A008478 -id:A008478 - cp's OEIS frontend

A062307 Numbers n such that n = A008477(A008477(n)) and n is not A008478.

8, 9, 25, 32, 49, 121, 125, 128, 169, 200, 243, 288, 289, 343, 361, 392, 500, 529, 675, 841, 864, 961, 968, 972, 1125, 1152, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1849, 1944, 2000, 2048, 2187, 2197, 2209, 2312, 2809, 2888, 3087, 3200, 3267, 3456, 3481

Offset: 1

Naohiro Nomoto, Mar 28 2002

From Bernard Schott, Mar 29 2021: (Start)
If m is a term, then A008477(m) = q is another term and A008477(q) = m.
The first such pairs (m, q) in lexicographic order are (8, 9), (25, 32), (49, 128), (121, 2048), (125, 243), (169, 8192), (200, 288), (289, 131072), ...
If f = A008477, terms k of this sequence are precisely the ones for which the sequence k, f(k), f(f(k)), f(f(f(k))), ... is periodic with period = 2 (see 1st comment in A008477); example for k = 8, this periodic sequence is 8, 9, 8, 9, 8, 9, ...
Prime powers p^r, p, r primes, p <> r are terms. (End)

			8 = 2^3, A008477(8) = 3^2 = 9 and A008477(9) = 2^3 = 8, so 8 and 9 are terms.
200 = 2^3*5^2, A008477(200) = 3^2*2^5 = 288 and A008477(288) = 2^3*5^2 = 200, so 200 and 288 are terms.

Michel Marcus, Table of n, a(n) for n = 1..4336

A114130 is a subsequence.

f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477
isok(m) = my(nm = f(m)); (nm != m) && (f(nm) == m); \\ Michel Marcus, Mar 29 2021

A048768 Numbers n such that A048767(n) = n.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 112, 125, 250, 352, 360, 675, 832, 1008, 1125, 1350, 1500, 2176, 2250, 2401, 3168, 3969, 4802, 4864, 7488, 7938, 11776, 14000, 19584, 21609, 28812, 29403, 29696, 43218, 43776, 44000, 58806, 63488, 75600, 96040, 104000, 105984, 123201, 126000

Offset: 1

Naohiro Nomoto

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions that are fixed points under the map described in A217605 (which interchanges the parts with their multiplicities). The enumeration of these partitions by sum is given by A217605. - Gus Wiseman, May 04 2019

			12 = (2^2)*(3^1) = (2nd prime)^pi(2) * (first prime)^pi(3).
From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   125: {3,3,3}
   250: {1,3,3,3}
   352: {1,1,1,1,1,5}
   360: {1,1,1,2,2,3}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
  1008: {1,1,1,1,2,2,4}
  1125: {2,2,3,3,3}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  2176: {1,1,1,1,1,1,1,7}
  2250: {1,2,2,3,3,3}
  2401: {4,4,4,4}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..128

A subsequence of A130091.

Cf. A008478, A048767, A048769, A056239, A098859, A109297, A109298, A112798, A118914, A217605, A325326, A325368.

wt[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];
Select[Range[1000],wt[#]==#&] (* Gus Wiseman, May 04 2019 *)

is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); #Set(e) == #e && prod(i = 1, #e, prime(e[i])^primepi(p[i])) == n;} \\ Amiram Eldar, Oct 20 2023

a(1) inserted and more terms added by Amiram Eldar, Oct 20 2023

A008477 If n = Product (p_j^k_j) then a(n) = Product (k_j^p_j).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 8, 1, 1, 4, 1, 1, 1, 16, 1, 8, 1, 4, 1, 1, 1, 9, 32, 1, 27, 4, 1, 1, 1, 25, 1, 1, 1, 32, 1, 1, 1, 9, 1, 1, 1, 4, 8, 1, 1, 16, 128, 32, 1, 4, 1, 27, 1, 9, 1, 1, 1, 4, 1, 1, 8, 36, 1, 1, 1, 4, 1, 1, 1, 72, 1

Offset: 1

Olivier Gérard

For any n, the sequence n, a(n), a(a(n)), a(a(a(n))), ... is eventually periodic with period <= 2 [Farrokhi]. - N. J. A. Sloane, Apr 25 2009
a(A005117(n)) = 1; a(A013929(n)) > 1; A010052(a(A122132(n))) = 1. - Reinhard Zumkeller, Feb 17 2012
From Bernard Schott, Mar 26 2021: (Start)
The study of some properties of this sequence was proposed in the 1st problem of Concours Général in 2012 in France (see links).
Terms are precisely the powerful numbers in A001694.
If m is a term, there is a term q such that a(q) = m.
a(a(n)) <= n (see examples). (End)

			For n = 24 = 2^3*3^1, a(24) = 3^2*1^3 = 9, so a(9) = 2^3 = 8 and a(a(24)) = 8 < 24.
For n = 243 = 3^5, a(243) = 5^3 = 125, so a(125) = 3^5 = 243 and a(a(243)) = 243.

M. F. Hasler, Table of n, a(n) for n = 1..10000
Annales Concours Général, Sujet Concours Général 2012 (in French, problems).
Annales Concours Général, Corrigé Concours Général 2012 (in French, solutions).
M. Farrokhi, The Prime Exponentiation of an Integer: Problem 11315, Amer. Math. Monthly, 116 (2009), 470. - from _N. J. A. Sloane_, Apr 25 2009
Index to sequences related to Olympiads and other Mathematical Competitions.

Cf. A001694, A027748, A124010.
Cf. A005117, A013929, A010052, A062307, A122132, A342551.
Cf. A008478 (fixed points).

a008477 n = product $ zipWith (^) (a124010_row n) (a027748_row n)
-- Reinhard Zumkeller, Feb 17 2012

A008477 := proc(n) local e,j; e := ifactors(n)[2]:
mul (e[j][2]^e[j][1], j=1..nops(e)) end:
seq (A008477(n), n=1..60);
# Peter Luschny, Jan 17 2010

Prepend[ Array[ Times @@ Map[ Power @@ RotateLeft[ #1, 1 ]&, FactorInteger[ # ] ]&, 100, 2 ], 1 ]
Table[Times@@(First[#]^Last[#]&/@Transpose[Reverse[ Transpose[ FactorInteger[ n]]]]),{n,80}] (* Harvey P. Dale, Jul 22 2014 *)

A008477(n)=factorback(factor(n)*[0,1;1,0]) \\ M. F. Hasler, May 20 2012

from sympy import factorint, prod
a = lambda n: prod([pk[1]**pk[0] for pk in factorint(n).items()])
print([a(n) for n in range(1,61)]) # Darío Clavijo, Nov 06 2023
(APL, Dyalog dialect) A008477 ← {×/{⍺*⍨≢⍵}⌸factors(⍵)} ⍝ Needs also factors function from https://dfns.dyalog.com/c_factors.htm - Antti Karttunen, Feb 16 2024

Multiplicative with a(p^e) = e^p. - David W. Wilson, Aug 01 2001

A080353 a(1)=5; for n>1, a(n)=a(n-1)+1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

Original entry on oeis.org

5, 7, 9, 11, 12, 14, 15, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89

Offset: 1

N. J. A. Sloane, Mar 20 2003

nonn

Conjecture: This sequence is equivalent to N \ A008478. - Michael De Vlieger, Jul 02 2025

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 13.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A080036, A080037. Differences give A080354.

a[1] = 5; a[n_] := a[n] = If[MemberQ[Array[a, n-1], n], a[n-1]+1, a[n-1]+2]; Array[a, 67] (* Jean-François Alcover, Oct 08 2018 *)

a(n) = n + floor(sqrt(6*n)) + O(1).

A342973 Let f = A008477; nonsquarefree numbers m such that f(m) <> m or f(f(m)) <> m.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192

Offset: 1

Bernard Schott, Apr 01 2021

nonn

Equivalently, with f = A008477, terms m of this sequence are precisely the nonsquarefree numbers for which the iterated sequence {m, f(m), f(f(m)), f(f(f(m))), ... } is not periodic.
The first sixteen terms are the same as A126706, then a(17) = 64 while A126706(17) = 68.
There exist only these 4 possibilities:
-> for every squarefree number m in A005117, f(m) = 1, and iterated sequence is for example:  (3, 1, 1, 1, 1, ...).
-> For m nonsquarefree fixed point of f in A008478, f(m) = m, iterated sequence has period = 1, as for example: (4, 4, 4, 4, 4, ...).
-> For m nonsquarefree in A062307, f(m) = q and f(q) = m, iterated sequence has period = 2, as for example: (8, 9, 8, 9, 8, 9, ...).
-> For m in this sequence, f(m) = k and m, k belong to an infinite iterated sequence, as for example: (..., 196, 512, 81, 64, ...) (see example).

			196 = 2^2*7^2 => A008477(196) = 2^2*2^7 = 2^9 = 512.
512 = 2^9 => A008477(512) = 9^2 = 81.
81 = 9^2 = 3^4 => A008477(81) = 4^3 = 64.
196, 512, 81, 64 are not terms of (A008478 U A062307), so they belong to this sequence.

Equals A013929 \ {A008478 U A062307}.

Cf. A008477, A342551.

fun[p_, e_] := e^p; f[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[200], !SquareFreeQ[#] && f[#] != # && f[f[#]] != # &] (* Amiram Eldar, Apr 01 2021 *)

f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477
isok(m) = if (!issquarefree(m), my(mm=f(m)); (mm != m) && (f(mm) != m)); \\ Michel Marcus, Apr 02 2021

A159836 Integers n such that the orbit n, f(n), f(f(n)), ... is eventually periodic with period 2, where f(n) = product(a(k)^p(k)) when n has the prime factorization n = product(p(k)^a(k)).

Original entry on oeis.org

8, 9, 18, 24, 25, 32, 36, 40, 45, 49, 50, 56, 63, 64, 75, 81, 88, 90, 96, 98, 99, 100, 104, 117, 120, 121, 125, 126, 128, 136, 144, 147, 150, 152, 153, 160, 162, 168, 169, 171, 175, 180, 184, 192, 196, 198, 200, 207, 216, 224, 225, 232, 234, 242, 243, 245, 248

Offset: 1

John W. Layman, Apr 23 2009

nonn

It is proved in the reference that for every positive integer n the orbit n, f(n), f(f(n)), ... is eventually periodic with period 1 or 2.

Includes all numbers whose prime exponents are distinct primes. If n is in this sequence and k is a squarefree number such that (k,n) = 1, then k*n is in this sequence. - Charlie Neder, May 16 2019

Paolo P. Lava, Table of n, a(n) for n = 1..5000
M. Farrokhi, The Prime Exponentiation of an Integer: Problem 11315, Amer. Math. Monthly, 116 (2009), 470.

Cf. A008477, A008478.

with(numtheory); P:=proc(q) local a0f,a1,a1f,a2,a2f,a3,a3f,a4,a4f,k,n;
for n from 1 to q do a0:=1;a1:=1;a2:=2;a3:=3;a4:=n;
  while not (a1=a3 and a2=a4) do a0f:=ifactors(a4)[2];
   a1:=mul(a0f[k][2]^a0f[k][1],k=1..nops(a0f)); a1f:=ifactors(a1)[2];
   a2:=mul(a1f[k][2]^a1f[k][1],k=1..nops(a1f)); a2f:=ifactors(a2)[2];
   a3:=mul(a2f[k][2]^a2f[k][1],k=1..nops(a2f)); a3f:=ifactors(a3)[2];
   a4:=mul(a3f[k][2]^a3f[k][1],k=1..nops(a3f)); od;
if a1<>a2 then print(n); fi; od; end: P(10^6); # Paolo P. Lava, Oct 24 2013

f[n_] := Module[{f = Transpose[FactorInteger[n]]}, Times @@ (f[[2]]^f[[1]])]; Select[Range[300], (x = NestWhileList[f, #, UnsameQ, All]; x[[-2]] != x[[-1]]) &] (* T. D. Noe, Oct 24 2013 *)

A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

Original entry on oeis.org

1, 8, 9, 72, 13440, 21120, 24960, 29568, 32640, 34944, 36480, 44160, 45696, 49280, 51072, 54912, 55680, 58240, 59520, 61824, 71040, 71808, 76160, 77952, 78720, 80256, 82560, 83328, 84864, 85120, 90240, 91520, 94848, 97152, 99456, 101760, 103040, 110208, 113280, 114816, 115584, 117120, 119680

Offset: 1

Ilya Gutkovskiy, May 12 2018

nonn

Numbers k such that A000005(k)*A007947(k) = k.
Fixed points of A304409.
All terms are refactorable numbers (A033950).

			13440 is a term because 13440 = 2^7*3*5*7 = 2*(7 + 1) * 3*(1 + 1) * 5*(1 + 1) * 7*(1 + 1).

Cf. A000005, A000026, A007947, A008478, A033950, A036762, A048768, A078779, A190473, A304409.

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Select[Range[120000], a[#] == # &]

isok(k) = {my(f = factor(k)); numdiv(f) * vecprod(f[, 1]) == k;} \\ Amiram Eldar, Jan 31 2025

A342551 a(n) is the smallest m such that A008477(m) is the n-th powerful number (A001694).

Original entry on oeis.org

1, 4, 9, 8, 16, 32, 27, 25, 64, 128, 81, 72, 512, 1024, 108, 2048, 243, 49, 4096, 8192, 16384, 288, 729, 32768, 125, 225, 200, 131072, 262144, 2187, 524288, 1152, 1048576, 432, 2097152, 4194304, 972, 196, 8388608, 648, 33554432, 4608, 864, 67108864, 19683, 268435456

Offset: 1

Bernard Schott, Mar 27 2021

nonn

As A008477 is not injective and terms A008477(n) are precisely the powerful numbers, this sequence lists the smallest preimage of each powerful number.
There are these three possibilities (see corresponding examples):
-> If A008477(q) = q is a fixed point in A008478 and if q = A001694(u) then a(u) = q.
-> If k and m are in A062307 and satisfy A008477(k) = m  and A008477(m) = k, if m = A001694(s) and k = A001694(t), then a(t) = m and a(s) = k;
-> If A008477(j) = v where v is a powerful number not in {A008478 U A062307} and j is the smallest preimage of v with v = A001694(z) then a(z) = j.

			-> A008477(16) = 16 is a fixed point and 16 is the 5th powerful number, so a(5) = 16.
-> 25 and 32 are in A062307 and satisfy A008477(25) = 32 and A008477(32) = 25, as 25 = A001694(6) and 32 = A001694(8), so a(6) = 32 and a(8) = 25.
-> A008477(81) = A008477(256) = 64 that is the 11th powerful number, since 81 is the smallest preimage of 64, so a(11) = 81.

Cf. A001694, A008477, A008478, A062307.

pwf(n) = my(k=1, nb=1); while (nb != n, k++; if (ispowerful(k), nb++)); k; \\ A001694
f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477
a(n) = my(k=1, p=pwf(n)); while (f(k) != p, k++); k; \\ Michel Marcus, Mar 28 2021

More terms from Amiram Eldar, Mar 27 2021

A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

Original entry on oeis.org

1, 2, 12, 56, 180, 304, 336, 936, 1696, 1824, 2484, 5040, 5328, 6664, 8384, 8512, 9900, 10176, 13176, 14040, 25632, 26208, 27360, 33372, 33712, 37260, 39808, 39984, 47488, 50304, 51072, 52200, 65232, 69552, 79920, 126900, 128448, 142272, 149184, 152640, 162648, 167776, 184064, 193752, 197640

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

Numbers k such that A007947(k)*A156061(k) = k or A156061(k) = A003557(k).

			9900 is a term because 9900 = 2^2 * 3^2 * 5^2 * 11 = prime(1)^2 * prime(2)^2 * prime(3)^2 * prime(5) = 1*prime(1) * 2*prime(2) * 3*prime(3) * 5*prime(5).

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A003557, A005117, A007947, A008478, A033286, A046022, A048768, A078779, A109298, A156061.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); a[1] = 1; Select[Range[200000], a[#] == # &]

isok(n) = {my(f=factor(n)); prod(k=1, #f~, primepi(f[k,1])*f[k,1]) == n;} \\ Michel Marcus, May 08 2018

A304253 Numbers k such that k = Product (p_j^e_j) = Sum (prime(p_j)^e_j).

Original entry on oeis.org

20, 68, 76, 92, 8248

Offset: 1

Ilya Gutkovskiy, May 09 2018

Fixed points of A304251.

			68 is a term because 68 = 2^2*17 = prime(1)^2*prime(7) = prime(prime(1))^2 + prime(prime(7)).
8248 is a term because 8248 = 2^3*1031 = prime(1)^3*prime(173) = prime(prime(1))^3 + prime(prime(173)).

Index entries for sequences computed from indices in prime factorization

Cf. A006450, A008478, A048768, A304194, A304251.

a[n_] := Plus @@ (Prime[#[[1]]]^#[[2]] & /@ FactorInteger[n]); Select[Range[10000], a[#] == # &]

isok(n) = my(f=factor(n)); n == sum(k=1, #f~, prime(f[k,1])^f[k,2]); \\ Michel Marcus, May 09 2018

cp's OEIS Frontend

A062307 Numbers n such that n = A008477(A008477(n)) and n is not A008478.

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A048768 Numbers n such that A048767(n) = n.

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A008477 If n = Product (p_j^k_j) then a(n) = Product (k_j^p_j).

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A080353 a(1)=5; for n>1, a(n)=a(n-1)+1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

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A342973 Let f = A008477; nonsquarefree numbers m such that f(m) <> m or f(f(m)) <> m.

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A159836 Integers n such that the orbit n, f(n), f(f(n)), ... is eventually periodic with period 2, where f(n) = product(a(k)^p(k)) when n has the prime factorization n = product(p(k)^a(k)).

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A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

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A342551 a(n) is the smallest m such that A008477(m) is the n-th powerful number (A001694).