A318833 -id:A318833 - cp's OEIS frontend

A340197 a(1) = 1; for n > 1, a(n) = Sum_{d|n, dA318833(n/d)-1) * A340197(d).

1, 0, 0, 2, 0, 7, 0, 6, 6, 13, 0, 13, 0, 19, 22, 18, 0, 19, 0, 23, 32, 31, 0, 53, 20, 37, 24, 33, 0, 21, 0, 54, 52, 49, 58, 110, 0, 55, 62, 95, 0, 29, 0, 53, 52, 67, 0, 185, 42, 53, 82, 63, 0, 139, 94, 137, 92, 85, 0, 321, 0, 91, 74, 162, 112, 45, 0, 83, 112, 45, 0, 403, 0, 109, 82, 93, 136, 53, 0, 331, 114, 121

Offset: 1

Antti Karttunen, Jan 05 2021

nonn

See comments and question in A340140.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A023900, A318833, A340140, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));
A340197(n) = if(1==n,1,sumdiv(n,d,if(dA318833(n/d)-1)*A340197(d),0)));

a(1) = 1; for n > 1, a(n) = Sum_{d|n, dA318833(n/d)-1) * A340197(d).

A340367 Dirichlet inverse of sequence b(n) = 1-A318833(n).

Original entry on oeis.org

-1, 0, 0, 2, 0, 7, 0, 6, 6, 13, 0, 13, 0, 19, 22, 10, 0, 19, 0, 23, 32, 31, 0, -3, 20, 37, 24, 33, 0, 21, 0, 6, 52, 49, 58, -36, 0, 55, 62, -9, 0, 29, 0, 53, 52, 67, 0, -87, 42, 53, 82, 63, 0, -29, 94, -15, 92, 85, 0, -219, 0, 91, 74, -22, 112, 45, 0, 83, 112, 45, 0, -257, 0, 109, 82, 93, 136, 53, 0, -165, 42, 121

Offset: 1

Antti Karttunen, Jan 05 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A023900, A318833, A340198.

Cf. also A340140, A340197 for similar definitions.

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));
v340367 = DirInverseCorrect(vector(up_to, n, 1-A318833(n)));
A340367(n) = v340367[n];

\\ Or as:
A340367(n) = if(1==n, -1, sumdiv(n, d, if(dA318833(n/d))*A340367(d), 0)));

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA318833(n/d)) * a(d).

A319340 Sum of Euler totient function and its Dirichlet inverse: a(n) = A000010(n) + A023900(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 3, 4, 8, 0, 6, 0, 12, 16, 7, 0, 8, 0, 12, 24, 20, 0, 10, 16, 24, 16, 18, 0, 0, 0, 15, 40, 32, 48, 14, 0, 36, 48, 20, 0, 0, 0, 30, 32, 44, 0, 18, 36, 24, 64, 36, 0, 20, 80, 30, 72, 56, 0, 8, 0, 60, 48, 31, 96, 0, 0, 48, 88, 0, 0, 26, 0, 72, 48, 54, 120, 0, 0, 36, 52, 80, 0, 12, 128, 84, 112, 50, 0, 16

Offset: 1

Antti Karttunen, Sep 17 2018

nonn

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

Cf. A000010, A023900, A051953, A318833, A319341.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));

a(n) = A000010(n) + A023900(n).

a(n) = A318833(n) - A051953(n).

A318841 a(n) = n - A173557(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 7, 6, 1, 10, 1, 8, 7, 15, 1, 16, 1, 16, 9, 12, 1, 22, 21, 14, 25, 22, 1, 22, 1, 31, 13, 18, 11, 34, 1, 20, 15, 36, 1, 30, 1, 34, 37, 24, 1, 46, 43, 46, 19, 40, 1, 52, 15, 50, 21, 30, 1, 52, 1, 32, 51, 63, 17, 46, 1, 52, 25, 46, 1, 70, 1, 38, 67, 58, 17, 54, 1, 76, 79, 42, 1, 72, 21, 44, 31, 78, 1, 82, 19, 70, 33, 48

Offset: 1

Antti Karttunen, Sep 16 2018

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

Cf. A173557, A307868, A318833.

a[n_] := n - Times @@ (FactorInteger[n][[;;, 1]] - 1); a[1] = 0; Array[a, 100] (* Amiram Eldar, Dec 16 2023 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A318841(n) = (n-A173557(n));

a(n) = n - A173557(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A307868 = 0.528319... . - Amiram Eldar, Dec 16 2023

A340140 a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

Original entry on oeis.org

-1, 0, 0, -2, 0, -7, 0, -6, -6, -13, 0, -13, 0, -19, -22, -22, 0, -19, 0, -23, -32, -31, 0, -81, -20, -37, -24, -33, 0, -21, 0, -78, -52, -49, -58, -183, 0, -55, -62, -147, 0, -29, 0, -53, -52, -67, 0, -321, -42, -53, -82, -63, 0, -223, -94, -213, -92, -85, 0, -591, 0, -91, -74, -278, -112, -45, 0, -83, -112, -45, 0, -733

Offset: 1

Antti Karttunen, Jan 05 2021

This seems to differ from 1-A318833(n) at the points given by A033987.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A023900, A033987, A318833, A340197, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));
A340197(n) = if(1==n,1,sumdiv(n,d,if(dA318833(n/d)-1)*A340197(d),0)));
A340140(n) = if(1==n, -1, sumdiv(n, d, if(dA340197(n/d)*A340140(d), 0)));

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

A340189 a(n) = n + A340187(n).

Original entry on oeis.org

2, 1, 1, 4, 1, 9, 1, 8, 11, 17, 1, 11, 1, 25, 27, 16, 1, 13, 1, 17, 41, 41, 1, 24, 37, 49, 25, 21, 1, 1, 1, 32, 69, 65, 79, 40, 1, 73, 83, 40, 1, -7, 1, 35, 21, 89, 1, 48, 79, 17, 111, 39, 1, 61, 131, 60, 125, 113, 1, 83, 1, 121, 27, 64, 145, -27, 1, 53, 153, -49, 1, 71, 1, 145, 23, 57, 193, -31, 1, 80, 83, 161, 1, 131

Offset: 1

Antti Karttunen, Dec 31 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A063994, A318828, A340187, A340188.

Cf. also A318833.

up_to = 65537;
A063994(n) = { my(f=factor(n)); prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA063994(n)));
A340187(n) = v340187[n];
A340189(n) = (n+A340187(n));

a(n) = n + A340187(n).

a(n) = A340188(n) + A318828(n).

cp's OEIS Frontend

A340197 a(1) = 1; for n > 1, a(n) = Sum_{d|n, dA318833(n/d)-1) * A340197(d).

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A340367 Dirichlet inverse of sequence b(n) = 1-A318833(n).

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A319340 Sum of Euler totient function and its Dirichlet inverse: a(n) = A000010(n) + A023900(n).

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A318841 a(n) = n - A173557(n).

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A340140 a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

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A340189 a(n) = n + A340187(n).

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