A307746 -id:A307746 - cp's OEIS frontend

A088387 Prime corresponding to largest prime power factor of n, a(1)=1.

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 2, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 2, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 2, 41, 7, 43, 11, 3, 23, 47, 2, 7, 5, 17, 13, 53, 3, 11, 2, 19, 29, 59, 5, 61, 31, 3, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 2, 3, 41, 83, 7, 17

Offset: 1

Reinhard Zumkeller, Sep 28 2003

Most significant prime factor of n: If n = (p_1^e_1)(p_2^e_2)(p_3^e_3)... and max(p_1^e_1,p_2^e_2,...) = p_k^e_k then a(n) = p_k.

			a(6) = a(2*3) = 3 because 3^1 > 2^1;
a(36) = a((2^2)(3^2)) = 3 because 3^2 > 2^2;
a(12) = a((2^2)*3) = 2 because 2^2 > 3^1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034699, A088388, A307746, A307641.

f[n_] := Sort[ {#[[1]]^#[[2]], #[[1]]} & /@ FactorInteger@ n][[ -1, 2]]; Array[f, 85] (* Robert G. Wilson v, Nov 05 2007 *)
a[n_] := MaximalBy[FactorInteger[n], Power @@ # &][[1, 1]];
Array[a, 85] (* Jean-François Alcover, Jun 27 2019 *)

A088387(n) = if(1==n,1,my(f=factor(n),p=0); isprimepower(vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2])),&p); (p)); \\ Antti Karttunen, Jul 22 2018

from sympy import factorint
def A088387(n): return max(((p**e,p) for p, e in factorint(n).items()), default=(0,1))[1] # Chai Wah Wu, Apr 17 2023

A034699(n) = a(n)^A088388(n).

a(n*a(n)) = a(n). - Sam Alexander, Dec 15 2003

More terms from Ray Chandler, Dec 20 2003

Edited by N. J. A. Sloane at the suggestion of Stefan Steinerberger, Nov 04 2007

A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 5, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

I. V. Serov, Apr 19 2019

i=Product_{j=1..i} T(i,j). This is an adjusted formulation of the fundamental theorem of arithmetic with the fixed order of the prime-or-one factors, as well as with the regular length i of the factorization of i.
Remove all 1's except for n = 1 to get irregular triangle A307746.
A307723 is a quasi-logarithmic binary encoding of this triangle.

			Triangle begins:
  1,
  1, 2,
  1, 1, 3,
  1, 2, 1, 2,
  1, 1, 1, 1, 5,
  1, 2, 3, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 7,
  1, 2, 1, 2, 1, 1, 1, 2,
  1, 1, 3, 1, 1, 1, 1, 1, 3,
  1, 2, 1, 1, 5, 1, 1, 1, 1, 1,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,11,
  1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1,
  ...

I. V. Serov, Rows n=1..131 of triangle, flattened

Cf. A027746, A027748, A014963, A100995, A307662, A307723, A307742, A307743, A307746.

Table[Map[Which[PrimeNu@ # > 1, 1, And[PrimeQ@ #, Mod[n, #] == 0], #, Mod[n, #] == 0, FactorInteger[#][[1, 1]], True, 1] &, Range@ n], {n, 13}] // Flatten (* Michael De Vlieger, Apr 23 2019 *)

w(n) = my(t=isprimepower(n)); if (t, t, 0);
row(n) = vector(n, k, mnk = if ((n % k) == 0, k, 1); if (t=w(k), sqrtnint(mnk, t), 1)); \\ Michel Marcus, Apr 21 2019

T(i,j) = A307662(i,j)^w(j), where w(j)=0 if A100995(j)=0; otherwise w(j)=1/A100995(j), for 1 <= j <= n.

A307723 Naturally ordered prime factorization of n as a quasi-logarithmic word over the binary alphabet {1,0}.

Original entry on oeis.org

10, 1100, 1010, 110100, 101100, 11011000, 101010, 11001100, 10110100, 1101101000, 10110010, 1101100100, 1011011000, 1100110100, 10101010, 1101010100, 1011001100, 110110011000, 1010110100, 110011011000

Offset: 2

I. V. Serov, Apr 24 2019

Let m(n) be the number of digits (letters) in a(n).
m(n) = 2*A064097(n) = 2*(A073933(n)-1).
Split the word a(n) into two parts of equal length. The number of 1's in the left part equals the number of 0's in the right part and vice versa.

			The sequence begins:
   n a(n)
  -- -----------
   1
   2 10
   3 1100
   4 1010
   5 110100
   6 101100
   7 11011000
   8 101010
   9 11001100
  10 10110100
  11 1101101000
  12 10110010
  ...

I. V. Serov, Table of n, a(n) for n = 2..10000

Cf. A010051, A088387, A307641, A307746, A064097, A073933.

a(1) is empty.
a(n) = concatenation(1, a(n-1), 0) if n is prime.
a(n) = concatenation_{k=1..A001222(n)} a(A307746(n,k)) if n is composite.
a(n) = concatenation(a(n/A088387(n)), a(A088387(n))) if n is composite.

cp's OEIS Frontend

A088387 Prime corresponding to largest prime power factor of n, a(1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A307641 Triangle T(i,j=1..i) read by rows which contain the naturally ordered prime-or-one factorization of the row number i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A307723 Naturally ordered prime factorization of n as a quasi-logarithmic word over the binary alphabet {1,0}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula