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A329041
Square array read by antidiagonals: A(n, k) = A327936(A276086(n) * A276086(k)).
15
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 2, 1, 6, 3, 6, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, 1
OFFSET
0,5
COMMENTS
Array A(n, k) is symmetric, and is read as (n,k) = (0, 0), (1, 0), (0, 1), (2, 0), (1, 1), (0, 2), (3, 0), (2, 1), (1, 2), (0, 3), ...
If A(n, k) is 1, it tells that adding of n and k do not generate any carries, when done in primorial base (A049345). If A(n, k) is larger than one, then its prime factors indicate in which specific moduli (digit positions) the sum was larger than allowed for that position.
FORMULA
A(n, k) = A327936(A276086(n) * A276086(k)).
For all n, A(A328841(n), A328842(n)) = 1 and A(A328770(n), A328770(n)) = 1.
EXAMPLE
The top left corner of the array:
0 1 2 3 4 5 6 7 8 9 10 11 12
+--------------------------------------
0: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1: | 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
2: | 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, ...
3: | 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, ...
4: | 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, ...
5: | 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, ...
6: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
7: | 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
8: | 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, ...
9: | 1, 2, 1, 2, 3, 6, 1, 2, 1, 2, 3, 6, 1, ...
10: | 1, 1, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 1, ...
11: | 1, 2, 3, 6, 3, 6, 1, 2, 3, 6, 3, 6, 1, ...
12: | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
...
A(11,25) = A(25,11) = 10 because 11 is written in primorial base representation (A049345) as "121" and 25 as "401", and when these are added together digit by digit, we see that the maximal allowed digits "421" for the rightmost three positions are exceeded in positions 1 and 3, with the 1st and 3rd primes 2 and 5 as their moduli, thus A(11,25) = 2*5 = 10.
PROG
(PARI)
up_to = 105;
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));
A329041list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, if(i++ > up_to, return(v)); v[i] = A329041sq(a-col, col))); (v); };
v329041 = A329041list(up_to);
A329041(n) = v329041[1+n];
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Nov 03 2019
STATUS
approved