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.

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A065642 a(1) = 1; for n > 1, a(n) = Min {m > n | m has same prime factors as n ignoring multiplicity}.

Original entry on oeis.org

1, 4, 9, 8, 25, 12, 49, 16, 27, 20, 121, 18, 169, 28, 45, 32, 289, 24, 361, 40, 63, 44, 529, 36, 125, 52, 81, 56, 841, 60, 961, 64, 99, 68, 175, 48, 1369, 76, 117, 50, 1681, 84, 1849, 88, 75, 92, 2209, 54, 343, 80, 153, 104, 2809, 72, 275, 98, 171, 116, 3481, 90, 3721
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 03 2001

Keywords

Comments

After the initial 1, a permutation of the nonsquarefree numbers A013929. The array A284457 is obtained as a dispersion of this sequence. - Antti Karttunen, Apr 17 2017
Numbers such that a(n)/n is not an integer are listed in A284342.

Examples

			a(10) = a(2 * 5) = 2 * 2 * 5 = 20; a(12) = a(2^2 * 3) = 2 * 3^2 = 18.

Links

Crossrefs

Cf. A005117, A007947, A013929, A020639, A000040, A001221, A081382.
Cf. A285328 (a left inverse).
Cf. also arrays A284457 & A284311, A285321 and permutations A284572, A285112, A285332.
Cf. A084968, A084969, A084970, A284342.

Programs

  • Haskell
    a065642 1 = 1
a065642 n = head [x | let rad = a007947 n, x <- [n+1..], a007947 x == rad]
-- Reinhard Zumkeller, Jun 12 2015, Jul 27 2011
  • Mathematica
    ffi[x_]:= Flatten[FactorInteger[x]]; lf[x_]:= Length[FactorInteger[x]]; ba[x_]:= Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; cor[x_]:= Apply[Times, ba[x]]; Join[{1}, Table[Min[Flatten[Position[Table[cor[w], {w, n+1, n^2}]-cor[n], 0]]+n], {n, 2, 100}]] (* This code is suitable since prime factor set is invariant iff squarefree kernel is invariant. *) (* G. C. Greubel, Oct 31 2018 *)
Array[If[# == 1, 1, Function[{n, c}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &]] @@ {#, Times @@ FactorInteger[#][[All, 1]]}] &, 61] (* Michael De Vlieger, Oct 31 2018 *)
  • PARI
    A065642(n)={ my(r=A007947(n)); if(1==n,n, n += r; while(A007947(n) <> r, n += r); n)} \\ Antti Karttunen, Apr 17 2017
  • PARI
    a(n)=if(n<2, return(1)); my(f=factor(n),r,mx,mn,t); if(#f~==1, return(f[1,1]^(f[1,2]+1))); f=f[,1]; r=factorback(f); mn=mx=n*f[1]; forvec(v=vector(#f,i,[1,logint(mx/r,f[i])+1]), t=prod(i=1,#f, f[i]^v[i]); if(tn, mn=t)); mn \\ Charles R Greathouse IV, Oct 18 2017
  • Python
    from sympy import primefactors, prod
def a007947(n): return 1 if n < 2 else prod(primefactors(n))
def a(n):
    if n==1: return 1
    r=a007947(n)
    n += r
    while a007947(n)!=r:
        n+=r
    return n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 17 2017
  • Scheme
    (define (A065642 n) (if (= 1 n) n (let ((k (A007947 n))) (let loop ((n (+ n k))) (if (= (A007947 n) k) n (loop (+ n k))))))) ;; (Semi-naive implementation) - Antti Karttunen, Apr 17 2017

Formula

A007947(a(n)) = A007947(n); a(A007947(n)) = A007947(n) * A020639(n), where A007947 is the squarefree kernel (radical), A020639 is the least prime factor (lpf).
a(A000040(n)^k) = A000040(n)^(k+1); A001221(a(n)) = A001221(n).
A285328(a(n)) = n. - Antti Karttunen, Apr 17 2017
n < a(n) <= n*lpf(n) <= n^2. - Charles R Greathouse IV, Oct 18 2017

A285112 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 8, 7, 25, 14, 27, 10, 12, 13, 16, 11, 49, 39, 125, 22, 28, 42, 81, 15, 20, 19, 18, 21, 169, 26, 32, 17, 121, 79, 343, 65, 117, 205, 625, 35, 44, 43, 56, 69, 84, 133, 243, 23, 45, 33, 40, 31, 361, 30, 24, 34, 63, 277, 2197, 41, 52, 53, 64, 29, 289, 199, 1331, 130, 6241, 563, 2401, 106, 325, 193, 351, 335, 1025, 1030, 3125, 58
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Comments

Note the indexing: the domain starts from 0, while the range excludes zero.
This sequence can be represented as a binary tree. Each left hand child is produced as A005117(1+n), and each right hand child as A065642(n), when the parent node contains n >= 2:
1
|
...................2...................
3 4
5......../ \........9 6......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
7 25 14 27 10 12 13 16
11 49 39 125 22 28 42 81 15 20 19 18 21 169 26 32
etc.

Links

Crossrefs

Inverse: A285111.
Cf. A005117, A065642.
Similar or related permutations: A243344, A243346, A252753, A277696, A284572.
Cf. also arrays A284457 & A284311.

Formula

a(0) = 1, a(1) = 2, a(2n) = A005117(1+a(n)), a(2n+1) = A065642(a(n)).

A284571 Permutation of natural numbers: a(1) = 1, a(A005117(1+n)) = 2*a(n), a(A065642(1+n)) = 1 + 2*a(n).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 9, 5, 12, 32, 17, 18, 10, 24, 33, 64, 65, 34, 11, 36, 20, 48, 129, 7, 66, 19, 37, 128, 130, 68, 49, 22, 72, 40, 97, 96, 258, 14, 69, 132, 38, 74, 73, 21, 256, 260, 81, 13, 29, 136, 15, 98, 521, 44, 39, 144, 80, 194, 257, 192, 516, 23, 137, 28, 138, 264, 45, 76, 148, 146, 197, 42, 512, 147, 193, 520, 162, 26, 27
Offset: 1

Views

Author

Antti Karttunen, Apr 17 2017

Keywords

Links

Crossrefs

Inverse: A284572.
Cf. A005117, A008683, A013928, A065642, A285328.
Similar or related permutations: A243343, A243345, A277695, A285111.

Programs

  • Python
    from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a285328(n):
    if core(n) == n: return 1
    k=n - 1
    while k>0:
        if a007947(k) == a007947(n): return k
        else: k-=1
def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)
def a(n):
    if n==1: return 1
    if core(n)==n: return 2*a(a013928(n))
    else: return 1 + 2*a(a285328(n) - 1)
[a(n) for n in range(1, 121)] # Indranil Ghosh, Apr 17 2017

Formula

a(1) = 1, for n > 1, if A008683(n) <> 0 [when n is squarefree], a(n) = 2*a(A013928(n)), otherwise a(n) = 1 + 2*a(A285328(n)-1).
Showing 1-3 of 3 results.

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