n],divQ[lessQ[n]]]],True];
%t A144688 Select[Range[0,180],pdQ[#]&] (* _Ivan N. Ianakiev_, Aug 23 2016 *)
%o A144688 (Python)
%o A144688 def agen(): # generator of terms
%o A144688     yield 0
%o A144688     magic, biggermagic, digits = list(range(1, 10)), [], 2
%o A144688     while len(magic) > 0:
%o A144688         yield from magic
%o A144688         for i in magic:
%o A144688             for d in range(10):
%o A144688                 t = 10*i + d
%o A144688                 if t%digits == 0:
%o A144688                     biggermagic.append(t)
%o A144688         magic, biggermagic, digits = biggermagic, [], digits+1
%o A144688 print([an for an in agen()][:70]) # _Michael S. Branicky_, Feb 07 2022
%Y A144688 A subsequence of A098952.
%Y A144688 Cf. A082399, A051883, A143671, A214437.
%Y A144688 Cf. A050278, A050289.
%K A144688 base,nonn,fini,full
%O A144688 1,3
%A A144688 _N. J. A. Sloane_, based on email from _Roberto Bosch Cabrera_, Feb 02 2009