A097246 -id:A097246 - cp's OEIS frontend

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

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

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

a(n) = r(n,m) with m such that r(n,m)=r(n,m+1), where r(n,k) = A097246(r(n,k-1)), r(n,0)=n. (The original definition.)
A097248(n) = r(n,a(n)).
From Antti Karttunen, Nov 15 2016: (Start)
The above remark could be interpreted to mean that A097249(n) <= a(n).
All terms are squarefree, and the squarefree numbers are the fixed points.
These are also fixed points eventually reached when iterating A277886.
(End)

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

Range of values is A005117.
Cf. A008683, A019565, A048675, A097246, A097247, A097249, A277905, A332824.
A003961, A225546, A277885, A277886, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A007913, A260443, A329050, A329332.
See comments/formulas in A283475, A283478, A331751 giving their relationship to this sequence.

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, n], {n, 75}] (* Michael De Vlieger, Mar 18 2017 *)

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
\\ Antti Karttunen, Mar 18 2017

from sympy import factorint, nextprime
from operator import mul
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k # Indranil Ghosh, May 15 2017

;; with memoization-macro definec
;; Two implementations:
(definec (A097248 n) (if (not (zero? (A008683 n))) n (A097248 (A097246 n))))
(definec (A097248 n) (if (zero? (A277885 n)) n (A097248 (A277886 n))))
;; Antti Karttunen, Nov 15 2016

a(A005117(n)) = A005117(n).
From Antti Karttunen, Nov 15 2016: (Start)
If A008683(n) <> 0 [when n is squarefree], a(n) = n, otherwise a(n) = a(A097246(n)).
If A277885(n) = 0, a(n) = n, otherwise a(n) = a(A277886(n)).
A007913(a(n)) = a(n).
a(A007913(n)) = A007913(n).
A048675(a(n)) = A048675(n).
a(A260443(n)) = A019565(n).
(End)
From Peter Munn, Feb 06 2020: (Start)
a(1) = 1; a(p) = p, for prime p; a(m*k) = A331590(a(m), a(k)).
a(A331590(m,k)) = A331590(a(m), a(k)).
a(n^2) = a(A003961(n)) = A003961(a(n)).
a(A225546(n)) = a(n).
a(n) = A225546(2^A048675(n)) = A019565(A048675(n)).
a(A329050(n,k)) = prime(n+k-1) = A000040(n+k-1).
a(A329332(n,k)) = A019565(n * k).
Equivalently, a(A019565(n)^k) = A019565(n * k).
(End)
From Antti Karttunen, Feb 22-25 & Mar 01 2020: (Start)
a(A019565(x)*A019565(y)) = A019565(x+y).
a(A332461(n)) = A332462(n).
a(A332824(n)) = A019565(n).
a(A277905(n,k)) = A277905(n,1) = A019565(n), for all n >= 1, and 1 <= k <= A018819(n).
(End)

Name changed and the original definition moved to the Comments section by Antti Karttunen, Nov 15 2016

A097249 a(n) is the number of times we must iterate A097246, starting at n, before the result is squarefree.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

a(n) = Min{k: r(n,k)=r(n,k+1)}, where r(n,k)=A097246(r(n,k-1)), r(n,0)=n;

a(A005117(n))=0; a(A097250(n))=n and a(m)A097250(n).

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

Cf. A005117, A097246, A097248, A097250, A277899.

f[n_] := Product[{p, e} = pe; NextPrime[p]^Quotient[e, 2] p^Mod[e, 2], {pe, FactorInteger[n]}];
a[n_] := (NestWhileList[f, n, !SquareFreeQ[#]&] // Length) - 1;
Array[a, 105] (* Jean-François Alcover, Nov 18 2021 *)

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097249(n) = if(issquarefree(n),0,1+A097249(A097246(n))); \\ Antti Karttunen, Jul 29 2018

If A008966(n) = 1 [when n is in A005117], a(n) = 0, otherwise a(n) = 1 + a(A097246(n)). - Antti Karttunen, Jul 29 2018

Edited by Sam Alexander, Jan 05 2005

A322808 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is a squarefree number > 2, and f(n) = A097246(n) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 3, 3, 5, 6, 3, 3, 7, 3, 3, 3, 7, 3, 8, 3, 9, 3, 3, 3, 10, 11, 3, 9, 12, 3, 3, 3, 10, 3, 3, 3, 9, 3, 3, 3, 13, 3, 3, 3, 14, 15, 3, 3, 16, 17, 18, 3, 19, 3, 13, 3, 20, 3, 3, 3, 21, 3, 3, 22, 16, 3, 3, 3, 23, 3, 3, 3, 13, 3, 3, 12, 24, 3, 3, 3, 21, 15, 3, 3, 25, 3, 3, 3, 26, 3, 27, 3, 28, 3, 3, 3, 29, 3, 30, 31, 12, 3, 3, 3, 32, 3

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

For all i, j: a(i) = a(j) => A097249(i) = A097249(j).

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

Cf. A005117, A097246, A097249, A305980.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A322808aux(n) = if((n>2)&&issquarefree(n),0,A097246(n));
v322808 = rgs_transform(vector(up_to,n,A322808aux(n)));
A322808(n) = v322808[n];

A323078 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd prime, and f(n) = A097246(n) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 03 2019

nonn

For all i, j: a(i) = a(j) => A322869(i) = A322869(j).

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

Cf. A097246, A305801, A322808, A322869.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A323078aux(n) = if((n>2)&&isprime(n),0,A097246(n));
v323078 = rgs_transform(vector(up_to,n,A323078aux(n)));
A323078(n) = v323078[n];

A097247 A097246(A097246(n)).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 5, 13, 14, 15, 5, 17, 10, 19, 15, 21, 22, 23, 10, 7, 26, 15, 21, 29, 30, 31, 10, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 7, 46, 47, 15, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 25, 61, 62, 35, 15, 65, 66, 67, 51, 69, 70, 71, 30, 73, 74

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

This sequence is not multiplicative, see example.

			m=20 and n=63 are coprime:
a(20) = A097246(A097246(2*2*5)) = A097246(3*5)=15,
a(63) = A097246(A097246(3*3*7)) = A097246(5*7)=35 and
a(20*63) = A097246(A097246(2*2*3*3*5*7)) = A097246(3*5*5*7) =
3*7*7 = 147 <> a(20)*a(63) = 15*35 = 525: the sequence is not multiplicative.

Cf. A097248.

A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

Original entry on oeis.org

2, 6, 27, 28, 84, 270, 496, 1053, 1120, 1488, 1625, 1638, 3360, 3780, 4875, 8128, 10530, 24384, 66960, 147420, 167400, 406224, 611226, 775000, 872960, 943250, 1097280, 1245699, 1255338, 1303533, 1464320, 1686400, 1740024, 1922375, 1952500, 2011625, 2193408, 2325000, 2611440, 2618880, 2829750, 2941029, 4392960

Offset: 1

Antti Karttunen, Feb 05 2020

nonn

Numbers k such that A097248(sigma(k)) is equal to A097248(2*k).
Numbers k such that A331750(k) is equal to 1+A048675(k), which in turn is equal to A048675(A225546(2*k)) = A048675(2*A225546(k)).
Among the first 60 terms, 15 are odd: 27, 1053, 1625, 4875, 1245699, 1303533, 1922375, 2011625, 2941029, 5767125, 6034875, 12733875, 17137575, 26316675, 29362905, and only 1053 = 3^4 * 13 is in A228058.
Note that the condition A090880(sigma(k)) == A090880(2*k) appears to be much more constrained.

			For n = 1053 = 3^4 * 13^1, A331750(1053) = A331750(81) + A331750(13) = 32+9 = 41, while A048675(2*1053) = A048675(2)+A048675(81)+A048675(13) = 1+8+32 = 41 also, thus 1053 is included in this sequence.
For n = 3360 = 2^5 * 3^1 * 5^1 * 7^1, A331750(3360) = A331750(32)+A331750(3)+A331750(5)+A331750(7) = 12+2+3+3 = 20, while A048675(2*3360) = A048675(2)+A048675(32)+A048675(3)+A048675(5)+A048675(7) = 1+5+2+4+8 = 20 also, thus 3360 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..60 (up to the term 33550336).
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A048675, A090880, A097246, A097248, A331750, A331752, A332208.

Cf. A000396 (a subsequence).

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i, 1]+1)^(f[i, 2]\2))*((f[i, 1])^(f[i, 2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
isA331751(n) = (A097248(2*n)==A097248(sigma(n)));

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A277886 If n is squarefree, a(n) = n, else a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 9, 13, 14, 15, 12, 17, 10, 19, 15, 21, 22, 23, 18, 7, 26, 15, 21, 29, 30, 31, 24, 33, 34, 35, 27, 37, 38, 39, 30, 41, 42, 43, 33, 25, 46, 47, 36, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 45, 61, 62, 35, 48, 65, 66, 67, 51, 69, 70, 71, 54, 73, 74, 21, 57, 77, 78, 79, 60, 45

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

If n has non-unitary prime divisors, then divide it by the square of the smallest of them and multiply by a single instance of the next larger prime.

This differs from related A097246 for the first time at n=16. For both sequences A097248 gives the eventual stable points reached when starting iterating from n.

			For n = 12 = 2*2*3, the smallest non-unitary prime divisor (and in this case the only one) is 2, thus we divide with 2^2 and multiply with the next larger prime 3, to get ((2^2 * 3)/(2^2))*3 = 3*3, thus a(12) = 9.
For n = 16 = 2^4, we divide two instances of 2 out and multiply by a single instance of 3 to get 2*2*3 = 12.

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

Cf. A000040, A048675, A097246, A097248, A249739, A277885, A277887, A277888, A277896.

Table[If[SquareFreeQ@ n, n, Prime[1 + PrimePi@ Min[Select[FactorInteger[n][[All, 1]], ! CoprimeQ[#, n/#] &] /. {} -> 0]] (n/If[SquareFreeQ@ n, 1, p = 2; While[! Divisible[n, p^2], p = NextPrime@ p]; p]^2)], {n, 81}] (* Michael De Vlieger, Nov 15 2016 *)

(define (A277886 n) (if (zero? (A277885 n)) n (* (A000040 (+ 1 (A277885 n))) (/ n (expt (A249739 n) 2)))))

If A277885(n) = 0 [when n is squarefree], then a(n) = n, otherwise a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).
Other identities. For all n >= 1:
A048675(a(n)) = A048675(n).

A277899 a(n) = A097249(A260443(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 2, 0, 1, 0, 0, 0, 3, 2, 3, 1, 3, 2, 2, 0, 2, 1, 2, 0, 1, 0, 0, 0, 4, 3, 4, 2, 4, 3, 3, 1, 3, 3, 3, 2, 3, 2, 2, 0, 3, 2, 3, 1, 3, 2, 2, 0, 2, 1, 2, 0, 1, 0, 0, 0, 5, 4, 5, 3, 5, 4, 4, 2, 4, 4, 4, 3, 4, 3, 4, 1, 4, 3, 4, 3, 4, 3, 4, 2, 4, 3, 4, 2, 3, 2, 2, 0, 4, 3, 4, 2, 4, 3, 3, 1, 3, 3, 3, 2, 3, 2, 3, 0, 3, 2, 3, 1, 3, 2, 2, 0

Offset: 0

Antti Karttunen, Nov 15 2016

nonn

a(n) = number of times we must iterate A097246, starting at A260443(n), before the result is squarefree.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A097246, A097249, A260443, A277905.

Cf. A023758 (positions of zeros).

(define (A277899 n) (A097249_for_coeff_list (A260443as_coeff_list n)))
(define (A097249_for_coeff_list nums) (let loop ((nums nums) (s 0)) (if (<= (reduce max 0 nums) 1) s (loop (A097246_for_coeff_list nums) (+ 1 s)))))
(define (A097246_for_coeff_list nums) (add_two_lists (map A000035 nums) (cons 0 (map A004526 nums))))
;; For the other required functions, see A260443.

a(n) = A097249(A260443(n)).

A283478 a(n) = A097248(A108951(n)).

Original entry on oeis.org

1, 2, 6, 3, 30, 5, 210, 6, 15, 7, 2310, 10, 30030, 11, 21, 5, 510510, 30, 9699690, 14, 33, 13, 223092870, 15, 105, 17, 14, 22, 6469693230, 42, 200560490130, 10, 39, 19, 165, 7, 7420738134810, 23, 51, 21, 304250263527210, 66, 13082761331670030, 26, 70, 29, 614889782588491410, 30, 1155, 210, 57, 34, 32589158477190044730, 21, 195, 33, 69, 31

Offset: 1

Antti Karttunen, Mar 16 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A019565, A097248, A108951, A283475.

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, #] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], {n, 58}] (* Michael De Vlieger, Mar 18 2017 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015
A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
A283478(n) = A097248(A108951(n));

from sympy import primerange, factorint, nextprime
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a097248(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k
def a(n): return a097248(a108951(n)) # Indranil Ghosh, May 15 2017

(define (A283478 n) (A097248 (A108951 n)))

a(n) = A097248(A108951(n)).
Other identities:
For all n >= 0, a(A019565(n)) = A283475(n).

cp's OEIS Frontend

A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

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A097249 a(n) is the number of times we must iterate A097246, starting at n, before the result is squarefree.

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A322808 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is a squarefree number > 2, and f(n) = A097246(n) for all other numbers.

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A323078 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd prime, and f(n) = A097246(n) for all other numbers.

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A097247 A097246(A097246(n)).

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A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

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A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A277886 If n is squarefree, a(n) = n, else a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).

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