A319340 -id:A319340 - cp's OEIS frontend

A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

1, 1, 1, 1, 1, -1, 1, -1, -2, -5, 1, -8, 1, -9, -13, -9, 1, -16, 1, -22, -21, -17, 1, -28, -14, -21, -20, -36, 1, -43, 1, -31, -37, -29, -45, -49, 1, -33, -45, -62, 1, -67, 1, -64, -64, -41, 1, -69, -34, -64, -61, -78, 1, -68, -77, -96, -69, -53, 1, -88, 1, -57, -96, -79, -93, -115, 1, -106, -85, -123, 1, -95, 1, -69

Offset: 1

Antti Karttunen, Jan 05 2021

Conversely, the Dirichlet inverse of this sequence yields a sequence which is one less than A319340, i.e., pointwise sum s(n) = A109606(n) + A023900(n).

a(9796) = 0 is the only zero among the first 2^22 terms.

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

Cf. A000010, A023900, A109606, A319340, A340090, A340094, A340197, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));
A340198(n) = if(1==n,1,-sumdiv(n,d,if(dA319340(n/d)-1)*A340198(d),0)));

a(1) = 1, for n > 1, a(n) = -Sum_{d|n, dA319340(n/d)-1) * a(d).

A322581 Sum of A003958 and its Dirichlet inverse: a(n) = A003958(n) + A097945(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 8, 0, 2, 0, 12, 16, 1, 0, 4, 0, 4, 24, 20, 0, 2, 16, 24, 8, 6, 0, 0, 0, 1, 40, 32, 48, 4, 0, 36, 48, 4, 0, 0, 0, 10, 16, 44, 0, 2, 36, 16, 64, 12, 0, 8, 80, 6, 72, 56, 0, 8, 0, 60, 24, 1, 96, 0, 0, 16, 88, 0, 0, 4, 0, 72, 32, 18, 120, 0, 0, 4, 16, 80, 0, 12, 128, 84, 112, 10, 0, 16, 144, 22, 120, 92, 144

Offset: 1

Antti Karttunen, Dec 17 2018

nonn

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

Cf. A003958, A097945, A322582.

Cf. also A319340.

a[1] = 2; a[n_] := Times @@ ((First[#] - 1)^Last[#] & /@ FactorInteger[n]) + MoebiusMu[n] * EulerPhi[n]; Array[a, 60] (* Amiram Eldar, Dec 17 2018 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A097945(n) = (moebius(n)*eulerphi(n));
A322581(n) = (A003958(n)+A097945(n));

a(n) = A003958(n) + A097945(n).

A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 9, 16, 36, 0, 12, 0, 48, 48, 27, 0, 24, 0, 18, 64, 72, 0, 60, 36, 84, 32, 24, 0, 0, 0, 45, 96, 108, 96, 84, 0, 120, 112, 90, 0, 0, 0, 36, 48, 144, 0, 84, 64, 72, 144, 42, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 64, 99, 168, 0, 0, 54, 192, 0, 0, 132, 0, 228, 96, 60, 192, 0, 0, 126, 112, 252, 0, 288, 216, 264, 240, 180, 0

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

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

Cf. A001615, A323363.

Cf. also A319340, A322581, A323365, A323372, A323399.

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
A323363(n) = if(1==n,1,-sumdiv(n,d,if(dA001615(n/d)*A323363(d),0)));
A323364(n) = (A001615(n) + A323363(n));

A323371 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A295886(n) for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = 0 when n is an odd prime, and f(n) = [A003557(n), A023900(n)] for all other numbers.
For all i, j:
  A323370(i) = A323370(j) => a(i) = a(j),
  A323405(i) = A323405(j) => a(i) = a(j),
  a(i) = a(j) => A092248(i) = A092248(j),
  a(i) = a(j) => A319340(i) = A319340(j),
  a(i) = a(j) => A322587(i) = A322587(j).

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

Cf. A003557, A023900, A092248, A295886, A305801, A319340, A322587, A323370, A323405.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
Aux323371(n) = if((n>2)&&isprime(n),0,[A003557(n), A023900(n)]);
v323371 = rgs_transform(vector(up_to, n, Aux323371(n)));
A323371(n) = v323371[n];

A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = 0 when n is an odd prime, and f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323367(i) = A323367(j),
  a(i) = a(j) => A323371(i) = A323371(j).

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

Cf. A000035, A003557, A023900, A092248, A295886, A305801, A319340, A322587, A323367, A323371.

Differs from A323405 for the first time at n=78, where a(78) = 52, while A323405(78) = 58.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
Aux323370(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A023900(n)]);
v323370 = rgs_transform(vector(up_to, n, Aux323370(n)));
A323370(n) = v323370[n];

A323399 Sum of Jordan function J_2(n), A007434 and its Dirichlet inverse, A046970.

Original entry on oeis.org

2, 0, 0, 9, 0, 48, 0, 45, 64, 144, 0, 120, 0, 288, 384, 189, 0, 240, 0, 360, 768, 720, 0, 408, 576, 1008, 640, 720, 0, 0, 0, 765, 1920, 1728, 2304, 888, 0, 2160, 2688, 1224, 0, 0, 0, 1800, 1920, 3168, 0, 1560, 2304, 1872, 4608, 2520, 0, 1968, 5760, 2448, 5760, 5040, 0, 1728, 0, 5760, 3840, 3069, 8064, 0, 0, 4320, 8448, 0, 0, 3480, 0, 8208, 4992, 5400

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

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

Cf. A007434, A046970.

Cf. also A319340, A322581, A323364.

A007434(n) = sumdiv(n, d, d*d*moebius(n/d));
A046970(n) = if(1==n,n,my(f=factor(n)); for(i=1, #f~, f[i,1] = 1-(f[i,1]^2)); factorback(f[,1]));
A323399(n) = (A007434(n) + A046970(n));

a(n) = A007434(n) + A046970(n).

A319341 a(n) = A000010(n) - A173557(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 0, 7, 0, 4, 0, 4, 0, 0, 0, 6, 16, 0, 16, 6, 0, 0, 0, 15, 0, 0, 0, 10, 0, 0, 0, 12, 0, 0, 0, 10, 16, 0, 0, 14, 36, 16, 0, 12, 0, 16, 0, 18, 0, 0, 0, 8, 0, 0, 24, 31, 0, 0, 0, 16, 0, 0, 0, 22, 0, 0, 32, 18, 0, 0, 0, 28, 52, 0, 0, 12, 0, 0, 0, 30, 0, 16, 0, 22, 0, 0, 0, 30, 0, 36, 40, 36, 0, 0, 0, 36, 0

Offset: 1

Antti Karttunen, Sep 17 2018

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

Cf. A000010, A051953, A173557, A318841, A319340, A126864.

Cf. A013661, A307868.

a[n_] := EulerPhi[n] - Times @@ (FactorInteger[n][[;;, 1]] - 1); a[1] = 0; Array[a, 100] (* Amiram Eldar, Dec 21 2023 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A319341(n) = (eulerphi(n)-A173557(n));

a(n) = A000010(n) - A173557(n).
a(n) = A318841(n) - A051953(n).
a(A005117(n)) = 0. - Ivan N. Ianakiev, Sep 18 2018
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A059956 - A307868 = 0.136246... . - Amiram Eldar, Dec 21 2023

A323403 Sum of sigma and its Dirichlet inverse: a(n) = A000203(n) + A046692(n).

Original entry on oeis.org

2, 0, 0, 9, 0, 24, 0, 15, 16, 36, 0, 20, 0, 48, 48, 31, 0, 30, 0, 30, 64, 72, 0, 60, 36, 84, 40, 40, 0, 0, 0, 63, 96, 108, 96, 97, 0, 120, 112, 90, 0, 0, 0, 60, 60, 144, 0, 124, 64, 78, 144, 70, 0, 120, 144, 120, 160, 180, 0, 216, 0, 192, 80, 127, 168, 0, 0, 90, 192, 0, 0, 195, 0, 228, 104, 100, 192, 0, 0, 186, 121, 252, 0, 288, 216, 264, 240, 180, 0, 288

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)

Cf. A000203, A046692.

Cf. also A319340, A322581, A323364, A323399, A323408.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
v046692 = DirInverse(vector(up_to,n,sigma(n)));
A046692(n) = v046692[n];
A323403(n) = (sigma(n)+A046692(n));

a(n) = A000203(n) + A046692(n).

A323408 Sum of unitary phi and its Dirichlet inverse: a(n) = A047994(n) + A323407(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 5, 4, 8, 0, 10, 0, 12, 16, 15, 0, 12, 0, 20, 24, 20, 0, 18, 16, 24, 24, 30, 0, 0, 0, 35, 40, 32, 48, 32, 0, 36, 48, 36, 0, 0, 0, 50, 48, 44, 0, 30, 36, 32, 64, 60, 0, 28, 80, 54, 72, 56, 0, 8, 0, 60, 72, 71, 96, 0, 0, 80, 88, 0, 0, 64, 0, 72, 64, 90, 120, 0, 0, 60, 88, 80, 0, 12, 128, 84, 112, 90, 0, 16, 144, 110, 120

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

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

Cf. A047994, A323407.

Cf. also A319340, A322581, A323364, A323399, A323403.

up_to = 65537;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
v323407 = DirInverse(vector(up_to,n,A047994(n)));
A323407(n) = v323407[n];
A323408(n) = (A047994(n) + A323407(n));

A323913 Sum of A083254 (2*phi(n) - n) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, 0, -4, 0, 0, 10, 0, 0, 0, 9, 0, 5, 0, 0, -16, 0, 0, 18, 0, 30, -4, 0, 0, 22, 0, 0, -24, 0, 0, 11, 0, 0, 0, 25, -12, 30, 0, 0, -18, 54, 0, 34, 0, 0, -24, 0, 0, 21, 0, 66, -40, 0, 0, 42, -32, 0, 0, 0, 0, 9, 0, 90, -48, 0, 0, 19, 0, 0, -40, 90, 0, 54, 0, 0, -38, 110, 0, 58, 0, 102, 0, 0, -20

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A083254, A319340, A323911, A323912.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];
A323913(n) = (A083254(n)+A323912(n));

cp's OEIS Frontend

A340198 Dirichlet inverse of sequence f(n) = A319340(n)-1 = (A000010(n) + A023900(n) - 1), where A000010 is Euler Totient function phi, and A023900 is its Dirichlet inverse.

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A322581 Sum of A003958 and its Dirichlet inverse: a(n) = A003958(n) + A097945(n).

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A323364 Sum of Dedekind's psi, A001615, and its Dirichlet inverse, A323363.

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A323371 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A295886(n) for all other numbers, except f(n) = 0 for odd primes.

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A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

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A323399 Sum of Jordan function J_2(n), A007434 and its Dirichlet inverse, A046970.

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A319341 a(n) = A000010(n) - A173557(n).

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A323403 Sum of sigma and its Dirichlet inverse: a(n) = A000203(n) + A046692(n).

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A323408 Sum of unitary phi and its Dirichlet inverse: a(n) = A047994(n) + A323407(n).

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