A067240 -id:A067240 - cp's OEIS frontend

A374066 a(n) is the number of terms in the trajectory when the map x -> A067240(x) is iterated, starting from x = n until x = 0.

2, 3, 4, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 7, 6, 6, 7, 7, 8, 6, 6, 8, 9, 6, 7, 7, 8, 6, 7, 7, 8, 7, 6, 8, 7, 6, 7, 9, 8, 6, 7, 7, 8, 6, 7, 10, 11, 7, 8, 7, 8, 8, 9, 9, 8, 7, 7, 8, 9, 6, 7, 9, 6, 8, 7, 7, 8, 8, 7, 8, 9, 7, 8, 8, 9, 7, 7, 7, 8, 6, 10, 8, 9, 7, 7, 9, 8, 8, 9

Offset: 1

Rafik Khalfi, Jun 27 2024

nonn

			For n=11, the trajectory from n down to 0 comprises a(11) = 7 terms: 11 -> 10 -> 5 -> 4 -> 2 -> 1 -> 0.

Cf. A000010, A067240.

f := proc(n)
    local e, j:
    e := ifactors(n)[2]:
    add((e[j][1] - 1) * e[j][1]^(e[j][2] - 1), j = 1 .. nops(e))
end proc:
 A374066:= proc(n)
    local count, current:
    count := 1:
    current := n:
    while current <> 0 do
        current := f(current):
        count := count + 1
    end do:
    return count
end proc:
map(A374066, [$1..200]);

f[p_, e_] := (p - 1)*p^(e - 1); s[n_] := s[n] = Plus @@ f @@@ FactorInteger[n]; a[n_] := Length[NestWhileList[s, n, # > 0 &]]; Array[a, 100] (* Amiram Eldar, Jun 27 2024 *)

A080737 a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.

Original entry on oeis.org

0, 0, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 6, 8, 16, 6, 18, 6, 8, 10, 22, 6, 20, 12, 18, 8, 28, 6, 30, 16, 12, 16, 10, 8, 36, 18, 14, 8, 40, 8, 42, 12, 10, 22, 46, 10, 42, 20, 18, 14, 52, 18, 14, 10, 20, 28, 58, 8, 60, 30, 12, 32, 16, 12, 66, 18, 24, 10, 70, 10, 72, 36, 22, 20, 16, 14

Offset: 1

N. J. A. Sloane, Mar 08 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.
Savinien Kreczman, Luca Prigioniero, Eric Rowland, and Manon Stipulanti, Magic numbers in periodic sequences, Univ. Liège (Belgium, 2023). See p. 7.

Cf. A080736, A080738, A080739, A080740, A067240, A000010, A141809.

See A152455 for another version.

a080737 n = a080737_list !! (n-1)
a080737_list = 0 : (map f [2..]) where
f n | mod n 4 == 2 = a080737 $ div n 2
| otherwise = a067240 n
-- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012

a[1] = a[2] = 0; a[p_?PrimeQ] := a[p] = p-1; a[n_] := a[n] = If[Length[fi = FactorInteger[n]] == 1, EulerPhi[n], Total[a /@ (fi[[All, 1]]^fi[[All, 2]])]]; Table[a[n], {n, 1, 78}] (* Jean-François Alcover, Jun 20 2012 *)

for(n=1,78,k=0; if(n>1,f=factor(n); k=sum(j=1,matsize(f)[1],eulerphi(f[j,1]^f[j,2])); if(f[1,1]==2&&f[1,2]==1,k--)); print1(k,",")) \\ Klaus Brockhaus, Mar 10 2003

For n > 2, a(2^r) = 2^(r-1) with r>1, a(p^r) = phi(p^r) with p > 2 prime, r >= 1, where phi is Euler's function A000010; in general if a(Product p_i^e_i) = Sum a(p_i^e_i).

More terms from Klaus Brockhaus, Mar 10 2003

A005417 Maximal period of an n-stage shift register.

Original entry on oeis.org

2, 6, 12, 30, 60, 120, 210, 420, 840, 1260, 2520, 2520, 5040, 9240, 13860, 27720, 32760, 55440, 65520, 120120, 180180, 360360, 360360, 720720, 720720, 942480, 1113840

Offset: 0

N. J. A. Sloane

Maximal order of an element of finite order in GL(2n, Z) or GL(2n+1, Z).

a(n) is the max of the first n numbers in A080742.

H. Lüneburg, Galoisfelder, Kreisteilungskörper und Schieberegisterfolgen. B. I. Wissenschaftsverlag, Mannheim, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Benjamin Grossmann, What is the largest (finite) order of an element of GL(10,Q)?
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.

Cf. A000793, A080742, A080743.

(* b,c = a080737 *)
nmax = 26;
kmax = 1200000; (* kmax increased by 100000 until results do not change *)
b[1] = b[2] = 0; b[p_?PrimeQ] := b[p] = p-1; b[k_] := b[k] = If[Length[f = FactorInteger[k]]==1, EulerPhi[k], Total[b /@ (f[[All, 1]]^f[[All, 2]])] ];
orders = Table[{k, b[k]}, {k, 1, kmax}];
c[0] = 2; c[n_] := c[n] = Select[orders, 2n-1 <= #[[2]] <= 2n&][[-1, 1]];
a[n_] := Table[c[m], {m, 0, n}] // Max;
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 17 2017 *)

a(n) = max m such that A067240(m) <= 2n + 1. E.g., a(2) = 12 since 12 is largest m such that A067240(m) <= 5.

Entry revised by N. J. A. Sloane, Mar 10 2002

A171453 a(n) = sum_i p_i^(e_i-1) where n = product_i p_i^e_i is the prime number decomposition of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 2, 8, 1, 4, 1, 3, 2, 2, 1, 5, 5, 2, 9, 3, 1, 3, 1, 16, 2, 2, 2, 5, 1, 2, 2, 5, 1, 3, 1, 3, 4, 2, 1, 9, 7, 6, 2, 3, 1, 10, 2, 5, 2, 2, 1, 4, 1, 2, 4, 32, 2, 3, 1, 3, 2, 3, 1, 7, 1, 2, 6, 3, 2, 3, 1, 9, 27, 2, 1, 4, 2, 2, 2, 5, 1, 5, 2, 3, 2, 2, 2, 17, 1, 8, 4, 7

Offset: 1

R. J. Mathar, Dec 09 2009

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

Cf. A008475, A067240.

A171453 := proc(n) add( op(1,f)^(op(2,f)-1),f =ifactors(n)[2]) ; end proc:
seq(A171453(n),n=1..100) ;

A171453(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^(f[i,2]-1))); }; \\ Antti Karttunen, Sep 24 2017

from sympy import factorint
def A171453(n): return sum(p**(e-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 01 2024

a(n) = A008475(n) - A067240(n).

A333698 G.f.: Sum_{k>=1} phi(k) * x^prime(k) / (1 - x^prime(k)).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 1, 1, 3, 4, 2, 2, 3, 3, 1, 6, 2, 4, 3, 3, 5, 6, 2, 2, 3, 1, 3, 4, 4, 10, 1, 5, 7, 4, 2, 4, 5, 3, 3, 12, 4, 6, 5, 3, 7, 8, 2, 2, 3, 7, 3, 8, 2, 6, 3, 5, 5, 16, 4, 6, 11, 3, 1, 4, 6, 18, 7, 7, 5, 8, 2, 12, 5, 3, 5, 6, 4, 10, 3, 1, 13, 22, 4, 8

Offset: 1

Ilya Gutkovskiy, Apr 02 2020

nonn

			a(63) = a(3^2 * 7) = a(prime(2)^2 * prime(4)) = A000010(2) + A000010(4) = 1 + 2 = 3.

Index entries for sequences computed from indices in prime factorization

Cf. A000010, A000720, A055631, A066328, A067240.

nmax = 85; CoefficientList[Series[Sum[EulerPhi[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (EulerPhi[PrimePi[#[[1]]]] & /@ FactorInteger[n]); Table[a[n], {n, 85}]

a(n) = my(f=factor(n)); sum(k=1, #f~, eulerphi(primepi(f[k,1]))); \\ Michel Marcus, Apr 03 2020

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

cp's OEIS Frontend

A374066 a(n) is the number of terms in the trajectory when the map x -> A067240(x) is iterated, starting from x = n until x = 0.

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A080737 a(1) = a(2) = 0; for n > 2, the least dimension of a lattice possessing a symmetry of order n.

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A005417 Maximal period of an n-stage shift register.

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A171453 a(n) = sum_i p_i^(e_i-1) where n = product_i p_i^e_i is the prime number decomposition of n.

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A333698 G.f.: Sum_{k>=1} phi(k) * x^prime(k) / (1 - x^prime(k)).

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Keywords

Examples

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Programs

Mathematica

PARI

Formula