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A007496
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Numbers n such that the decimal expansions of 2^n and 5^n contain no 0's (probably 33 is last term).
(Formerly M0497)
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12
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0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33
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refs;
listen;
history;
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OFFSET
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1,3
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COMMENTS
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Intersection of A007377 and A008839. - Lekraj Beedassy, Jul 27 2004
Comments from Jonathan Vos Post, Jul 20 2005 (Start): Equivalently, numbers n such that 10^n is the product of two integers without any zero digits.
The first number must be some power of 2, the second the same power of 5. Such a pair is the only positive solution because neither factor could have both a 2 and a 5 in its prime factorization, or else it would be a multiple of 10 and would thus have a 0 as its last digit, which is ruled out. "We may wonder what powers of 10 are products of two integers without any zero digits. For large numbers, this is very unlikely because there will normally be 10% of zeros among many random digits... In fact, there seems to be only 11 possibilities... The probability is roughly (0.9)^n that the n-th power of 10 would yield a solution. So the expected number of solutions above the n-th power of 10 is someting like 10*(0.9)^n. Since we've actually checked that there's no other solution below n = 1500, we can be very confident that we've not missed anything..."
10^0 = 1 * 1
10^1 = 2 * 5
10^2 = 4 * 25
10^3 = 8 * 125
10^4 = 16 * 625
10^5 = 32 * 3125
10^6 = 64 * 15625
10^7 = 128 * 78125
10^9 = 512 * 1953125
10^18 = 262144 * 3814697265625
10^33 = 8589934592 * 116415321826934814453125 (End)
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REFERENCES
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Leroy C. Dalton & Henry D. Snyder, Topics for Mathematics Clubs, pp. 68-69, NCTM Reston VA 1983.
J. S. Madachy, Madachy's Mathematical Recreation, "#2. Number Toughies", pp. 126-8, Dover NY 1979.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory. Oxford Univ. Press, 1966, p. 89.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=1..11.
G. P. Michon, What two integers without zero digits have a product of 1000000000?
W. Schneider, NoZeros: Powers n^k without Digit Zero [Cached copy]
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CROSSREFS
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Sequence in context: A048319 A037405 A048333 * A082274 A029804 A084690
Adjacent sequences: A007493 A007494 A007495 * A007497 A007498 A007499
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KEYWORD
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fini,nonn,full,base
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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EXTENSIONS
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Edited by N. J. A. Sloane, Oct 24 2009 at the suggestion of M. F. Hasler
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STATUS
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approved
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