A331590 -id:A331590 - cp's OEIS frontend

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

Antti Karttunen, Dec 13 2021

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

			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,

Index entries for sequences computed from indices in prime factorization

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.

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

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

A332382 If n = Sum (2^e_k) then a(n) = Product (prime(e_k + 2)).

Original entry on oeis.org

1, 3, 5, 15, 7, 21, 35, 105, 11, 33, 55, 165, 77, 231, 385, 1155, 13, 39, 65, 195, 91, 273, 455, 1365, 143, 429, 715, 2145, 1001, 3003, 5005, 15015, 17, 51, 85, 255, 119, 357, 595, 1785, 187, 561, 935, 2805, 1309, 3927, 6545, 19635, 221, 663, 1105, 3315, 1547, 4641, 7735, 23205

Offset: 0

Ilya Gutkovskiy, Feb 10 2020

nonn

Permutation of odd squarefree numbers (A056911).

a(n) is the n-th power of 3 in the monoid defined in A331590. - Peter Munn, May 02 2020

			21 = 2^0 + 2^2 + 2^4 so a(21) = prime(2) * prime(4) * prime(6) = 3 * 7 * 13 = 273.

Index entries for sequences related to binary expansion of n

Bisection of A019565.
Cf. A002110, A029930, A048675, A056911, A121663, A225546.
A003961, A003987, A059897, A331590, A334748 are used to express relationship between terms of this sequence.

a:= n-> (l-> mul(ithprime(i+1)^l[i], i=1..nops(l)))(convert(n, base, 2)):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 10 2020

nmax = 55; CoefficientList[Series[Product[(1 + Prime[k + 2] x^(2^k)), {k, 0, Floor[Log[2, nmax]]}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := Prime[Floor[Log[2, n]] + 2] a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 55}]

a(n) = my(b=Vecrev(binary(n))); prod(k=1, #b, if (b[k], prime(k+1), 1)); \\ Michel Marcus, Feb 10 2020

G.f.: Product_{k>=0} (1 + prime(k+2) * x^(2^k)).
a(0) = 1; a(n) = prime(floor(log_2(n)) + 2) * a(n - 2^floor(log_2(n))).
a(2^(k-1)-1) = A002110(k)/2 for k > 0.
From Peter Munn, May 02 2020: (Start)
a(2n) = A003961(a(n)).
a(2n+1) = 3 * a(2n).
a(n) = A225546(4^n).
a(n+k) = A331590(a(n), a(k)).
a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
A048675(a(n)) = 2n.
(End)
a(n+1) = A334748(a(n)). - Peter Munn, Mar 04 2022

cp's OEIS Frontend

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

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