A356300 Square array read by antidiagonals. A(n,k) is the nearest common ancestor of n and k in the binary tree depicted in A253563.

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

Offset: 1

Author

Antti Karttunen, Aug 03 2022

Keywords

• nonn
• tabl

Comments

Array is symmetric and is read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... .

Also the nearest common ancestor of n and k in the tree depicted in A253565 (the mirror image of the A253563-tree).

Examples

			The top left 21x21 corner of the array:
n/k  |  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21
-----+----------------------------------------------------------------------------
   1 |  1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
   2 |  1, 2, 2, 2, 2, 2, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,
   3 |  1, 2, 3, 2, 3, 2, 3, 2, 3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3,
   4 |  1, 2, 2, 4, 2, 4, 2, 4, 2,  4,  2,  4,  2,  4,  2,  4,  2,  4,  2,  4,  2,
   5 |  1, 2, 3, 2, 5, 2, 5, 2, 3,  2,  5,  2,  5,  2,  3,  2,  5,  2,  5,  2,  3,
   6 |  1, 2, 2, 4, 2, 6, 2, 4, 2,  6,  2,  4,  2,  6,  2,  4,  2,  6,  2,  4,  2,
   7 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2,  7,  2,  7,  2,  3,  2,  7,  2,  7,  2,  3,
   8 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2,  8,  2,  4,  2,  8,  2,  4,  2,  8,  2,
   9 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2,  9,  2,  3,  2,  3,  2,  9,
  10 |  1, 2, 2, 4, 2, 6, 2, 4, 2, 10,  2,  4,  2, 10,  2,  4,  2,  6,  2,  4,  2,
  11 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 11,  2,  3,  2, 11,  2, 11,  2,  3,
  12 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2, 12,  2,  4,  2,  8,  2,  4,  2, 12,  2,
  13 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 13,  2, 13,  2,  3,
  14 |  1, 2, 2, 4, 2, 6, 2, 4, 2, 10,  2,  4,  2, 14,  2,  4,  2,  6,  2,  4,  2,
  15 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2, 15,  2,  3,  2,  3,  2, 15,
  16 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2,  8,  2,  4,  2, 16,  2,  4,  2,  8,  2,
  17 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 17,  2, 17,  2,  3,
  18 |  1, 2, 2, 4, 2, 6, 2, 4, 2,  6,  2,  4,  2,  6,  2,  4,  2, 18,  2,  4,  2,
  19 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 17,  2, 19,  2,  3,
  20 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2, 12,  2,  4,  2,  8,  2,  4,  2, 20,  2,
  21 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2, 15,  2,  3,  2,  3,  2, 21,
.
A(3,6) = A(6,3) = 2 because the nearest common ancestor of 3 and 6 in the tree described in A253563 (and in A253565) is 2.
A(4,6) = A(6,4) = 4 because 6 occurs as a descendant of 4 in A253563-tree, thus their nearest common ancestor is 4 itself.
		

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A253553, A253563, A253565, A356301.

Cf. also A348041.

Programs

PARI

up_to = 105;
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A356300sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, stemlen, h=y); while(k>1, listput(lista,k); k = A253553(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A253553(h)));
A356300list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A356300sq(col,(a-(col-1))))); (v); };
v356300 = A356300list(up_to);
A356300(n) = v356300[n];