A374433 -id:A374433 - cp's OEIS frontend

A374431 Row product of A374433.

1, 1, 2, 3, 4, 5, 72, 7, 16, 27, 800, 11, 5184, 13, 6272, 30375, 256, 17, 373248, 19, 640000, 750141, 247808, 23, 26873856, 3125, 1384448, 19683, 39337984, 29, 30233088000000, 31, 65536, 235782657, 37879808, 1313046875, 139314069504, 37, 189267968, 3502727631

Offset: 0

Peter Luschny, Jul 10 2024

nonn

Michael De Vlieger, Log log scatterplot of Log_10(a(n)), n = 0..1024, showing odd indexed terms in red, even indexed terms in blue.

Cf. A374433, A374430 (odd bisection).

seq(mul(A374433(n, k), k = 0..n), n=0..40);

nn = 39; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Array[Product[Times @@ Intersection[s[k], s[#]], {k, 0, #}] &, nn + 1, 0] (* Michael De Vlieger, Jul 11 2024 *)

from math import prod
print([prod([A374433(n, k) for k in range(n + 1)]) for n in range(40)])

A374432 Row sums of A374433.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 16, 14, 13, 16, 28, 22, 31, 26, 40, 46, 25, 34, 46, 38, 55, 66, 64, 46, 61, 46, 76, 46, 79, 58, 136, 62, 49, 106, 100, 118, 91, 74, 112, 126, 109, 82, 196, 86, 127, 136, 136, 94, 121, 92, 136, 166, 151, 106, 136, 190, 157, 186, 172, 118, 271

Offset: 0

Peter Luschny, Jul 10 2024

nonn

Cf. A374433, A176345 (a(n + 1) - 1).

seq(add(A374433(n, k), k = 0..n), n=0..60);

nn = 120; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1};
Table[Sum[Times @@ Intersection[s[k], s[n]], {k, 0, n}], {n, 0, nn}] (* Michael De Vlieger, Jul 11 2024 *)

print([sum([A374433(n, k) for k in range(n + 1)]) for n in range(61)])

A374434 Triangle read by rows: T(n, k) = Product_{p in PF(n) symmetric difference PF(k)} p, where PF(a) is the set of the prime factors of a.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 6, 1, 2, 2, 1, 6, 1, 5, 5, 10, 15, 10, 1, 6, 6, 3, 2, 3, 30, 1, 7, 7, 14, 21, 14, 35, 42, 1, 2, 2, 1, 6, 1, 10, 3, 14, 1, 3, 3, 6, 1, 6, 15, 2, 21, 6, 1, 10, 10, 5, 30, 5, 2, 15, 70, 5, 30, 1, 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1

Offset: 0

Peter Luschny, Jul 10 2024

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  1;
  [ 3]  3,  3,  6,  1;
  [ 4]  2,  2,  1,  6,  1;
  [ 5]  5,  5, 10, 15, 10,  1;
  [ 6]  6,  6,  3,  2,  3, 30,  1;
  [ 7]  7,  7, 14, 21, 14, 35, 42,  1;
  [ 8]  2,  2,  1,  6,  1, 10,  3, 14,  1;
  [ 9]  3,  3,  6,  1,  6, 15,  2, 21,  6,  1;
  [10] 10, 10,  5, 30,  5,  2, 15, 70,  5, 30,   1;
  [11] 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 1;

Family: A374433 (intersection), this sequence (symmetric difference), A374435 (difference), A374436 (union).

Cf. A007947 (column 0), A000034 (central terms), A050873 (gcd).

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374434 := (n, k) -> mul(symmdiff(PF(n), PF(k))):
seq(print(seq(A374434(n, k), k = 0..n)), n = 0..11);

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Times @@ SymmetricDifference[s[k], s[n]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

# Function A374434 defined in A374433.
for n in range(11): print([A374434(n, k) for k in range(n + 1)])

From Michael De Vlieger, Jul 11 2024: (Start)
T(0,0) = T(n,0) = rad(n)/rad(0) = 1 where rad = A007947;
T(n,k) = rad(k*n)/rad(gcd(k,n))
       = A007947(k*n)/A007947(S(n,k)) where S = A050873
       = A374436(n,k)/A374433(n,k). (End)

A374435 Triangle read by rows: T(n, k) = Product_{p in PF(n) difference PF(k)} p, where PF(a) is the set of the prime factors of a.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 2, 2, 1, 2, 1, 5, 5, 5, 5, 5, 1, 6, 6, 3, 2, 3, 6, 1, 7, 7, 7, 7, 7, 7, 7, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 10, 10, 5, 10, 5, 2, 5, 10, 5, 10, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1

Offset: 0

Peter Luschny, Jul 10 2024

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  1;
  [ 3]  3,  3,  3,  1;
  [ 4]  2,  2,  1,  2,  1;
  [ 5]  5,  5,  5,  5,  5,  1;
  [ 6]  6,  6,  3,  2,  3,  6,  1;
  [ 7]  7,  7,  7,  7,  7,  7,  7,  1;
  [ 8]  2,  2,  1,  2,  1,  2,  1,  2,  1;
  [ 9]  3,  3,  3,  1,  3,  3,  1,  3,  3,  1;
  [10] 10, 10,  5, 10,  5,  2,  5, 10,  5, 10,  1;
  [11] 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1;

Family: A374433 (intersection), A374434 (symmetric difference), this sequence (difference), A374436 (union).

Cf. A007947 (column 0), A000034 (central terms).

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374435 := (n, k) -> mul(PF(n) minus PF(k)):
seq(print(seq(A374435(n, k), k = 0..n)), n = 0..11);

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Apply[Times, Complement[s[n], s[k]]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

# Function A374435 defined in A374433.
for n in range(12): print([A374435(n, k) for k in range(n + 1)])

A374436 Triangle read by rows: T(n, k) = Product_{p in PF(n) union PF(k)} p, where PF(a) is the set of the prime factors of a.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 6, 3, 2, 2, 2, 6, 2, 5, 5, 10, 15, 10, 5, 6, 6, 6, 6, 6, 30, 6, 7, 7, 14, 21, 14, 35, 42, 7, 2, 2, 2, 6, 2, 10, 6, 14, 2, 3, 3, 6, 3, 6, 15, 6, 21, 6, 3, 10, 10, 10, 30, 10, 10, 30, 70, 10, 30, 10, 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 11

Offset: 0

Peter Luschny, Jul 10 2024

			  [ 0]  1;
  [ 1]  1,  1;
  [ 2]  2,  2,  2;
  [ 3]  3,  3,  6,  3;
  [ 4]  2,  2,  2,  6,  2;
  [ 5]  5,  5, 10, 15, 10,  5;
  [ 6]  6,  6,  6,  6,  6, 30,  6;
  [ 7]  7,  7, 14, 21, 14, 35, 42,  7;
  [ 8]  2,  2,  2,  6,  2, 10,  6, 14,  2;
  [ 9]  3,  3,  6,  3,  6, 15,  6, 21,  6,  3;
  [10] 10, 10, 10, 30, 10, 10, 30, 70, 10, 30,  10;
  [11] 11, 11, 22, 33, 22, 55, 66, 77, 22, 33, 110, 11;

Family: A374433 (intersection), A374434 (symmetric difference), A374435 (difference), this sequence (union).

Cf. A007947 (column 0, main diagonal), A099985 (central terms).

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
A374436 := (n, k) -> mul(PF(n) union PF(k)):
seq(print(seq(A374436(n, k), k = 0..n)), n = 0..11);

nn = 12; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; s[0] = {1}; Table[Apply[Times, Union[s[k], s[n]]], {n, 0, nn}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 11 2024 *)

# Function A374436 defined in A374433.
for n in range(12): print([A374436(n, k) for k in range(n + 1)])

T(0,0) = T(n,0) = 1; T(n,k) = rad(k*n) where rad = A007947. - Michael De Vlieger, Jul 11 2024

A374430 Odd bisection of A374431.

Original entry on oeis.org

1, 3, 5, 7, 27, 11, 13, 30375, 17, 19, 750141, 23, 3125, 19683, 29, 31, 235782657, 1313046875, 37, 3502727631, 41, 43, 28025208984375, 47, 823543, 634465620819, 53, 7863818359375, 7971951402153, 59, 61, 422112982235053221, 453238525390625

Offset: 0

Peter Luschny, Jul 10 2024

nonn

Cf. A374431, A374433.

PF := n -> ifelse(n = 0, {}, NumberTheory:-PrimeFactors(n)):
T := (n, k) -> mul(PF(n) intersect PF(k)):
seq(mul(T(2*n+1, k), k = 0..2*n+1), n = 0..20);

nn = 65; Do[Set[s[i], FactorInteger[i][[All, 1]]], {i, 0, nn}]; Table[Product[Times @@ Intersection[s[k], s[n]], {k, 0, n}], {n, 1, nn, 2}] (* Michael De Vlieger, Jul 11 2024 *)

A374443 Triangle read by rows: T(n, k) = rad(gcd(n, k)) if n, k > 0, T(0, 0) = 1, where rad = A007947 and gcd = A109004.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 1, 3, 2, 1, 2, 1, 2, 5, 1, 1, 1, 1, 5, 6, 1, 2, 3, 2, 1, 6, 7, 1, 1, 1, 1, 1, 1, 7, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6

Offset: 0

Peter Luschny and Michael De Vlieger, Jul 12 2024

			Triangle starts:
  [ 0]  1;
  [ 1]  1, 1;
  [ 2]  2, 1, 2;
  [ 3]  3, 1, 1, 3;
  [ 4]  2, 1, 2, 1, 2;
  [ 5]  5, 1, 1, 1, 1, 5;
  [ 6]  6, 1, 2, 3, 2, 1, 6;
  [ 7]  7, 1, 1, 1, 1, 1, 1, 7;
  [ 8]  2, 1, 2, 1, 2, 1, 2, 1, 2;
  [ 9]  3, 1, 1, 3, 1, 1, 3, 1, 1, 3;
  [10] 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10;
  [11] 11, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 11;

Variant: A374433.

Cf. A374442 (row sums), A007947, A109004.

rad := n -> ifelse(n = 0, 1, NumberTheory:-Radical(n)):
T := (n, k) -> rad(igcd(n, k)); seq(seq(T(n, k), k = 0..n), n = 0..11);

rad[n_] := If[n == 0, 1, Product[p, {p, Select[Divisors[n], PrimeQ]}]];
T[n_, k_] := rad[GCD[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten

from math import gcd, prod
from sympy.ntheory import primefactors
def T(n, k) -> int: return prod(primefactors(gcd(n, k)))
for n in range(16): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Jun 22 2025

cp's OEIS Frontend

A374431 Row product of A374433.

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A374432 Row sums of A374433.

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A374434 Triangle read by rows: T(n, k) = Product_{p in PF(n) symmetric difference PF(k)} p, where PF(a) is the set of the prime factors of a.

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A374435 Triangle read by rows: T(n, k) = Product_{p in PF(n) difference PF(k)} p, where PF(a) is the set of the prime factors of a.

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A374436 Triangle read by rows: T(n, k) = Product_{p in PF(n) union PF(k)} p, where PF(a) is the set of the prime factors of a.

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A374430 Odd bisection of A374431.

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A374443 Triangle read by rows: T(n, k) = rad(gcd(n, k)) if n, k > 0, T(0, 0) = 1, where rad = A007947 and gcd = A109004.

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