A318836 -id:A318836 - cp's OEIS frontend

A318837 Restricted growth sequence transform of A318836.

1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 2, 5, 2, 6, 7, 8, 2, 9, 2, 7, 10, 11, 2, 12, 4, 13, 9, 10, 2, 14, 2, 15, 16, 17, 18, 19, 2, 20, 21, 22, 2, 23, 2, 16, 24, 25, 2, 26, 6, 27, 28, 21, 2, 29, 30, 31, 32, 33, 2, 34, 2, 35, 36, 37, 38, 39, 2, 28, 40, 41, 2, 42, 2, 43, 44, 32, 45, 46, 2, 47, 29, 48, 2, 49, 50, 51, 52, 53, 2, 54, 55, 40, 56, 57, 58, 59, 2, 60, 61, 44, 2

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

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

Cf. A007431, A062790, A318836.

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; };
A007431(n) = sumdiv(n,d,moebius(n/d)*eulerphi(d));
A318836(n) = { my(m=1); fordiv(n,d,if((dA007431(d)!=0),m *= prime(A007431(d)))); (m); }; \\
v318837 = rgs_transform(vector(up_to,n,A318836(n)));
A318837(n) = v318837[n];

For all i, j: a(i) = a(j) => A062790(i) = A062790(j).

A062790 Moebius transform of the cototient function A051953.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 4, 1, 3, 1, 6, 5, 4, 1, 6, 1, 5, 7, 10, 1, 6, 4, 12, 6, 7, 1, 8, 1, 8, 11, 16, 9, 8, 1, 18, 13, 10, 1, 12, 1, 11, 12, 22, 1, 12, 6, 20, 17, 13, 1, 18, 13, 14, 19, 28, 1, 13, 1, 30, 16, 16, 15, 20, 1, 17, 23, 24, 1, 16, 1, 36, 24, 19, 15, 24, 1, 20, 18, 40, 1, 19

Offset: 1

Labos Elemer, Jul 19 2001

nonn

			n = 255, its divisors are {1,3,5,25,17,51,85,255}, A051953(255/d) = {127,21,19,1,7,1,1,0}, mu(d) = {1,-1,-1,1,-1,1,1,-1}, the sum is a(255) = 127-21-19+1-7+1+1+0 = 130-47 = 83.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 1 .. 2000 from Harry J. Smith)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A000010, A001065, A007431, A051953, A056239, A318836.

Table[DirichletConvolve[MoebiusMu[n], n-EulerPhi[n], n, k], {k, 100}]  (* Amiram Eldar, Nov 24 2018 *)

A062790(n)={
    local(a=0) ;
    fordiv(n,d,
        a += moebius(d)*(n/d-eulerphi(n/d)) ;
    ) ;
    return(a) ;
} \\ R. J. Mathar, Mar 24 2012

A062790(n) = sumdiv(n,d,moebius(n/d)*(d-eulerphi(d))); \\ Antti Karttunen, Nov 24 2018

a(n) = Sum f(n/d)*mu(d), where d divides n and f(x) = x-phi(x) = A051953(x).
a(n) = A056239(A318836(n)). - Antti Karttunen, Nov 24 2018
From Amiram Eldar, Dec 15 2023: (Start)
a(n) = A000010(n) - A007431(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 6/Pi^2 - 36/Pi^4. (End)

OFFSET changed from 0 to 1 by Harry J. Smith, Aug 11 2009

A318838 Product_{d|n, A007431(d) > 0} prime(A007431(d)), where A007431 is the Möbius transform of Euler's totient function.

Original entry on oeis.org

2, 2, 4, 4, 10, 4, 22, 12, 28, 10, 46, 16, 62, 22, 100, 84, 94, 28, 118, 100, 484, 46, 146, 144, 530, 62, 1036, 484, 206, 100, 218, 1596, 2116, 94, 5170, 784, 298, 118, 3844, 3900, 334, 484, 358, 2116, 25900, 146, 394, 7056, 3322, 530, 8836, 3844, 466, 1036, 23690, 42108, 13924, 206, 538, 10000, 554, 218, 240548

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

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

Cf. A000010, A007431, A318839 (rgs-transform).

Cf. also A318836.

A007431(n) = sumdiv(n,d,moebius(n/d)*eulerphi(d));
A318838(n) = { my(m=1); fordiv(n,d,if((A007431(d)!=0),m *= prime(A007431(d)))); (m); };

a(n) = product_{d|n} A008578(1+A007431(d)).

For all n >= 1, A056239(a(n)) = A000010(n).

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A318837 Restricted growth sequence transform of A318836.

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A062790 Moebius transform of the cototient function A051953.

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A318838 Product_{d|n, A007431(d) > 0} prime(A007431(d)), where A007431 is the Möbius transform of Euler's totient function.

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