A352001 -id:A352001 - cp's OEIS frontend

A351995 Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(ki) where n = Sum_{ i >= 0 } b_i 2^i.

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 3, 4, 1, 0, 2, 4, 5, 8, 1, 0, 2, 5, 16, 9, 16, 1, 0, 3, 6, 17, 64, 17, 32, 1, 0, 1, 7, 20, 65, 256, 33, 64, 1, 0, 2, 8, 21, 72, 257, 1024, 65, 128, 1, 0, 2, 9, 64, 73, 272, 1025, 4096, 129, 256, 1, 0, 3, 10, 65, 512, 273, 1056, 4097, 16384, 257, 512, 1, 0

Offset: 0

Rémy Sigrist, Feb 27 2022

In other words, in binary expansion of n, replace 2^i by 2^(k*i).

			Square array A(n, k) begins:
  n\k|  0  1   2   3    4     5     6      7      8       9       10
  ------------------------------------------------------------------
    0|  0  0   0   0    0     0     0      0      0       0        0
    1|  1  1   1   1    1     1     1      1      1       1        1
    2|  1  2   4   8   16    32    64    128    256     512     1024
    3|  2  3   5   9   17    33    65    129    257     513     1025
    4|  1  4  16  64  256  1024  4096  16384  65536  262144  1048576
    5|  2  5  17  65  257  1025  4097  16385  65537  262145  1048577
    6|  2  6  20  72  272  1056  4160  16512  65792  262656  1049600
    7|  3  7  21  73  273  1057  4161  16513  65793  262657  1049601

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Index entries for sequences related to binary expansion of n

Cf. A000120, A000695, A033045, A033052, A352001.

A351995[n_, k_] := If[n <= 1, n, Total[2^(k*(Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1))]];
Table[A351995[n - k, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Aug 26 2025 *)

A(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=2^(k*e)); v }

A(A(n, k), k') = A(n, k*k') for k, k' > 0.
A(n, 0) = A000120(n).
A(n, 1) = n.
A(n, 2) = A000695(n).
A(n, 3) = A033045(n).
A(n, 4) = A033052(n).
A(0, k) = 0.
A(1, k) = 1.
A(2, k) = 2^k.
A(3, k) = 2^k + 1.

A352028 a(n) = Product p_{n*i}^e_i if the prime factorization of n is Product p_i^e_i.

Original entry on oeis.org

1, 3, 13, 49, 47, 481, 107, 6859, 3721, 3277, 257, 121841, 397, 11309, 22261, 7890481, 653, 1390861, 881, 1416521, 78373, 47479, 1279, 157208087, 143641, 92011, 15813251, 7018237, 1889, 14701639, 2293, 38579489651, 309709, 207527, 461939, 2938615681, 3119

Offset: 1

Alois P. Heinz, Mar 01 2022

nonn

Or replace prime(i) in n by prime(n*i).

All terms are odd.

			a(1) = 1 because 1 is the empty product.
a(2) = 3 = prime(2) = prime(2*1) because 2 = prime(1).
a(3) = 13 = prime(6) = prime(3*2) because 3 = prime(2).
a(4) = 49 = 7^2 = prime(4)^2 = prime(4*1)^2 because 4 = prime(1)^2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Main diagonal of A352001.

Cf A000040, A033286, A228529.

a:= n-> mul(ithprime(n*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..45);

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(n*primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Mar 02 2022

a(n) = A352001(n,n).

a(prime(n)) = A228529(n) = A000040(A033286(n)).

cp's OEIS Frontend

A351995 Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(ki) where n = Sum_{ i >= 0 } b_i 2^i.

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A352028 a(n) = Product p_{n*i}^e_i if the prime factorization of n is Product p_i^e_i.

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cp's OEIS Frontend

A351995 Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(k*i) where n = Sum_{ i >= 0 } b_i * 2^i.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A352028 a(n) = Product p_{n*i}^e_i if the prime factorization of n is Product p_i^e_i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A351995 Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(ki) where n = Sum_{ i >= 0 } b_i 2^i.