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A144688
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"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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13
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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
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OFFSET
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1,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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102 has three digits, 102 is divisible by 3, and 10 is also magic, so 102 is a member.
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MAPLE
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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:
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MATHEMATICA
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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];
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PROG
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(Python)
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
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CROSSREFS
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KEYWORD
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base,nonn,fini,full
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AUTHOR
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STATUS
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approved
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