OFFSET
1,4
LINKS
FORMULA
a(n) = Sum_{d|n} [d <= (n/d) and A329041(d,n/d) == 1], where [ ] is the Iverson bracket, and the dyadic function A329041 returns 1 only when its two arguments do not generate any carries when added together in the primorial base.
For all n >= 1, a(n) <= A038548(n) [see A358671 for the indices where the equality is attained] and a(n) <= A358236(n).
For all n >= 1, a(2n-1) = 0, a(4n-2) = A358236(4n-2).
EXAMPLE
a(6) = 2, because 6 has only two factor pairs, {1, 6} and {2, 3}, and for both of those pairs the criterion is satisfied, as we have A329041(1, 6) = 1 and 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(8) = 1, because while there are two ways to factor 8 into two factors, as 1*8 and 2*4, only 1+8 yields a carry-free sum ("1" and "110" added together gives "111" = 9 in primorial base, A049345), while 2+4 (= "10" + "20") is not carry-free, as 2 is the max. allowed digit in the second rightmost place.
PROG
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Antti Karttunen, Nov 26 2022
STATUS
approved