A085079 -id:A085079 - cp's OEIS frontend

A046523 Smallest number with same prime signature as n.

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 12, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2

Offset: 1

N. J. A. Sloane

			If p,q,... are different primes, a(p)=2, a(p^2)=4, a(pq)=6, a(p^2*q)=12, etc.
n = 108 = 2*2*3*3*3 is replaced by a(n) = 2*2*2*3*3 = 72;
n = 105875 = 5*5*5*7*11*11 is represented by a(n) = 2*2*2*3*3*5 = 360.
Prime-powers are replaced by corresponding powers of 2, primes by 2.
Factorials, primorials and lcm[1..n] are in the sequence.
A000005(a(n)) = A000005(n) remains invariant; least and largest prime factors of a(n) are 2 or p[A001221(n)] resp.

T. D. Noe, Table of n, a(n) for n = 1..10000

A025487 gives range of values of this sequence.
Cf. A000142, A002110, A003418, A001221, A000040, A000005, A124010, A071364, A085079, A089247.
Cf. A181819, A181821.

import Data.List (sort)
a046523 = product .
          zipWith (^) a000040_list . reverse . sort . a124010_row
-- Reinhard Zumkeller, Apr 27 2013

a:= n-> (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)))
        (sort(ifactors(n)[2], (x, y)->x[2]>y[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 18 2014

Table[Apply[Times, p[w]^Reverse[Sort[ex[w]]]], {w, 1, 1000}] p[x_] := Table[Prime[w], {w, 1, lf[x]}] ex[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]]
ps[n_] := Sort[Last /@ FactorInteger[n]]; Join[{1}, Table[i = 2; While[ps[n] != ps[i], i++]; i, {n, 2, 89}]] (* Jayanta Basu, Jun 27 2013 *)

a(n)=my(f=vecsort(factor(n)[,2],,4),p);prod(i=1,#f,(p=nextprime(p+1))^f[i]) \\ Charles R Greathouse IV, Aug 17 2011

A046523(n)=factorback(primes(#n=vecsort(factor(n)[,2],,4)),n) \\ M. F. Hasler, Oct 12 2018, improved Jul 18 2019

from sympy import factorint
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1 # Indranil Ghosh, May 05 2017

from math import prod
from sympy import factorint, prime
def A046523(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n).values(),reverse=True))) # Chai Wah Wu, Feb 04 2022

In prime factorization of n, replace most common prime by 2, next most common by 3, etc.
a(n) = A124859(A124859(n)) = A181822(A124859(n)). - Matthew Vandermast, May 19 2012
a(n) = A181821(A181819(n)). - Alois P. Heinz, Feb 17 2020

Corrected and extended by Ray Chandler, Mar 11 2004

A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 18, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 18, 6, 12, 2, 54, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 18, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24

Offset: 1

Reinhard Zumkeller, May 21 2002

nonn

A046523(a(n))=A046523(n); A046523(n)<=a(n)<=n; A001221(a(n))=A001221(n), A001222(a(n))=A001222(n); A020639(a(n))=2, A006530(a(n))=A000040(A001221(n))<=A006530(n); A000005(a(n))=A000005(n);
a(a(n))=a(n); a(n)=2^k iff n=p^k, p prime, k>0 (A000961); if n>1 is not a prime power, then a(n) mod 6 = 0; range of values = A055932, as distinct prime factors of a(n) are consecutive: a(n)=n iff n=A055932(k) for some k;
a(A003586(n))=A003586(n).

			a(105875) = a(5*5*5*7*11*11) = 2*2*2*3*5*5 = 600.

Daniel Forgues, Table of n, a(n) for n=1..100000

Cf. A071365, A071366.
Cf. A000040.
Cf. A085079, A089247.
The range is A055932.
The reversed version is A331580.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.

a071364 = product . zipWith (^) a000040_list . a124010_row
-- Reinhard Zumkeller, Feb 19 2012

Table[ e = Last /@ FactorInteger[n]; Product[Prime[i]^e[[i]], {i, Length[e]}], {n, 88}] (* Ray Chandler, Sep 23 2005 *)

a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ Michel Marcus, Jun 13 2014

from math import prod
from sympy import prime, factorint
def A071364(n): return prod(prime(i+1)**p[1] for i,p in enumerate(sorted(factorint(n).items()))) # Chai Wah Wu, Sep 16 2022

In prime factorization of n, replace least prime by 2, next least by 3, etc.

a(n) = product(A000040(k)^A124010(k): k=1..A001221(n)). - Reinhard Zumkeller, Apr 27 2013

Extended by Ray Chandler, Sep 23 2005

A112769 Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is less than the previous exponent when read from left to right.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212, 220

Offset: 1

Ray Chandler, Sep 23 2005, based on a suggestion from Leroy Quet

nonn

This sequence lists the integers x such that A085079(x) > x. - Michel Marcus, Jun 25 2025 and Jul 30 2015

			90 = 2^1 * 3^2 * 5^1 and 2 > 1, so 90 is in sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A085079, A097318, A097319, A097320, A304678.

mopfQ[n_]:=Module[{e=FactorInteger[n][[All,2]]},Length[e]>1&&Min[ Differences[ e]]<0]; Select[Range[300],mopfQ] (* Harvey P. Dale, May 30 2018 *)

isok(n) = {f = factor(n)[,2]; if (#f > 1, for (k=2, #f, if (f[k] < f[k-1], return (1)););); return (0);} \\ Michel Marcus, Jul 30 2015

A089247 a(n) = smallest number obtainable by permuting the exponents in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 12, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 20, 51, 52, 53, 24, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Sam Alexander, Dec 11 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027748, A124010, A046523, A071364, A085079.

import Data.List (sort)
a089247 n = product $ zipWith (^)
                      (a027748_row n) (reverse $ sort $ a124010_row n)
-- Reinhard Zumkeller, Apr 27 2013

a(n)=my(f=factor(n)); f[,2]=vecsort(f[,2],,4); factorback(f) \\ Charles R Greathouse IV, Sep 02 2015

A340302 Numbers k such that k and the least number that is larger than k and has the same prime signature as k also has the same set of distinct prime divisors as k.

Original entry on oeis.org

12, 72, 144, 420, 432, 540, 864, 1728, 1800, 2000, 2268, 2520, 2592, 5184, 5400, 6300, 7020, 10125, 10368, 10692, 10800, 11340, 12600, 15120, 15552, 16200, 17640, 20000, 20736, 21168, 21600, 24000, 24948, 25200, 26460, 31104, 37800, 40500, 41472, 42750, 43200

Offset: 1

Amiram Eldar, Jan 03 2021

nonn

Numbers k such that A007947(k) = A007947(A081761(k)).

This sequence is infinite since it includes all the numbers of the form 2*6^k for k>=1.

			12 = 2^2 * 3 is a term since the least number that is larger than 12 and has the same prime signature as 12 is 18 = 2 * 3^2 which also has the same set of distinct prime divisors as 12, {2, 3}.

Amiram Eldar, Table of n, a(n) for n = 1..500
Index entries for sequences related to prime signature

Cf. A007947, A081761, A085079, A089247, A118914, A167747.

sig[n_] := Sort@FactorInteger[n][[;; , 2]]; nextsig[n_] := Module[{sign = sig[n], k = n + 1}, While[sig[k] != sign, k++]; k]; rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; Select[Range[2, 1000], rad[#] == rad[nextsig[#]] &]

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A046523 Smallest number with same prime signature as n.

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A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

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A112769 Numbers with more than one prime factor and, in the ordered factorization, at least one exponent is less than the previous exponent when read from left to right.

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A089247 a(n) = smallest number obtainable by permuting the exponents in the prime factorization of n.

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A340302 Numbers k such that k and the least number that is larger than k and has the same prime signature as k also has the same set of distinct prime divisors as k.

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