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A163143
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).
2
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
OFFSET
1,1
COMMENTS
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.
From Ray Chandler, Apr 30 2010: (Start)
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)
LINKS
EXAMPLE
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.
CROSSREFS
Cf. A177196.
Sequence in context: A066631 A090896 A131637 * A154586 A371190 A185615
KEYWORD
nonn,base
AUTHOR
Yossi Elran (yossi.elran(AT)weizmann.ac.il) and Royi Lachmi, Jul 21 2009
EXTENSIONS
Edited by N. J. A. Sloane, Jul 26 2009
a(8) and following from Ray Chandler, Apr 30 2010
STATUS
approved