A350066 Symmetric square array A(n,k) = A122111(A122111(n) * A122111(k)), n >= 1, k >= 1, read by antidiagonals.

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 6, 7, 10, 10, 7, 6, 7, 10, 11, 9, 11, 10, 7, 8, 11, 14, 14, 14, 14, 11, 8, 9, 12, 13, 15, 13, 15, 13, 12, 9, 10, 15, 20, 22, 22, 22, 22, 20, 15, 10, 11, 14, 21, 18, 17, 21, 17, 18, 21, 14, 11, 12, 13, 22, 25, 28, 26, 26, 28, 25, 22, 13, 12, 13, 20, 17, 21, 33, 30, 19, 30, 33, 21, 17, 20, 13

Offset: 1

Author

Antti Karttunen, Dec 13 2021

Keywords

• nonn
• tabl

Comments

A122111 is a self-inverse permutation, so this array represents a binary operation A(.,.) over the positive integers that is isomorphic to multiplication. Its primes are the positive powers of 2 (as defined by standard multiplication): 2, 4, 8, 16, 32, ... . The positive powers of 2, as defined by A(.,.), are the prime numbers as we usually understand them: 2, 3, 5, 7, 11, ... . - Peter Munn, Aug 04 2022

Examples

			The top left 15 X 15 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  12, 13, 14,  15,
   2,  3,  5,  6,  7, 10, 11, 12, 15, 14, 13,  20, 17, 22,  21,
   3,  5,  7, 10, 11, 14, 13, 20, 21, 22, 17,  28, 19, 26,  33,
   4,  6, 10,  9, 14, 15, 22, 18, 25, 21, 26,  30, 34, 33,  35,
   5,  7, 11, 14, 13, 22, 17, 28, 33, 26, 19,  44, 23, 34,  39,
   6, 10, 14, 15, 22, 21, 26, 30, 35, 33, 34,  42, 38, 39,  55,
   7, 11, 13, 22, 17, 26, 19, 44, 39, 34, 23,  52, 29, 38,  51,
   8, 12, 20, 18, 28, 30, 44, 27, 50, 42, 52,  45, 68, 66,  70,
   9, 15, 21, 25, 33, 35, 39, 50, 49, 55, 51,  70, 57, 65,  77,
  10, 14, 22, 21, 26, 33, 34, 42, 55, 39, 38,  66, 46, 51,  65,
  11, 13, 17, 26, 19, 34, 23, 52, 51, 38, 29,  68, 31, 46,  57,
  12, 20, 28, 30, 44, 42, 52, 45, 70, 66, 68,  63, 76, 78, 110,
  13, 17, 19, 34, 23, 38, 29, 68, 57, 46, 31,  76, 37, 58,  69,
  14, 22, 26, 33, 34, 39, 38, 66, 65, 51, 46,  78, 58, 57,  85,
  15, 21, 33, 35, 39, 55, 51, 70, 77, 65, 57, 110, 69, 85,  91,
		

Links

• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A122111, A297002 (main diagonal), A253550 (after its initial term, gives row 2 / column 2 from the second term onward).

See the formula section for the relationships with A003961, A061142.

Cf. also A003991, A129595, A331590.

Programs

PARI

up_to = 105;
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A350066sq(n,k) = A122111(A122111(n)*A122111(k));
A350066list(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] = A350066sq(col,(a-(col-1))))); (v); };
v350066 = A350066list(up_to);
A350066(n) = v350066[n]; \\ Antti Karttunen, Dec 13 2021

Formula

A(n, A061142(n)) = A003961(n). - Peter Munn, Aug 04 2022