A008477 -id:A008477 - cp's OEIS frontend

A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

1, 2, 4, 3, 8, 8, 16, 5, 9, 16, 32, 12, 64, 32, 32, 7, 128, 18, 256, 24, 64, 64, 512, 20, 27, 128, 25, 48, 1024, 64, 2048, 11, 128, 256, 128, 27, 4096, 512, 256, 40, 8192, 128, 16384, 96, 72, 1024, 32768, 28, 81, 54, 512, 192, 65536, 50, 256, 80, 1024, 2048

Offset: 1

Naohiro Nomoto

If the prime power factors p^e of n are replaced by prime(e)^pi(p), then the prime terms q in the sequence pertain to 2^m with m > 1, since pi(2) = 1. - Michael De Vlieger, Apr 25 2017

Also the Heinz number of the integer partition obtained by applying the map described in A217605 (which interchanges the parts with their multiplicities) to the integer partition with Heinz number n, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The image of this map (which is the union of this sequence) is A130091. - Gus Wiseman, May 04 2019

			For n=6, 6 = (2^1)*(3^1), a(6) = ([first prime]^pi(2))*([first prime]^pi(3)) = (2^1)*(2^2) = 8.
From _Gus Wiseman_, May 04 2019: (Start)
For n = 1..20, the prime indices of n together with the prime indices of a(n) are the following:
   1: {} {}
   2: {1} {1}
   3: {2} {1,1}
   4: {1,1} {2}
   5: {3} {1,1,1}
   6: {1,2} {1,1,1}
   7: {4} {1,1,1,1}
   8: {1,1,1} {3}
   9: {2,2} {2,2}
  10: {1,3} {1,1,1,1}
  11: {5} {1,1,1,1,1}
  12: {1,1,2} {1,1,2}
  13: {6} {1,1,1,1,1,1}
  14: {1,4} {1,1,1,1,1}
  15: {2,3} {1,1,1,1,1}
  16: {1,1,1,1} {4}
  17: {7} {1,1,1,1,1,1,1}
  18: {1,2,2} {1,2,2}
  19: {8} {1,1,1,1,1,1,1,1}
  20: {1,1,3} {1,1,1,2}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.

Cf. A008477, A048768, A048769, A064988.

Cf. A056239, A098859, A109298, A112798, A118914, A130091, A217605, A320348, A325326, A325337.

A048767 := proc(n)
    local a,p,e,f;
    a := 1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        e := op(2,f) ;
        a := a*ithprime(e)^numtheory[pi](p) ;
    end do:
    a ;
end proc: # R. J. Mathar, Nov 08 2012

Table[{p, k} = Transpose@ FactorInteger[n]; Times @@ (Prime[k]^PrimePi[p]), {n, 58}] (* Ivan Neretin, Jun 02 2016 *)
Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; e >= 0 :> Prime[e]^PrimePi[p]] &, 65] (* Michael De Vlieger, Apr 25 2017 *)

a(1)=1 prepended by Alois P. Heinz, Jul 26 2015

A122132 Squarefree numbers multiplied by binary powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Reinhard Zumkeller, Aug 21 2006

These numbers are called "oddly squarefree" in Banks and Luca. - Michel Marcus, Mar 14 2016

The asymptotic density of this sequence is 8/Pi^2 (A217739). - Amiram Eldar, Sep 21 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
William D. Banks and Florian Luca, Roughly squarefree values of the Euler and Carmichael functions, Acta Arithmetica, Vol. 120, No. 3 (2005), pp. 211-230.

Subsequences: A000079, A005117, A007283, A051916, A056911, A029747.
Complement: A038838.
Cf. A217739.

a122132 n = a122132_list !! (n-1)
a122132_list = filter ((== 1) . a008966 . a000265) [1..]
-- Reinhard Zumkeller, Jan 24 2012

Select[Range@ 85, SquareFreeQ[#/2^IntegerExponent[#, 2]] &] (* Michael De Vlieger, Mar 15 2020 *)

is(n)=issquarefree(n>>valuation(n,2)); \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A007947(a(n)) * A006519(a(n)) / (2 - a(n) mod 2);
A007947(a(n)) = A000265(a(n)) * (2 - a(n) mod 2).
A008966(A000265(a(n))) = 1. - Reinhard Zumkeller, Jan 24 2012
A010052(A008477(a(n))) = 1. - Reinhard Zumkeller, Feb 17 2012

A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

Original entry on oeis.org

1, 4, 16, 27, 72, 108, 432, 800, 3125, 6272, 12500, 21600, 30375, 50000, 84375, 121500, 169344, 225000, 247808, 337500, 486000, 750141, 823543, 1350000, 1384448, 3000564, 3294172, 6690816, 12002256, 13176688, 19600000, 22235661, 37380096, 37879808, 59295096, 88942644

Offset: 1

Olivier Gérard

nonn

Fixed points of A008477.

a(3) = 16 is the only term of the form p^q with p <> q. - Bernard Schott, Mar 28 2021

			16 = 2^4 = 4^2.
27 = 3^3.
108 = 2^2*3^3.
6272 = 2^7*7^2.
121500 = 2^2 * 3^5*5^3.

Cf. A000040, A008477.

Some subsequences: p_i^p_i (A051674), Product_i {p_i^p_i} (A048102), Product_(j,k)(p_j^p_k * p_k^p_j) with p_j < p_k (A082949) (see examples).

f[n_] := Product[{p, e} = pe; e^p, {pe, FactorInteger[n]}];
Reap[For[n = 1, n <= 10^8, n++, If[f[n] == n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 29 2021 *)

for(n=2,10^8,if(n==prod(i=1,omega(n), component(component(factor(n),2),i)^component(component(factor(n),1),i)),print1(n,",")))

More terms from David W. Wilson

a(34)-a(36) from Jean-François Alcover, Mar 29 2021

A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 0, 3, 10, 11, 3, 13, 14, 15, 0, 17, 6, 19, 5, 21, 22, 23, 0, 10, 26, 1, 7, 29, 30, 31, 0, 33, 34, 35, 3, 37, 38, 39, 0, 41, 42, 43, 11, 15, 46, 47, 0, 21, 20, 51, 13, 53, 2, 55, 0, 57, 58, 59, 15, 61, 62, 21, 0, 65, 66

Offset: 1

Peter Luschny, Jan 16 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Cf. A000026, A001414, A008473, A008474, A008475, A008476, A008477, A028310, A069799.

Cf. A027748, A124010, A007318, A328745.

a182938 n = product $ zipWith a007318'
   (a027748_row n) (map toInteger $ a124010_row n)
-- Reinhard Zumkeller, Feb 18 2012

A182938 := proc(n) local e,j; e := ifactors(n)[2]:
mul (binomial(e[j][1], e[j][2]), j=1..nops(e)) end:
seq (A182938(n), n=1..100);

a[n_] := Times @@ (Map[Binomial @@ # &, FactorInteger[n], 1]);
Table[a[n], {n, 1, 100}] (* Kellen Myers, Jan 16 2011 *)

a(n)=prod(i=1,#n=factor(n)~,binomial(n[1,i],n[2,i])) \\ M. F. Hasler

for(n=1, 100, print1(direuler(p=2, n, (1 + X)^p)[n], ", ")) \\ Vaclav Kotesovec, Mar 28 2025

a(A185359(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
Dirichlet g.f.: Product_{p prime} (1 + p^(-s))^p. - Ilya Gutkovskiy, Oct 26 2019
Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.33754... - Vaclav Kotesovec, Mar 28 2025

Given terms checked with new PARI code by M. F. Hasler, Jan 16 2011

A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

Original entry on oeis.org

1, 1, 1, 4, 1, 32, 1, 9, 8, 128, 1, 243, 1, 512, 256, 16, 1, 243, 1, 2187, 1024, 8192, 1, 1024, 32, 32768, 27, 19683, 1, 59049, 1, 25, 16384, 524288, 4096, 1024, 1, 2097152, 65536, 16384, 1, 531441, 1, 1594323, 6561, 33554432, 1, 3125, 128, 2187, 1048576, 14348907, 1, 1024, 65536

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

			a(48) = a(2^4 * 3^1) = (4 + 1)^(2 + 3) = 5^5 = 3125.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000142, A000312, A001222, A002110, A007504, A008472, A008474, A008477, A022559, A034387, A039697, A088865, A263653, A285769.

Join[{1}, Table[PrimeOmega[n]^DivisorSum[n, # &, PrimeQ[#] &], {n, 2, 55}]]

a(n) = my(f=factor(n)); vecsum(f[,2])^vecsum(f[,1]); \\ Michel Marcus, Apr 21 2018

a(n) = bigomega(n)^sopf(n) = A001222(n)^A008472(n).
a(p^k) = k^p where p is a prime.
a(A000312(k)) = a(k)*k^A008472(k).
a(A000142(k)) = A022559(k)^A034387(k).
a(A002110(k)) = k^A007504(k).

A303278 If n = Product_j p_j^k_j where the p_j are distinct primes then a(n) = (Product_j k_j)^(Product_j p_j).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 8, 1, 1, 64, 1, 1, 1, 16, 1, 64, 1, 1024, 1, 1, 1, 729, 32, 1, 27, 16384, 1, 1, 1, 25, 1, 1, 1, 4096, 1, 1, 1, 59049, 1, 1, 1, 4194304, 32768, 1, 1, 4096, 128, 1024, 1, 67108864, 1, 729, 1, 4782969, 1, 1, 1, 1073741824, 1, 1, 2097152, 36, 1, 1, 1, 17179869184, 1, 1, 1, 46656, 1, 1, 32768

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

This is different from A008477, which is Product_j k_j^p_j. - N. J. A. Sloane, May 01 2021

			a(36) = a(2^2 * 3^2) = (2*2)^(2*3) = 4^6 = 4096.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000026, A000142, A005117 (indices of ones), A005361, A007947, A008477, A034386, A039696, A135291, A285769, A303277.

Table[Times@@Transpose[FactorInteger[n]][[2]]^Last[Select[Divisors[n], SquareFreeQ]], {n, 75}]

a(n) = my(f=factor(n)); factorback(f[, 2])^factorback(f[, 1]); \\ Michel Marcus, Apr 21 2018

a(n) = tau(n/rad(n))^rad(n) = A005361(n)^A007947(n).
a(p^k) = k^p where p is a prime.
a(A000142(k)) = A135291(k)^A034386(k).

Definition clarified by N. J. A. Sloane, May 01 2021

A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 6, 10, 11, 6, 13, 14, 15, 4, 17, 12, 19, 10, 21, 22, 23, 18, 10, 26, 3, 14, 29, 30, 31, 10, 33, 34, 35, 12, 37, 38, 39, 30, 41, 42, 43, 22, 30, 46, 47, 12, 14, 20, 51, 26, 53, 6, 55, 42, 57, 58, 59, 30, 61, 62, 42, 6, 65, 66, 67, 34, 69, 70, 71

Offset: 1

Paolo Xausa, Feb 28 2025

			a(18) = 12 because 18 = 2^1*3^2, lcm(2,1) = 2, lcm(3,2) = 6 and 2*6 = 12.
a(300) = 30 because 300 = 2^2*3^1*5^2, lcm(2,2) = 2, lcm(3,1) = 3, lcm(5,2) = 10 and 2*3*10 = 60.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008473, A008477, A035306, A144338 (fixed points), A369008 (analogous for gcd).

A381588[n_] := Times @@ LCM @@@ FactorInteger[n];
Array[A381588, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, lcm(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

a(p) = p, for p prime.

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 *)

A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).

Original entry on oeis.org

1, 4, 9, 8, 25, 36, 49, 32, 27, 100, 121, 72, 169, 196, 225, 128, 289, 108, 361, 200, 441, 484, 529, 288, 125, 676, 243, 392, 841, 900, 961, 2048, 1089, 1156, 1225, 216, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 675, 2116, 2209, 1152, 343, 500, 2601, 1352, 2809, 972, 3025

Offset: 1

Ilya Gutkovskiy, May 09 2018

			a(12) = a(2^2*3^1) = 2^prime(2)*3^prime(1) = 2^3*3^2 = 72.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000040, A001248, A002110, A003963, A008477, A030078, A048767, A050997, A056166 (records), A061742, A181819.

Cf. A064988 (apply prime to p), A321874 (apply prime to both p & e).

a:= n-> mul(i[1]^ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 20 2021

a[n_] := Times @@ (#[[1]]^Prime[#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 55}]

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

apply( A304203(n)=factorback((n=factor(n))[,1],apply(prime,n[,2])), [1..50]) \\ M. F. Hasler, Nov 20 2018

a(prime(i)^k) = prime(i)^prime(k).
a(A000040(k)) = A001248(k).
a(A001248(k)) = A030078(k).
a(A030078(k)) = A050997(k).
a(A002110(k)) = A061742(k).
Multiplicative with a(p^e) = p^prime(e). - M. F. Hasler, Nov 20 2018
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^prime(k)) = 1.80728269690724154161... . - Amiram Eldar, Jan 20 2024

A381613 If n = Product (p_j^k_j) then a(n) = Product (min(p_j, k_j)), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1

Offset: 1

Paolo Xausa, Mar 01 2025

First differs from A323308 at n = 27.

			a(18) = 2 because 18 = 2^1*3^2, min(2,1) = 1, min(3,2) = 2 and 1*2 = 2.
a(300) = 4 because 300 = 2^2*3^1*5^2, min(2,2) = 2, min(3,1) = 1, min(5,2) = 2 and 2*1*2 = 4.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008473, A008477, A035306, A323308, A369008, A381588, A381614.

A381613[n_] := Times @@ Min @@@ FactorInteger[n];
Array[A381613, 100]

a(n) = my(f=factor(n)); prod(i=1, #f~, min(f[i,1], f[i,2])); \\ Michel Marcus, Mar 02 2025

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (1/p - 1/p^p)/(p-1)) = 1.59383299054679951264... . - Amiram Eldar, Mar 07 2025

cp's OEIS Frontend

A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

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A122132 Squarefree numbers multiplied by binary powers.

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A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

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A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

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A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

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A303278 If n = Product_j p_j^k_j where the p_j are distinct primes then a(n) = (Product_j k_j)^(Product_j p_j).

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A381588 If n = Product (p_j^k_j) then a(n) = Product (lcm(p_j, k_j)), with a(1) = 1.

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