cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A046523 Smallest number with same prime signature as n.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Haskell
    import Data.List (sort)
a046523 = product .
          zipWith (^) a000040_list . reverse . sort . a124010_row
-- Reinhard Zumkeller, Apr 27 2013
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    A046523(n)=factorback(primes(#n=vecsort(factor(n)[,2],,4)),n) \\ M. F. Hasler, Oct 12 2018, improved Jul 18 2019
  • Python
    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
  • Python
    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

Formula

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

Extensions

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

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Author

Reinhard Zumkeller, May 21 2002

Keywords

Comments

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).

Examples

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

Links

Crossrefs

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.

Programs

  • Haskell
    a071364 = product . zipWith (^) a000040_list . a124010_row
-- Reinhard Zumkeller, Feb 19 2012
  • Mathematica
    Table[ e = Last /@ FactorInteger[n]; Product[Prime[i]^e[[i]], {i, Length[e]}], {n, 88}] (* Ray Chandler, Sep 23 2005 *)
  • PARI
    a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ Michel Marcus, Jun 13 2014
  • Python
    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

Formula

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

Extensions

Extended by Ray Chandler, Sep 23 2005

A085079 Largest number with the prime signature of n using prime divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 13, 14, 15, 16, 17, 18, 19, 50, 21, 22, 23, 54, 25, 26, 27, 98, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 49, 50, 51, 338, 53, 54, 55, 686, 57, 58, 59, 150, 61, 62, 147, 64, 65, 66, 67, 578, 69, 70
Offset: 1

Views

Author

Amarnath Murthy, Jul 01 2003

Keywords

Comments

Though a large number of initial terms match, this sequence is different from A069799. Example: a(1500) = a(2^2*3*5^3) = 5^3*3^2*2 = 2250, whereas A069799(1500) = 5^2*3*2^3 = 600.
The first term that is different from A069799 is a(18). - Ivan Neretin, Jul 29 2015

Examples

			20 = 2^2*5, hence a(20) = 5^2*2 = 50.

Links

Crossrefs

Cf. A069799.
Cf. A027748, A124010, A089247, A071364, A046523.

Programs

  • Haskell
    import Data.List (sort)
a085079 n = product $ zipWith (^) (a027748_row n) (sort $ a124010_row n)
-- Reinhard Zumkeller, Apr 27 2013
  • Mathematica
    Table[Times @@ ((ar = Transpose[FactorInteger[n]])[[1]]^Sort[ar[[2]]]), {n, 70}] (* Ivan Neretin, Jul 29 2015*)
fise[n_]:=Module[{fi=FactorInteger[n]},Times@@(fi[[All,1]]^Sort[ fi[ [All,2]]])]; Array[fise,100] (* Harvey P. Dale, Sep 18 2017 *)

Formula

a(n) >= n. - Michel Marcus, Jul 30 2015

Extensions

Corrected and extended by Ray Chandler, Aug 17 2003

A087315 a(n) = Product_{k=1..n} prime(k)^prime(n-k+1).

Original entry on oeis.org

1, 4, 72, 21600, 190512000, 580909190400000, 428616352408083840000000, 859278392084450410309036800000000000, 2097197194438629126172451944256706311040000000000000
Offset: 0

Views

Author

Amarnath Murthy, Sep 03 2003

Keywords

Examples

			a(3) = 2^5*3^3*5^2 = 21600.

Links

Crossrefs

Cf. A006939, A002110, A025487, A004394, A002093, A002182, A055932, A025487, A089247, A071364, A097320, A046523, A056166, A114129, A053810.

Programs

  • Magma
    [1] cat [(&*[NthPrime(k)^(NthPrime(n-k+1)): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018
  • Maple
    seq(product(ithprime(k)^ithprime(n-k+1), k=1..n), n=0..10);
  • Mathematica
    Table[Product[Prime[k]^Prime[n - k + 1], {k, 1, n}], {n, 0, 10}] (* G. C. Greubel, Oct 14 2018 *)
  • PARI
    for(n=0, 10, print1(prod(k=1,n, prime(k)^prime(n-k+1)), ", ")) \\ G. C. Greubel, Oct 14 2018
  • Sage
    [prod(nth_prime(i)^nth_prime(k-i+1) for i in (1..k)) for k in (0..10)] # Giuseppe Coppoletta, Nov 03 2014

Extensions

More terms from Jorge Coveiro, Dec 22 2004
Corrected by David Wasserman, May 02 2005

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

Views

Author

Amiram Eldar, Jan 03 2021

Keywords

Comments

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.

Examples

			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}.

Links

Crossrefs

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

Programs

  • Mathematica
    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[#]] &]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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