login
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

%I #14 Aug 05 2023 21:18:37

%S 4,8,25,26,27,36,44,48,54,56,62,64,65,68,75,80,84,92,96,98,108,121,

%T 125,128,129,143,147,155,156,164,168,176,182,183,184,188,189,192,195,

%U 206,216,224,242,248,256,258,260,264,270,276,278,284,288,294,296,308,318

%N 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).

%C 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.

%C 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.

%C 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.

%C From _Ray Chandler_, Apr 30 2010: (Start)

%C Triple zippers:

%C 25 = 5^2, 26 = 2^1*13^1, 27 = 3^1*3^2;

%C 182 = 13^1*14^1, 183 = 3^1*61^1, 184 = 2^1*2^2*23^1;

%C 735 = 7^1*105^1, 736 = 2^3*4^1*23^1, 737 = 11^1*67^1;

%C 902 = 22^1*41^1, 903 = 21^1*43^1, 904 = 2^1*2^2*113^1.

%C 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)

%H Ray Chandler, <a href="/A163143/b163143.txt">Table of n, a(n) for n = 1..253</a>

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

%e The number 36 can be factored as 36=2^2*3^2, and 3+6 = 9 = 2+2+3+2.

%e The number 121 can be factored as 121=11^2.

%Y Cf. A177196.

%K nonn,base

%O 1,1

%A Yossi Elran (yossi.elran(AT)weizmann.ac.il) and Royi Lachmi, Jul 21 2009

%E Edited by _N. J. A. Sloane_, Jul 26 2009

%E a(8) and following from _Ray Chandler_, Apr 30 2010