cp's OEIS Frontend

A358233 Number of ways n can be expressed as an unordered product of two natural numbers that do not generate any carries when added together in the primorial base.

%I A358233 #25 Sep 02 2023 19:27:55
%S A358233 0,1,0,2,0,2,0,1,0,1,0,2,0,2,0,2,0,3,0,1,0,1,0,4,0,2,0,3,0,3,0,1,0,1,
%T A358233 0,4,0,2,0,2,0,4,0,1,0,1,0,4,0,2,0,3,0,4,0,2,0,1,0,5,0,2,0,3,0,3,0,1,
%U A358233 0,2,0,6,0,2,0,3,0,4,0,1,0,1,0,4,0,2,0,2,0,5,0,1,0,1,0,6,0,3,0,3,0,3,0,2,0
%N A358233 Number of ways n can be expressed as an unordered product of two natural numbers that do not generate any carries when added together in the primorial base.
%H A358233 Antti Karttunen, <a href="/A358233/b358233.txt">Table of n, a(n) for n = 1..65537</a>
%H A358233 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A358233 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.
%F A358233 For all n >= 1, a(n) <= A038548(n) [see A358671 for the indices where the equality is attained] and a(n) <= A358236(n).
%F A358233 For all n >= 1, a(A100484(n)) = A358235(A100484(n)).
%F A358233 For all n >= 1, a(2n-1) = 0, a(4n-2) = A358236(4n-2).
%e A358233 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).
%e A358233 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.
%o A358233 (PARI)
%o A358233 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A358233 A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
%o A358233 A329041sq(row,col) = A327936(A276086(row)*A276086(col));
%o A358233 A358233(n) = sumdiv(n, d, ((d <= (n/d)) && 1==A329041sq(d,n/d)));
%Y A358233 Cf. A038548, A049345, A100484, A276086, A329041, A358234 (even bisection), A358671.
%Y A358233 Cf. also A358235, A358236.
%K A358233 nonn,base
%O A358233 1,4
%A A358233 _Antti Karttunen_, Nov 26 2022