A323911 -id:A323911 - cp's OEIS frontend

A323910 Dirichlet inverse of the deficiency of n, A033879.

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2

Offset: 1

Antti Karttunen, Feb 12 2019

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Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323910, Sequence Machine.

Cf. A033879, A323911, A323912, A359549 (parity of terms).

Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A346250 Sum of -A252748 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, -3, 1, 6, 0, -11, 0, 6, 6, -17, 0, -23, 0, -17, 6, 18, 0, -39, 9, 18, -15, -25, 0, -48, 0, -51, 18, 30, 18, -49, 0, 30, 18, -77, 0, -72, 0, -35, -61, 34, 0, -85, 9, -31, 30, -43, 0, -123, 54, -97, 30, 54, 0, -117, 0, 50, -77, -89, 54, -96, 0, -53, 34, -104, 0, -19, 0, 66, -55, -61, 54, -120, 0

Offset: 1

Antti Karttunen, Jul 19 2021

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Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A252748, A346248.

Cf. also A323911, A346236, A346247, A346255.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A252748(n) = (A003961(n) - (2*n));
v346248 = DirInverseCorrect(vector(up_to,n,-A252748(n)));
A346248(n) = v346248[n];
A346250(n) = (A346248(n)-A252748(n));

a(n) = A346248(n) - A252748(n).

A346247 Sum of A344587 (the deficiency of prime shifted n) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 16, 0, 12, 16, 24, 0, 16, 0, 40, 48, 37, 0, 28, 0, 28, 80, 48, 0, 36, 36, 64, 88, 52, 0, -48, 0, 114, 96, 72, 120, 54, 0, 88, 128, 68, 0, -64, 0, 64, 116, 112, 0, 92, 100, 68, 144, 88, 0, 124, 144, 132, 176, 120, 0, -12, 0, 144, 204, 349, 192, -72, 0, 100, 224, -72, 0, 128, 0, 160, 160, 124, 240, -88, 0, 182

Offset: 1

Antti Karttunen, Jul 19 2021

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Cf. A000203, A003961, A323911, A344587, A346246.

Cf. also A346250.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };
v346246 = DirInverseCorrect(vector(up_to,n,A344587(n)));
A346246(n) = v346246[n];
A346247(n) = (A344587(n)+A346246(n));

a(n) = A344587(n) + A346246(n).

a(n) = A323911(A003961(n)).

A346478 Sum of A346476 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 2, 0, -3, 1, 6, 0, -11, 0, 6, 6, -5, 0, -23, 0, -29, 6, 18, 0, -3, 9, 18, -15, -37, 0, -60, 0, -9, 18, 30, 18, 23, 0, 30, 18, 1, 0, -84, 0, -83, -61, 34, 0, -13, 9, -67, 30, -91, 0, 45, 54, 5, 30, 54, 0, 75, 0, 50, -77, -5, 54, -184, 0, -137, 34, -176, 0, -13, 0, 66, -55, -145, 54, -188, 0, -37, 49

Offset: 1

Antti Karttunen, Jul 30 2021

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Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences generated by sieves

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346477.

Cf. also A323911, A346250, A346480.

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];
A346478(n) = (A346476(n)+A346477(n));

a(n) = A346476(n) + A346477(n).

a(1) = 2; and for n > 2, a(n) = -Sum_{d|n, 1A346476(n/d) * A346477(d).

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

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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));

A324044 a(n) = A003958(n) - A033879(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, -1, 2, 0, 6, 0, 2, 2, 0, 0, 7, 0, 6, 2, 2, 0, 14, -3, 2, -6, 6, 0, 20, 0, 0, 2, 2, 2, 23, 0, 2, 2, 14, 0, 24, 0, 6, 4, 2, 0, 30, -5, 9, 2, 6, 0, 20, 2, 14, 2, 2, 0, 56, 0, 2, 2, 0, 2, 32, 0, 6, 2, 28, 0, 55, 0, 2, 6, 6, 2, 36, 0, 30, -25, 2, 0, 68, 2, 2, 2, 14, 0, 70, 2, 6, 2, 2, 2, 62, 0, 11, -2, 33, 0, 44, 0, 14, 30

Offset: 1

Antti Karttunen, Feb 13 2019

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

Cf. A003958, A033879.

Cf. also A319687, A323911.

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A033879(n) = (2*n-sigma(n));
A324044(n) = (A003958(n)-A033879(n));

a(n) = A003958(n) - A033879(n).

cp's OEIS Frontend

A323910 Dirichlet inverse of the deficiency of n, A033879.

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A346250 Sum of -A252748 and its Dirichlet inverse.

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A346247 Sum of A344587 (the deficiency of prime shifted n) and its Dirichlet inverse.

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A346478 Sum of A346476 and its Dirichlet inverse.

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

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A324044 a(n) = A003958(n) - A033879(n).

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