A008473 -id:A008473 - cp's OEIS frontend

A039779 Numbers k such that phi(k) is equal to A008473(k).

1, 54, 99, 112, 540, 3344, 4743, 6720, 7644, 8307, 11088, 12852, 15225, 20300, 22320, 83160, 86304, 94944, 129504, 160208, 186992, 200640, 205712, 207264, 266266, 280592, 331731, 364941, 383724, 404550, 441232, 445050, 447876, 449072, 454575, 458052, 497781

Offset: 1

Olivier Gérard

nonn

			phi(54) = 18, 54 = 2^1*3^3, (2+1)*(3+3) = 18.

Amiram Eldar, Table of n, a(n) for n = 1..866 (terms below 10^10)

Cf. A000010, A008473.

Reap[For[n = 1, n < 500000, n++, If[EulerPhi[n] == Times @@ Plus @@@ FactorInteger[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, May 06 2017 *)

is(k) = {my(f = factor(k)); eulerphi(f) == prod(i = 1, #f~, f[i, 1] + f[i, 2]);} \\ Amiram Eldar, Dec 04 2024

A039780 Numbers k such that phi(k) is equal to A008473(k-1).

Original entry on oeis.org

5, 25, 1176, 1625, 7385, 18633, 21726, 30276, 32923, 36905, 46025, 50809, 55449, 60726, 89257, 102425, 151657, 185745, 278226, 289961, 301017, 319145, 373176, 394121, 459513, 779817, 815676, 827926, 1053019, 1055719, 1170153, 1399123, 1657865, 1663209, 1667326

Offset: 1

Olivier Gérard

nonn

			phi(1176)=336, 1175=5^2*47^1, (5+2)*(47+1)=336.

Cf. A000010, A008473.

Cf. A039779, A039781.

b(n) = my(f = factor(n)); for (k=1, #f~, f[k, 1] = f[k, 1] + f[k, 2]; f[k, 2] = 1; ); factorback(f); \\ A008473
isok(k) = (k>1) && (eulerphi(k) == b(k-1)); \\ Michel Marcus, Feb 25 2021

Title corrected and more terms from Sean A. Irvine, Feb 24 2021

A039781 Numbers k such that phi(k) is equal to A008473(k+1).

Original entry on oeis.org

55, 174, 183, 341, 407, 1274, 5424, 6887, 18903, 22167, 27559, 53847, 66711, 68237, 77957, 78155, 91524, 132791, 133574, 138471, 149435, 191575, 220759, 274224, 339024, 413424, 432233, 493724, 505735, 543221, 684167, 694823, 703824, 711774, 747175, 883463

Offset: 1

Olivier Gérard

nonn

			phi(174)=56, 175=5^2*7^1, (5+2)*(7+1)=56.

Cf. A000010, A008473.

Cf. A039779, A039780.

b(n)=my(f = factor(n)); for (k=1, #f~, f[k, 1] = f[k, 1] + f[k, 2]; f[k, 2] = 1; ); factorback(f); \\ A008473
isok(k) = eulerphi(k) == b(k+1); \\ Michel Marcus, Feb 25 2021

Title corrected and more terms from Sean A. Irvine, Feb 24 2021

A376299 Fixed points of A008473.

Original entry on oeis.org

1, 4, 90, 120

Offset: 1

Darío Clavijo, Sep 19 2024

These are the numbers such that the sum of the powerfree parts of the divisors of n equals n.
These fixed points appeared in A299352 and the linked Combo Class video. The former notes there are no more fixed points <= 10^8. - Michael S. Branicky, Sep 19 2024
Any further terms are > 10^11. - Lucas A. Brown, Oct 19 2024

Lucas A. Brown, Python program.
Combo Class, The Mysterious Pattern I Found Within Prime Factorizations, Sep 12 2024.

Cf. A008473, A299352.

from sympy import factorint, prod
A008473 = lambda n: prod(sum(pk) for pk in factorint(n).items())
isok = lambda n: A008473(n) == n
print([n for n in range(1, 10**6) if isok(n)])

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

A203908 Multiplicative with a(p^e) = abs(p-e).

Original entry on oeis.org

1, 1, 2, 0, 4, 2, 6, 1, 1, 4, 10, 0, 12, 6, 8, 2, 16, 1, 18, 0, 12, 10, 22, 2, 3, 12, 0, 0, 28, 8, 30, 3, 20, 16, 24, 0, 36, 18, 24, 4, 40, 12, 42, 0, 4, 22, 46, 4, 5, 3, 32, 0, 52, 0, 40, 6, 36, 28, 58, 0, 60, 30, 6, 4, 48, 20, 66, 0, 44, 24, 70, 1, 72, 36

Offset: 1

José María Grau Ribas, Jan 07 2012

Density of nonzero terms is 0.85317570460439... = Product(1 - p^-p + p^-(p+1)) where p runs over the primes. - Charles R Greathouse IV, Jan 23 2012 [corrected by Amiram Eldar, Jan 14 2023]

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. A100717 (n such that a(n)=0).

Cf. A008473, A027748, A124010.

a203908 n = product $ map abs $
            zipWith (-) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Dec 24 2013

ar[p_,s_] := Abs[p-s]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100] (* José María Grau Ribas, Jan 25 2012 *)

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} ((1 - 1/p^5 + 2/p^4 + 2/p^3 - 4/p^2)*(1 - p - 3*p^2 + p^3 + p^4 + 2*p^(2-2*p))/(1 - p - 3*p^2 + p^3 + p^4)) = 0.2228124152... . - Amiram Eldar, Jan 14 2023

A299352 For x=n, iterate the map x -> Product_{k is a prime dividing x} (k + (multiplicity of k)), a(n) is the number of steps to see a repeated term for the first time.

Original entry on oeis.org

3, 2, 1, 4, 3, 6, 5, 5, 5, 4, 3, 4, 3, 3, 3, 5, 4, 3, 2, 8, 4, 3, 2, 7, 11, 4, 8, 10, 9, 8, 7, 4, 6, 4, 3, 12, 11, 12, 10, 11, 10, 5, 4, 10, 9, 4, 3, 6, 9, 9, 12, 6, 5, 9, 11, 5, 2, 11, 10, 11, 10, 11, 6, 8, 11, 10, 9, 10, 11, 9, 8, 14, 13, 9, 5, 10, 13, 5, 4

Offset: 2

Lars Blomberg, Feb 07 2018

nonn

It appears that all n end in one of the orbits (6,12,16) or (20,24) or one of the fixed points 4, 90, 120, verified to n=10^8.

			For n=2: 2=2^1 -> (2+1)=3=3^1 -> (3+1)=4=2^2 -> (2+2)=4; 4 is repeated so a(2)=3.
For n=12: 12=2^2*3^1 -> (2+2)*(3+1)=16=2^4 -> (2+4)=6=2^1*3^1 -> (2+1)*(3+1)=12; 12 is repeated so a(12)=3.

Lars Blomberg, Table of n, a(n) for n = 2..10000

Cf. A008473 (the map), A299351.

A322965 Numerator of Sum_{d | n} 1/rad(d).

Original entry on oeis.org

1, 3, 4, 2, 6, 2, 8, 5, 5, 9, 12, 8, 14, 12, 8, 3, 18, 5, 20, 12, 32, 18, 24, 10, 7, 21, 2, 16, 30, 12, 32, 7, 16, 27, 48, 10, 38, 30, 56, 3, 42, 16, 44, 24, 2, 36, 48, 4, 9, 21, 24, 28, 54, 3, 72, 20, 80, 45, 60, 16, 62, 48, 40, 4, 84, 24, 68, 36, 32, 72, 72, 25, 74, 57, 28

Offset: 1

David S. Metzler, Dec 31 2018

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. The sequence a(n) lists the numerators of the fractions f(n) in lowest terms.

If p is prime, then a(p^k) = p+k if p does not divide k, 1 + k/p if it does. In particular, a(p^p) = 2. - Robert Israel, Jan 25 2019

			The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3 and a(12) = 8. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007947 (radical), A322966 (denominators), A008473 (unreduced numerators, i.e., f(n)*rad(n)), A082695.

Numbers n where f(n) increases to a record: A322447.

rad:= n -> convert(numtheory:-factorset(n),`*`):
f:= proc(n) numer(add(1/rad(d),d=numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 25 2019

Array[Numerator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]]] &] &, 71] (* Michael De Vlieger, Jan 19 2019 *)

rad(n) = factorback(factor(n)[, 1]); \\ A007947
a(n) = numerator(sumdiv(n, d, 1/rad(d))); \\ Michel Marcus, Jan 10 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A322966(k) = zeta(2)*zeta(3)/zeta(6) (A082695). - Amiram Eldar, Dec 09 2023

More terms from Michel Marcus, Jan 19 2019

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.

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

A039779 Numbers k such that phi(k) is equal to A008473(k).

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A039780 Numbers k such that phi(k) is equal to A008473(k-1).

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A039781 Numbers k such that phi(k) is equal to A008473(k+1).

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A376299 Fixed points of A008473.

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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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A203908 Multiplicative with a(p^e) = abs(p-e).

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A299352 For x=n, iterate the map x -> Product_{k is a prime dividing x} (k + (multiplicity of k)), a(n) is the number of steps to see a repeated term for the first time.

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A322965 Numerator of Sum_{d | n} 1/rad(d).

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