A291784 -id:A291784 - cp's OEIS frontend

A318441 a(n) = Sum_{d|n} [moebius(n/d) > 0]*A033879(d).

1, 1, 2, 1, 4, 1, 6, 1, 5, 3, 10, -3, 12, 5, 7, 1, 16, -1, 18, -1, 11, 9, 22, -11, 19, 11, 14, 1, 28, -5, 30, 1, 19, 15, 23, -19, 36, 17, 23, -9, 40, -3, 42, 5, 14, 21, 46, -27, 41, 11, 31, 7, 52, -7, 39, -7, 35, 27, 58, -45, 60, 29, 24, 1, 47, 1, 66, 11, 43, 7, 70, -55, 72, 35, 30, 13, 59, 3, 78, -25, 41, 39, 82, -51, 63, 41, 55, -3, 88

Offset: 1

Antti Karttunen, Aug 26 2018

sign

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

Cf. A033879, A083254, A291784, A318325, A318442.

A318441(n) = sumdiv(n,d,(1==moebius(n/d))*(d+d-sigma(d)));

a(n) = Sum_{d|n} [A008683(n/d) == 1]*A033879(d).
a(n) = A291784(n) - A318325(n).
A083254(n) = a(n) - A318442(n).

A292786 a(n) = psi(n) - phi(n).

Original entry on oeis.org

0, 2, 2, 4, 2, 10, 2, 8, 6, 14, 2, 20, 2, 18, 16, 16, 2, 30, 2, 28, 20, 26, 2, 40, 10, 30, 18, 36, 2, 64, 2, 32, 28, 38, 24, 60, 2, 42, 32, 56, 2, 84, 2, 52, 48, 50, 2, 80, 14, 70, 40, 60, 2, 90, 32, 72, 44, 62, 2, 128, 2, 66, 60, 64, 36, 124, 2, 76, 52, 120, 2, 120, 2, 78, 80, 84, 36, 144

Offset: 1

Altug Alkan, Sep 23 2017

Even numbers that are not the terms of this sequence are 12, 102, 114, 130, ...

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

Cf. A000010, A001615, A051612, A069359, A070915, A088245, A291784.

psi[n_] := If[n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors@ n}]]; Array[psi@# - EulerPhi@# &, 87] (* Robert G. Wilson v, Sep 23 2017 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
a(n) = a001615(n) - eulerphi(n); \\ after Charles R Greathouse IV at A001615

a(n) = A001615(n) - A000010(n).
a(n) = 2 iff n is prime.
a(n) = 2*A069359(n) iff n is in A070915.
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c =  9/(2*Pi^2) = 0.455945... (A088245). - Amiram Eldar, Dec 05 2023

A319681 Restricted growth sequence transform of A319680.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 53, 54, 55, 49, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Antti Karttunen, Sep 29 2018

nonn

For all i, j: a(i) = a(j) => A291784(i) = A291784(j).

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

Cf. A319680, A319682.

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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319680(n) = { my(m=1); fordiv(n,d,if((1==moebius(n/d)), m *= A019565(d))); m; };
v319681 = rgs_transform(vector(up_to,n,A319680(n)));
A319681(n) = v319681[n];

A306369 a(n) = A000010(n) + A069359(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 14, 13, 15, 16, 16, 17, 21, 19, 22, 22, 23, 23, 28, 25, 27, 27, 30, 29, 39, 31, 32, 34, 35, 36, 42, 37, 39, 40, 44, 41, 53, 43, 46, 48, 47, 47, 56, 49, 55, 52, 54, 53, 63, 56, 60, 58, 59, 59, 78, 61, 63, 66, 64, 66, 81, 67, 70, 70, 83, 71, 84, 73, 75, 80

Offset: 1

Torlach Rush, Feb 10 2019

nonn

a(n) = A291784(n) iff A001221(n) < 3, that is, iff n is in A070915.

			1 is a term because A000010(1) + A069359(1) = 1 + 0.
7 is a term because A000010(6) + A069359(6) = 2 + 5 = 7 = 6 + 1 = A000010(7) + A069359(7).

Cf. A000010, A001221, A069359, A070915, A291784.

Cf. A059956, A085548.

A069359[n_] := n * Plus @@ (1/FactorInteger[n][[;; , 1]]); A069359[1] = 0; a[n_] := A069359[n] + EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

a(n) = eulerphi(n) + n*sumdiv(n, d, isprime(d)/d); \\ Michel Marcus, Feb 12 2019

a(n) = A000010(n) + A069359(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A059956 + A085548 = 1.0601745... . - Amiram Eldar, Dec 05 2023

cp's OEIS Frontend

A318441 a(n) = Sum_{d|n} [moebius(n/d) > 0]*A033879(d).

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A292786 a(n) = psi(n) - phi(n).

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A319681 Restricted growth sequence transform of A319680.

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A306369 a(n) = A000010(n) + A069359(n).

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