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A358233
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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.
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6
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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, 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, 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
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OFFSET
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1,4
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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PROG
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(PARI)
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); };
A358233(n) = sumdiv(n, d, ((d <= (n/d)) && 1==A329041sq(d, n/d)));
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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