A351995 -id:A351995 - cp's OEIS frontend

A352001 Square array A(n, k), n, k >= 1, read by antidiagonals upwards; A(n, k) = Product_{ i >= 1 } prime(k*i)^e_i where n = Product_{ i >= 1 } prime(i)^e_i (where prime(i) denotes the i-th prime number).

1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 9, 13, 7, 1, 6, 13, 25, 19, 11, 1, 7, 21, 23, 49, 29, 13, 1, 8, 19, 65, 37, 121, 37, 17, 1, 9, 27, 37, 133, 47, 169, 43, 19, 1, 10, 49, 125, 53, 319, 61, 289, 53, 23, 1, 11, 39, 169, 343, 71, 481, 73, 361, 61, 29, 1

Offset: 1

Rémy Sigrist, Feb 27 2022

In other words, in prime factorization of n, replace prime(i) by prime(k*i).

For any k >= 1, n -> A(n, k) is completely multiplicative.

			Square array A(n, k) begins:
  n\k|   1   2    3    4     5     6     7     8      9     10
  ------------------------------------------------------------
    1|   1   1    1    1     1     1     1     1      1      1
    2|   2   3    5    7    11    13    17    19     23     29
    3|   3   7   13   19    29    37    43    53     61     71
    4|   4   9   25   49   121   169   289   361    529    841
    5|   5  13   23   37    47    61    73    89    103    113
    6|   6  21   65  133   319   481   731  1007   1403   2059
    7|   7  19   37   53    71    89   107   131    151    173
    8|   8  27  125  343  1331  2197  4913  6859  12167  24389
    9|   9  49  169  361   841  1369  1849  2809   3721   5041
   10|  10  39  115  259   517   793  1241  1691   2369   3277

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (antidiagonals n = 1..150 flattened)

Cf. A000040, A001248, A031215, A051128, A297002, A351995.

Main diagonal gives A352028.

A:= (n, k)-> mul(ithprime(k*numtheory[pi](i[1]))^i[2], i=ifactors(n)[2]):
seq(seq(A(d+1-k, k), k=1..d), d=1..12);  # Alois P. Heinz, Feb 28 2022

Table[If[# == 1, 1, Times @@ Map[Prime[#3*PrimePi[#1]]^#2 & @@ Flatten[{#1, k}] &, FactorInteger[#]]] &[n - k + 1], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Feb 28 2022 *)

A(n,k) = { my (f=factor(n)); prod (i=1, #f~, prime(k * primepi(f[i,1])) ^ f[i,2]) }

A(A(n, k), k') = A(n, k*k').
A(n, 1) = n.
A(n, 2) = A297002(n).
A(1, k) = 1.
A(2, k) = prime(k) (the k-th prime number).
A(3, k) = prime(2*k) = A031215(k).
A(4, k) = A001248(k).

A361995 Order array of A361993, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 5, 10, 11, 8, 9, 16, 18, 14, 12, 15, 26, 29, 23, 19, 13, 24, 42, 46, 38, 31, 22, 17, 39, 68, 74, 62, 50, 36, 28, 20, 63, 110, 119, 100, 81, 59, 45, 32, 21, 102, 111, 192, 101, 131, 97, 73, 52, 35, 25, 165, 179, 310, 162, 212, 158, 118, 84

Offset: 1

Clark Kimberling, Apr 05 2023

This array is an interspersion (hence a dispersion, as in A114537 and A163255), so every positive integer occurs exactly once. See A333029 for the definition of order array.

			Corner:
   1    2    3    5    9   15   24 ...
   4    6   10   16   26   42   68 ...
   7   11   18   29   46   74  119 ...
   8   14   23   38   62  100  162 ...
  12   19   31   50   81  131  212 ...
  13   22   36   59   97  158  191 ...
  ...

Cf. A114537, A163255, A333029, A361993, A361996.

zz = 300; z = 40;
w[n_, k_] := w[n, k] = Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
b[h_, k_] := b[h, k] = w[2 h - 1, k] + w[2 h, k];
s = Flatten[Table[b[h, k], {h, 1, zz}, {k, 1, z}]];
r[h_, k_] := Length[Select[s, # <= b[h, k] &]]
TableForm[Table[r[h, k], {h, 1, 50}, {k, 1, 12}]]  (*A351995, array*)
v = Table[r[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (*A351995, sequence*)

A385674 Triangle read by rows: T(n,n) = 1 and T(n,k) = (T(n-1,k) | T(n-2,k) | ... | T(n-k,k)) + 1, where | is bitwise OR, (0<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 4, 2, 1, 1, 6, 8, 8, 4, 2, 1, 1, 7, 16, 15, 8, 4, 2, 1, 1, 8, 25, 16, 16, 8, 4, 2, 1, 1, 9, 26, 32, 31, 16, 8, 4, 2, 1, 1, 10, 28, 64, 32, 32, 16, 8, 4, 2, 1, 1, 11, 31, 113, 64, 63, 32, 16, 8, 4, 2, 1, 1, 12, 32, 114, 128, 64, 64, 32, 16, 8, 4, 2, 1

Offset: 0

Natalia L. Skirrow, Aug 04 2025

For the purposes of the recurrence, the region of the array above the triangle is filled with 0's.

In the cellular automaton, on binary representations, that starts with s = 2^n-1 on iteration t=0, then each iteration sets s := s XOR (s<<1 AND s<<2 AND ... AND s<

With respect to n, T(n,k) is in Theta(n^k); this corresponds with the k-th cellular automaton growing with width Theta(t^(1/k)).

Rows are not always unimodal! Row 18 (1 < 18 < 104 < 512 > 496 < 1986 < 2048 > 1024 > 512 = 512 > 256 > 128 > 64 > 32 > 16 > 8 > 4 > 2 > 1) is the first exception.

			Table begins
  n\k| 0  1  2   3   4   5   6   7   8  9 10 11 12 13 14 15
  ---+-----------------------------------------------------
   0 | 1
   1 | 1  1
   2 | 1  2  1
   3 | 1  3  2   1
   4 | 1  4  4   2   1
   5 | 1  5  7   4   2   1
   6 | 1  6  8   8   4   2   1
   7 | 1  7 16  15   8   4   2   1
   8 | 1  8 25  16  16   8   4   2   1
   9 | 1  9 26  32  31  16   8   4   2  1
  10 | 1 10 28  64  32  32  16   8   4  2  1
  11 | 1 11 31 113  64  63  32  16   8  4  2  1
  12 | 1 12 32 114 128  64  64  32  16  8  4  2  1
  13 | 1 13 64 116 256 128 127  64  32 16  8  4  2  1
  14 | 1 14 97 120 481 256 128 128  64 32 16  8  4  2  1
  15 | 1 15 98 127 482 512 256 255 128 64 32 16  8  4  2  1
Binary expansions of the k=2 column:
  . . . . . . . . . . . . 11111111111111111111111
  . . . . . . . . . . . .1 1111111111111111111111
  . . . . . . 11111111111 . . . . . . 11111111111
  . . . . . .1 1111111111 . . . . . .1 1111111111
  . . . 11111 . . . 11111 . . . 11111 . . . 11111
  . . .1 1111 . . .1 1111 . . .1 1111 . . .1 1111
  . .11 . .11 . .11 . .11 . .11 . .11 . .11 . .11
  . 1 1 . 1 1 . 1 1 . 1 1 . 1 1 . 1 1 . 1 1 . 1 1
  .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1
The recurrence emulates binary counting, albeit with k X k 'metabits' in this diagram; the bit being  carried to is filled along the diagonal, followed by a single number in which all metabits are full and the 0th bit is on.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Natalia L. Skirrow, on nearsighted binary counters, theorems 1 and 2 (in which T(k,n) is written as the function o(k) for the n-th cellular automaton).

Cf. A338888 (column k=2).

A351995[n_, k_] := If[n <= 1, n, Total[2^(k*(Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1))]];
A385674[n_, k_] := 2*(2^k - 1)*A351995[#[[1]], k] + If[#[[2]] == k, 1, 2*2^(k*IntegerExponent[#[[1]], 2])*(1 + 2^#[[2]] - 2^k)] & [QuotientRemainder[n, k + 1]];
Table[A385674[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Sep 04 2025 *)

lista(nn) = my(m=matrix(nn, nn)); for (n=1, nn, m[n,n] = 1; for (k=1, n-1, for (i=1, k-1, m[n,k] = bitor(m[n,k], m[n-i,k]);); m[n,k]++;);); my(vrows=vector(nn, i, vector(i, k, m[i,k]))); vrows; \\ Michel Marcus, Aug 04 2025

from functools import reduce
ORsum=lambda l: reduce(int._or_,l,0)
A351995=lambda n,k: ORsum(map(lambda i: (n>>i&1)<<(i*k),range(n.bit_length()))) if k else n.bit_count()
T=lambda n,k: 2*~(~0<A351995(d:=n//(k+1),k)+((r:=n%(k+1))==k or (d&-d)**k*2*(1+2**r-2**k))
#corollary 1:
T=lambda n,k: T(n-(k+1<<(l:=(d:=n//(k+1)).bit_length()-1)),k)|(~(~0<k else int(n==k)
bit=lambda n,k,i: (i%k==n%(k+1) if n%(k+1)>i//k&1) if i else n%(k+1)==k #returns i-th bit of T(n,k)

Let d = floor(n/(k+1)) and m = n mod (k+1), then
T(n,k) = 2 * (2^k-1) * A351995(d,k) + b(n,k) where b(n,k) = 1 if n==k (mod k+1), otherwise b(n,k) = 2*2^(k*val_2(d))*(1+2^m-2^k).
Bounds: 2*(n/(k+1))^k <= T(n,k) <= 2*(2^k-1) * ((n-k)/(k+1))^k+1, with upper equality when n is of form 2^(k*i+1) and lower when n is of form (2^k-1)*2^(k*i+1).
T(n,k) = T(n-(k+1)*2^floor(log_2(d)),k) + 2^(k*floor(log_2(d))+1)*(if d is a power of 2 and m != k then 2^m, else 2^k-1).
O.g.f. for k-th column: 2*(2^k-1) * (Sum_{i>=0} 2^(k*i) * x^((k+1)*2^i) / (1+x^((k+1)*2^i))) / (1-x) + x^k / (1-x^(k+1)) + 2 * ((1-2^k) * (1-x^k)/(1-x) + (1-(2*x)^k)/(1-2*x)) * Sum_{i>=0} 2^(k*i) * x^((k+1)*2^i) / (1-x^((k+1)*2^(i+1))).

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A352001 Square array A(n, k), n, k >= 1, read by antidiagonals upwards; A(n, k) = Product_{ i >= 1 } prime(k*i)^e_i where n = Product_{ i >= 1 } prime(i)^e_i (where prime(i) denotes the i-th prime number).

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A361995 Order array of A361993, read by descending antidiagonals.

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A385674 Triangle read by rows: T(n,n) = 1 and T(n,k) = (T(n-1,k) | T(n-2,k) | ... | T(n-k,k)) + 1, where | is bitwise OR, (0<=k<=n).

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