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A163143
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Numbers k whose digit sum is equal to the sum of the digits of the factors of k when written in a certain way as a product of numbers each raised to some power (the sum includes the digits of the exponents).
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2
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4, 8, 25, 26, 27, 36, 44, 48, 54, 56, 62, 64, 65, 68, 75, 80, 84, 92, 96, 98, 108, 121, 125, 128, 129, 143, 147, 155, 156, 164, 168, 176, 182, 183, 184, 188, 189, 192, 195, 206, 216, 224, 242, 248, 256, 258, 260, 264, 270, 276, 278, 284, 288, 294, 296, 308, 318
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OFFSET
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1,1
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COMMENTS
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We call these numbers "zipper numbers" because the factorization resembles a zipper both graphically and in the way one would go about summing the digits.
Zipper numbers are similar to vampire numbers, that is, there can be many ways to factor a number as a product of powers; e.g., 36=6^2, but one has to find the correct way, i.e., that will yield the same digit sum. Obviously, some restrictions must be made; e.g., the use of x^0 and 1^x is forbidden. Note that 8=4^1*2^1, 27=3^2*3^1 and 44=11^1*4^1 are not prime factorizations.
The consecutive numbers 25,26,27 can be called triple zippers or trip-zips; how many more are there? Prime numbers and powers of 10 can never be zippers.
Triple zippers:
25 = 5^2, 26 = 2^1*13^1, 27 = 3^1*3^2;
182 = 13^1*14^1, 183 = 3^1*61^1, 184 = 2^1*2^2*23^1;
735 = 7^1*105^1, 736 = 2^3*4^1*23^1, 737 = 11^1*67^1;
902 = 22^1*41^1, 903 = 21^1*43^1, 904 = 2^1*2^2*113^1.
Quadruple zipper: 782 = 2^1*391^1, 783 = 3^3*29^1, 784 = 2^3*7^1*14^1, 785 = 5^1*157^1. (End)
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LINKS
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EXAMPLE
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The number 8 is a zipper number as it can be written as 8 = 4^1*2^1 and 8 = 4+1+2+1.
The number 36 can be factored as 36=2^2*3^2, and 3+6 = 9 = 2+2+3+2.
The number 121 can be factored as 121=11^2.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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Yossi Elran (yossi.elran(AT)weizmann.ac.il) and Royi Lachmi, Jul 21 2009
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EXTENSIONS
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STATUS
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approved
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