

A163143


The sum of the digits of each number in the sequence is equal the sum of the digits of its factors 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
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OFFSET

1,1


COMMENTS

We call these numbers "zipper numbers", because the factorization resembles a zipper both graphically and 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 i.e.: 36=6^2, but one has to find the correct way, i.e., that that will yield the same digit sum. Obviously, some restrictions must be made, such as: 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 tripzips... 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. (Chandler)
Quadruple zipper: 782 = 2^1*391^1, 783 = 3^3*29^1, 784 = 2^3*7^1*14^1, 785 = 5^1*157^1. (Chandler)


LINKS

Ray Chandler, Table of n, a(n) for n=1..253


EXAMPLE

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. The number 8 is a nonprime zipper number as it can be written as: 8 = 4^1*2^1 and 8 = 4+1+2+1.


CROSSREFS

Cf. A177196.
Sequence in context: A066631 A090896 A131637 * A154586 A185615 A068367
Adjacent sequences: A163140 A163141 A163142 * A163144 A163145 A163146


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



