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

1,2

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

Links

Table of n, a(n) for n=1..91.

Index entries for sequences computed from indices in prime factorization

Formula

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

Example

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,

Prog

(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

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.

Sequence in context: A124882 A262518 A085578 * A360917 A135646 A360920

Adjacent sequences: A350063 A350064 A350065 * A350067 A350068 A350069

Keyword

nonn,tabl

Author

Antti Karttunen, Dec 13 2021

Status

approved