A284457 -id:A284457 - cp's OEIS frontend

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller, Mar 15 1996

Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]
a(n) = A128651(A129132(n-1) + 2) for n > 1. - Reinhard Zumkeller, Mar 30 2007
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length.  For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017

			G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandache-type multiplicative sequences
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
Neville Holmes, Integer Sequences [Broken link]
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
Wikipedia, Radical of an integer.

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
Range of values is A005117.
Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
Bisections: A099984, A099985.
Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
A003961, A059896 are used to express relationship between terms of this sequence.
Cf. A001615, A008683, A076479, A078615.

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
seq(A007947(i),i=1..69); # Peter Luschny, Mar 22 2011
A:= n -> convert(numtheory:-factorset(n),`*`):
seq(A(n),n=1..100); # Robert Israel, Aug 10 2014
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
From Antti Karttunen, Jun 18 2017: (Start)
a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
a(n) = A019565(A087207(n)). (End)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
From Peter Munn, Jan 01 2020: (Start)
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A003961(n)) = A003961(a(n)).
a(n^2) = a(n).
a(A225546(n)) = A019565(A267116(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

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

Reinhard Zumkeller, Dec 03 2001

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.

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

Reinhard Zumkeller (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..65537

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.

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

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 *)

A065642(n)={ my(r=A007947(n)); if(1==n,n, n += r; while(A007947(n) <> r, n += r); n)} \\ Antti Karttunen, Apr 17 2017

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

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

(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

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

A008479 Number of numbers <= n with same prime factors as n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 4, 1, 2, 1, 7, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 3, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 9

Offset: 1

Jeffrey Shallit, Olivier Gérard

For n > 1, a(n) gives the (one-based) index of the row where n is located in arrays A284311 and A285321 or respectively, index of the column where n is in A284457. A285329 gives the other index. - Antti Karttunen, Apr 17 2017

T. D. Noe, Table of n, a(n) for n = 1..10000
Paul Erdős and T. Motzkin, Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681. (Incorrect solution!)
H. N. Shapiro, Problem 5735, Amer. Math. Monthly, 97 (1990), 937.

Cf. A005117 (positions of ones), A005361, A007947, A008683, A010846, A284311, A284457, A285321, A285328, A285329.

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
V[1]:= 1:
for n from 2 to N do
  if V[n] = 0 then
   S:= {n};
   for p in numtheory:-factorset(n) do
     S := S union {seq(seq(s*p^k,k=1..floor(log[p](N/s))),s=S)};
   od:
   S:= sort(convert(S,list));
   for k from 1 to nops(S) do V[S[k]]:= k od:
fi
od:
convert(V,list); # Robert Israel, May 20 2016

PkTbl=Prepend[ Array[ Times @@ First[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ];1+Array[ Count[ Take[ PkTbl, #-1 ], PkTbl[ [ # ] ] ]&, Length[ PkTbl ] ]
Count[#, k_ /; k == Last@ #] & /@ Function[s, Take[s, #] & /@ Range@ Length@ s]@ Array[Map[First, FactorInteger@ #] &, 120] (* or *)
Table[Sum[(Floor[n^k/k] - Floor[(n^k - 1)/k]) (Floor[k^n/n] - Floor[(k^n - 1)/n]), {k, n}], {n, 120}] (* Michael De Vlieger, May 20 2016 *)

a(n)=my(f=factor(n)[,1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); s \\ Charles R Greathouse IV, Oct 19 2017

(define (A008479 n) (if (not (zero? (A008683 n))) 1 (+ 1 (A008479 (A285328 n))))) ;; Antti Karttunen, Apr 17 2017

a(n) = Sum_{k=1..n} (floor(n^k/k)-floor((n^k-1)/k))*(floor(k^n/n)-floor((k^n-1)/n)). - Anthony Browne, May 20 2016
If A008683(n) <> 0 [when n is squarefree, A005117], a(n) = 1, otherwise a(n) = 1+a(A285328(n)). - Antti Karttunen, Apr 17 2017
a(n) <= A010846(n), with equality if and only if n = 1. - Amiram Eldar, May 25 2025
a(m^(k+1)) = A010846(m^k) when m is squarefree. - Flávio V. Fernandes, Aug 20 2025

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

Original entry on oeis.org

2, 4, 3, 8, 9, 5, 16, 27, 25, 7, 32, 81, 125, 49, 11, 64, 243, 625, 343, 121, 13, 128, 729, 3125, 2401, 1331, 169, 17, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23

Offset: 1

Clark Kimberling, Dec 14 2010

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).
The monotonic ordering of this sequence, prefixed by 1, is A000961.
The joint-rank array of this sequence is A182869.
Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

			Square array A(i,j) begins:
  i \ j: 1      2      3      4      5  ...
  ---\-------------------------------------
  1:     2,     4,     8,    16,    32, ...
  2:     3,     9,    27,    81,   243, ...
  3:     5,    25,   125,   625,  3125, ...
  4:     7,    49,   343,  2401, 16807, ...
  ...
The triangle T(n,k) begins:
  n\k:  1     2     3     4     5     6  ...
  1:    2
  2:    4     3
  3:    8     9     5
  4:   16    27    25     7
  5:   32    81   125    49    11
  6:   64   243   625   343   121    13
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.

Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).
Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).
A319075 extends the array with 0th powers.
Subtable of A242378, A284457, A329332.

TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]

From Peter Munn, Dec 29 2019: (Start)
A(i,j) = A182945(j,i) = A319075(j,i).
A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).
A(i,i) = A062457(i).
(End)

Clarified in respect of alternate reading as a triangle by Peter Munn, Aug 28 2022

A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 6, 25, 27, 16, 7, 12, 125, 81, 32, 10, 49, 18, 625, 243, 64, 11, 20, 343, 24, 3125, 729, 128, 13, 121, 40, 2401, 36, 15625, 2187, 256, 14, 169, 1331, 50, 16807, 48, 78125, 6561, 512, 15, 28, 2197, 14641, 80, 117649, 54, 390625, 19683, 1024

Offset: 1

Bob Selcoe, Mar 24 2017

A permutation of the natural numbers > 1.

T(1,k)= A005117(m) with m > 1; terms in column k = T(1,k) * A162306(T(1,k)) only not bounded by T(1,k). Let T(1,k) = b. Terms in column k are multiples of b and numbers c such that c | b^e with e >= 0. Alternatively, terms in column k are multiples bc with c those numbers whose prime divisors p also divide b. - Michael De Vlieger, Mar 25 2017

			Array starts:
    2    3     5  6      7  10       11        13  14  15
    4    9    25 12     49  20      121       169  28  45
    8   27   125 18    343  40     1331      2197  56  75
   16   81   625 24   2401  50    14641    371293  98 135
   32  243  3125 36  16807  80   161051   4826809 112 225
   64  729 15625 48 117649 100  1771561  62748517 196 375
  128 2187 78125 54 823543 160 19487171 815730721 224 405
Column 6 is: T(1,6) = 2*5; T(2,6) = 2^2*5; T(3,6) = 2^3*5; T(4,6) = 2*5^2; T(5,6) = 2^4*5, etc.

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A005117 (squarefree numbers), A033845 (column 4), columns 1,2,3,5 are powers of primes, A033846 (column 6), A033847 (column 9), A033849 (column 10).
The columns that are powers of primes have indices A071403(n) - 1. - Michel Marcus, Mar 24 2017
See also A007947; the k-th column of the array corresponds to the numbers with radical A005117(k+1). - Rémy Sigrist, Mar 24 2017
Cf. A284457 (this sequence read by antidiagonals upwards), A285321 (a similar array, but columns come in different order).
Cf. A065642.
Cf. A008479 (index of the row where n is located), A285329 (of the column).

f[n_, k_: 1] := Block[{c = 0, sgn = Sign[k], sf}, sf = n + sgn; While[c < Abs[k], While[! SquareFreeQ@ sf, If[sgn < 0, sf--, sf++]]; If[sgn < 0, sf--, sf++]; c++]; sf + If[sgn < 0, 1, -1]] (* after Robert G. Wilson v at A005117 *); T[n_, k_] := T[n, k] = Which[And[n == 1, k == 1], 2, k == 1, f@ T[n - 1, k], PrimeQ@ T[n, 1], T[n, 1]^k, True, Module[{j = T[n, k - 1]/T[n, 1] + 1}, While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; Table[T[n - k + 1, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)

(define (A284311 n) (A284311bi  (A002260 n) (A004736 n)))
(define (A284311bi row col) (if (= 1 row) (A005117 (+ 1 col)) (A065642 (A284311bi (- row 1) col))))
;; Antti Karttunen, Apr 17 2017

From Antti Karttunen, Apr 17 2017: (Start)
A(1,k) = A005117(1+k), A(n,k) = A065642(A(n-1,k)).
A(A008479(n), A285329(n)) = n for all n >= 2.
(End)

A285329 a(n) = A013928(A007947(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 4, 5, 1, 2, 6, 7, 4, 8, 9, 10, 1, 11, 4, 12, 6, 13, 14, 15, 4, 3, 16, 2, 9, 17, 18, 19, 1, 20, 21, 22, 4, 23, 24, 25, 6, 26, 27, 28, 14, 10, 29, 30, 4, 5, 6, 31, 16, 32, 4, 33, 9, 34, 35, 36, 18, 37, 38, 13, 1, 39, 40, 41, 21, 42, 43, 44, 4, 45, 46, 10, 24, 47, 48, 49, 6, 2, 50, 51, 27, 52, 53, 54, 14, 55, 18, 56, 29, 57, 58, 59, 4, 60, 9, 20, 6

Offset: 1

Antti Karttunen, Apr 17 2017

nonn

For n > 1, a(n) gives the (one-based) index of the column where n is located in array A284311, or respectively, index of the row where n is in A284457. A008479 gives the other index.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A008479 (the other index).
Cf. A005117, A007947, A008683, A013928, A019565, A064273, A087207, A285328.
Cf. array A284311 (A284457).

from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
from functools import reduce
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a013928(n): return sum(1 for i in range(1, n) if core(i) == i)
print([a013928(a007947(n)) for n in range(1, 101)]) # Indranil Ghosh, Apr 18 2017

from math import prod, isqrt
from sympy import primefactors, mobius
def A285329(n):
    m=prod(primefactors(n))-1
    return sum(mobius(k)*(m//k**2) for k in range(1,isqrt(m)+1)) # Chai Wah Wu, May 12 2024

a(n) = A013928(A007947(n)).
Other identities. For all n >= 0:
If A008683(n) <> 0 [when n is squarefree, A005117], a(n) = A013928(n), otherwise a(n) = a(A285328(n)).
a(A019565(n)) = A064273(n).

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

Antti Karttunen, Apr 17 2017

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.

Antti Karttunen, Table of n, a(n) for n = 0..116
Index entries for sequences that are permutations of the natural numbers

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

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

A379713 Array read by downward antidiagonals: rows list practical numbers with the same progenitor primitive practical number (A267124).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 20, 16, 18, 40, 28, 32, 24, 80, 56, 30, 64, 36, 100, 112, 60, 42, 128, 48, 160, 196, 90, 84, 66, 256, 54, 200, 224, 120, 126, 132, 78, 512, 72, 320, 392, 150, 168, 198, 156, 88, 1024, 96, 400, 448, 180, 252, 264, 234, 176, 104, 2048, 108, 500, 784, 240, 294, 396, 312, 352, 208, 140, 4096, 144, 640, 896, 270, 336, 528, 468, 704, 416, 280, 204, 8192

Offset: 1

Frank M Jackson, Dec 30 2024

A permutation of the practical numbers.
This sequence is presented as an array of rows. The first row contains a single term of value 1. Subsequent rows are infinite sequences and are presented as a square array by listing the antidiagonals downwards that is 1: 2; 4,6; 8,12,20; etc.
The first column contains the primitive practical numbers A267124; each row lists all practical numbers (A005153) having the same primitive practicle progenitor and which is the first term in each row. See A379325 comments for further details. If T[1,m] is squarefree then the row is identical to the same squarefree row in A284457.
Every primitive practical number A267124(n) is the progenitor of a disjoint subsequence of the practical numbers. If the PP column represents the sequence of primitive practical numbers A267124, the table below give the 7 initial terms of the disjoint sequences of practical numbers A005153 generated by the initial 7 terms of the sequence of primitive practical numbers.
PP: Disjoint subsequence of A005153
--  -------------------------------
1:   1
2:   2,  4,  8, 16, 32, 64,128, . . .- A000079 with offset 1,1
6:   6, 12, 18, 24, 36, 48, 54, . . .- A033845
20: 20, 40, 80,100,160,200,320, . . .
28: 28, 56,112,196,224,392,448, . . .
30: 30, 60, 90,120,150,180,240, . . .- A143207
42: 42, 84,126,168,252,294,336
...
Row 1 is T[1,1] = 1 and only has one term in the subsequence.
Row 7 is T[1,7] = 2*3*7; T[2,7] = 2^2*3*7; T[3,7] = 2*3^2*7; T[4,7] = 2^3*3*7; T[5,7] = 2^2*3^2*7, etc.

			a(14) = 80 and it is T[3,4] = 2^4*5. Its primitive progenitor is 20 = 2^2*5 and its equivalence class are the terms of row 4.

Frank M Jackson, Mathematica program

Cf. A000079, A005117, A005153, A007947, A033845, A143207, A267124, A284457, A379325.

Mathematica
```
(* See link above *)
```

A284785 a(n) = rad(A280864(n)).

Original entry on oeis.org

1, 2, 2, 3, 6, 2, 5, 10, 6, 3, 7, 14, 2, 11, 22, 6, 15, 10, 6, 21, 14, 26, 13, 17, 34, 30, 15, 19, 38, 2, 23, 46, 6, 3, 5, 35, 42, 6, 29, 58, 10, 55, 33, 39, 26, 22, 77, 7, 31, 62, 10, 65, 78, 6, 37, 74, 14, 21, 51, 34, 30, 15, 41, 82, 2, 43, 86

Offset: 1

Bob Selcoe, Apr 02 2017

nonn

By definition, all terms are squarefree (see A007947); repeated terms here are the squarefree kernels of A280864(n).
All even squarefree numbers appear infinitely often.
1 appears only at a(1).
Even terms appear consecutively in pairs, each pair followed by one or more odd terms.
Conjecture: all odd squarefree numbers > 1 appear infinitely often. If so, then A280864 is a permutation of the natural numbers.
Theorem: a(n) = b(n-1)*b(n) where b = A280738. - N. J. A. Sloane, Apr 11 2017

			a(61) = 30 because A280864(61) = 60, and rad(60) = 30.

Cf. A280864, A284311, A284457, A007947, A280738.

A365656 Array T(n,k) read by antidiagonals (downward): T(n,1) = A005117(n) (squarefree numbers > 1); for k > 1, columns are nonsquarefree numbers (in descending order) with exactly the same prime factors as T(n,1).

Original entry on oeis.org

1, 2, 4, 3, 8, 9, 5, 16, 27, 25, 6, 32, 81, 125, 12, 7, 64, 243, 625, 18, 49, 10, 128, 729, 3125, 24, 343, 20, 11, 256, 2187, 15625, 36, 2401, 40, 121, 13, 512, 6561, 78125, 48, 16807, 50, 1331, 169, 14, 1024, 19683, 390625, 54, 117649, 80, 14641, 2197, 28, 15

Offset: 0

Michael De Vlieger, Nov 17 2023

Permutation of natural numbers.
Transpose of A284311 with a(0) = 1 prepended.
Essentially the same as A284457. - R. J. Mathar, Jan 23 2024

			Table T(n,k) for n = 1..12 and k = 1..6 shown below:
  n\k |  1    2     3       4        5         6 ...
  ----------------------------------------------
   1  |  1
   2  |  2    4     8      16       32        64
   3  |  3    9    27      81      243       729
   4  |  5   25   125     625     3125     15625
   5  |  6   12    18      24       36        48
   6  |  7   49   343    2401    16807    117649
   7  | 10   20    40      50       80       100
   8  | 11  121  1331   14641   161051   1771561
   9  | 13  169  2197   28561   371293   4826809
  10  | 14   28    56      98      112       196
  11  | 15   45    75     135      225       375
  12  | 17  289  4913   83521  1419857  24137569
  ...
Triangle begins:
   1;
   2;
   4,   3;
   8,   9,   5;
  16,  27,  25,  6;
  32,  81, 125, 12,  7;
  64, 243, 625, 18, 49, 10;
 ...

Michael De Vlieger, Table of n, a(n) for n = 0..11326 (rows 0..150, flattened)
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..64981 (i.e., 360 rows, flattened).
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..4096, showing primes in red, squarefree composites in green, composite prime powers in gold, and numbers neither squarefree nor prime powers in blue and purple; we show squareful numbers that are not prime powers in purple.

Cf. A002260, A005117, A007947, A065642, A284311, A284457.

f[n_, k_ : 1] := Block[{c = 0, s = Sign[k], m}, m = n + s;
    While[c < Abs[k], While[! SquareFreeQ@ m, If[s < 0, m--, m++]];
     If[s < 0, m--, m++]; c++];
    m + If[s < 0, 1, -1] ] (* after Robert G.Wilson v at A005117 *);
  T[n_, k_] := T[n, k] =
    Which[And[n == 1, k == 1], 2, k == 1, f@T[n - 1, k],
     PrimeQ@ T[n, 1], T[n, 1]^k, True,
     Module[{j = T[n, k - 1]/T[n, 1] + 1},
      While[PowerMod[T[n, 1], j, j] != 0, j++]; j T[n, 1]]]; {1}~Join~
   Table[T[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // TableForm

For prime n = p, T(p,k) = p^k.

cp's OEIS Frontend

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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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}.

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A008479 Number of numbers <= n with same prime factors as n.

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A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

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A284311 Array T(n,k) read by antidiagonals (downward): T(1,k) = A005117(k+1) (squarefree numbers > 1); for n > 1, columns are nonsquarefree numbers (in ascending order) with exactly the same prime factors as T(1,k).

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A285329 a(n) = A013928(A007947(n)).

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