OFFSET
1,4
COMMENTS
Number of ways n can be factored as x*y, with 1 <= x <= y, so that the sum (x * y') + (x' * y) is carry-free when the addition is done in the primorial base. Here u' stands for A003415(u), the arithmetic derivative of u.
LINKS
FORMULA
EXAMPLE
a(6) = 2 because 6 has only two factor pairs, {1, 6} and {2, 3}, and for both of those pairs the criteria is satisfied, as we have A329041(1*A003415(6), A003415(1)*6) = A329041(5, 0) = 1 and A329041(2*A003415(3), A003415(2)*3) = A329041(2, 3) = 1. In the latter case the primorial base expansions of 2 and 3 are "10" and "11" (see A049345), which can be added together cleanly (i.e., without carries) to obtain "21" = A049345(2+3).
a(24) = 3 because 24 can be factored into two factors in four possible ways: 1*24, 2*12, 3*8 and 4*6, of which all others, except pair {4,6} are carry-free: we have A003415(6)*4 = 20 and A003415(4)*6 = 24, with respective primorial base expansions "310" and "400", which when added together, yield a carry at the third digit position from the right, because 3+4 = 7 > 4 (which is the max. allowed digit in that place), and therefore a(24) = 4-1 = 3.
a(63) = 3 because 63 can be factored into two factors in three possible ways: 1*63, 3*21 and 7*9. The trivial factorization always yields a carryless sum, and in this case also the other two factorizations satisfy the criteria: A329041(3*A003415(21), A003415(3)*21) = A329041(30, 21) = 1 (as 21 is "311" and 30 is "1000" in primorial base), and A329041(7*A003415(9), A003415(7)*9) = A329041(42, 9) = 1 (as 9 is "111", 42 is "1200", and 9+42 = 51 is "1311" in primorial base).
PROG
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Nov 26 2022
STATUS
approved