A324351 Square array read by antidiagonals: A(x,y) is the result from writing x and y in primorial base (A049345) and starting from their least significant ends, always choosing a minimal digit from each digit position, and converting back to decimal.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 1, 0, 3, 4, 3, 0, 1, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 2, 1, 0, 5, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 0, 0, 0, 2, 2, 2, 2, 6, 6, 2, 2, 2, 2, 0, 0, 0, 1, 0, 3, 4, 3, 6, 7, 6, 3, 4, 3, 0, 1, 0

Offset

0,13

Links

Antti Karttunen, Table of n, a(n) for n = 0..65702 (the first 362 antidiagonals of the array)

Index entries for sequences related to primorial base

Formula

A(x,y) = A276085(A324350(x,y)) = A276085(gcd(A276086(x), A276086(y))).

Example

The array A begins:
       0   1   2   3   4   5   6   7   8   9  10  11  12
  x/y  ------------------------------------------------------
   0:  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
   1:  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0,  1,  0, ...
   2:  0,  0,  2,  2,  2,  2,  0,  0,  2,  2,  2,  2,  0, ...
   3:  0,  1,  2,  3,  2,  3,  0,  1,  2,  3,  2,  3,  0, ...
   4:  0,  0,  2,  2,  4,  4,  0,  0,  2,  2,  4,  4,  0, ...
   5:  0,  1,  2,  3,  4,  5,  0,  1,  2,  3,  4,  5,  0, ...
   6:  0,  0,  0,  0,  0,  0,  6,  6,  6,  6,  6,  6,  6, ...
   7:  0,  1,  0,  1,  0,  1,  6,  7,  6,  7,  6,  7,  6, ...
   8:  0,  0,  2,  2,  2,  2,  6,  6,  8,  8,  8,  8,  6, ...
   9:  0,  1,  2,  3,  2,  3,  6,  7,  8,  9,  8,  9,  6, ...
  10:  0,  0,  2,  2,  4,  4,  6,  6,  8,  8, 10, 10,  6, ...
  11:  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  6, ...
  12:  0,  0,  0,  0,  0,  0,  6,  6,  6,  6,  6,  6, 12, ...
etc.
In primorial base, 5 is written as "21" (as 5 = 2*2 + 1*1) and 10 is written as "120" (as 10 = 1*6 + 2*2 + 0*1). Aligning them digit by digit (from the least significant end), and then always choosing a lesser digit leaves us with digits "020", which is 4 written in primorial base as 2*2 + 0*1 = 4, thus A(5,10) = A(10,5) = 4.

Prog

(PARI)
up_to = 65703; \\ = binomial(362+1, 2)
A002110(n) = prod(i=1, n, prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A276086(n) = { my(i=0, m=1, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m*=(prime(i)^((n%nextpr)/pr)); n-=(n%nextpr)); pr=nextpr); m; };
A324351sq(row, col) = A276085(gcd(A276086(row), A276086(col)));
A324351list(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] = A324351sq(a-col, col))); (v); };
v324351 = A324351list(up_to);
A324351(n) = v324351[1+n]; \\ Antti Karttunen, Feb 25 2019

Crossrefs

Cf. A001477 (central diagonal), A002110, A049345, A276085, A276086, A324350.

Sequence in context: A073253 A004198 A350673 * A116402 A093323 A106278

Adjacent sequences: A324348 A324349 A324350 * A324352 A324353 A324354

Keyword

nonn,base,tabl

Author

Antti Karttunen, Feb 25 2019

Status

approved