A358235 Number of ways n' (the arithmetic derivative of n) can be formed as a sum (x * y') + (x' * y) from two factors x and y of n, with x <= y, so that the said sum does not involve any carries when the addition is done in the primorial base.

1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 3, 2, 1, 1, 1, 1, 1, 4, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2

Offset: 1

Author

Antti Karttunen, Nov 26 2022

Keywords

• nonn
• base

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.

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to primorial base

Crossrefs

Cf. A003415, A049345, A100484, A276086, A329041, A358672, A358673, A358674.

Cf. also A358233.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
A329041sq(row,col) = A327936(A276086(row)*A276086(col));
A358235(n) = sumdiv(n, d, ((d <= (n/d)) && 1==A329041sq((d*A003415(n/d)), (A003415(d)*(n/d)))));

Formula

a(n) = Sum_{d|n} [d <= (n/d) and A329041(d*A003415(n/d), A003415(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 A358673 for the indices where the equality is attained].

For all n >= 1, a(A100484(n)) = A358233(A100484(n)).