A344587 -id:A344587 - cp's OEIS frontend

A346246 Dirichlet inverse of A344587, 2*A003961(n) - sigma(A003961(n)).

1, -2, -4, -1, -6, 10, -10, -2, -3, 14, -12, 4, -16, 22, 26, -4, -18, 2, -22, 6, 42, 26, -28, 6, -5, 34, -6, 10, -30, -66, -36, -8, 50, 38, 62, 7, -40, 46, 66, 10, -42, -106, -46, 12, 14, 58, -52, 8, -9, 2, 74, 16, -58, -2, 74, 18, 90, 62, -60, -18, -66, 74, 26, -16, 98, -126, -70, 18, 114, -150, -72, 18, -78, 82, 12, 22

Offset: 1

Antti Karttunen, Jul 19 2021

Dirichlet inverse of the deficiency of prime shifted n.

Cf. A000203, A003961, A003973, A323910, A344587, A346247, A346251 (positions of zeros).

Cf. also A346235, A346248, A346254.

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];

a(n) = A323910(A003961(n)).

a(n) = A346247(n) - A344587(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)).

A153881 1 followed by -1, -1, -1, ... .

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Mats Granvik, Jan 03 2009

Dirichlet inverse of A074206.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

If prefixed by initial 0, we get A134824.
Cf. A074206 (Dirichlet inverse).
Cf. A033879, A055615, A057427, A063524, A344587, A346234.

[1 - 2*Sign(n-1) : n in [1..100]]; // Wesley Ivan Hurt, Jun 20 2014

A153881:=n->`if`(n=1, 1, -1); seq(A153881(n), n=1..100); # Wesley Ivan Hurt, Jun 20 2014

Table[1 - 2 Sign[n - 1], {n, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)

a(n)=if(n>1,-1,1) \\ Charles R Greathouse IV, Sep 24 2015

G.f: x*(1-2*x)/(1-x). - Mats Granvik, Mar 09 2009, rewritten R. J. Mathar, Mar 31 2010
a(n) = (-1)^A000040(n). - Juri-Stepan Gerasimov, Sep 10 2009
G.f.: x / (1 + x / (1 - 2*x)). - Michael Somos, Apr 02 2012
From Wesley Ivan Hurt, Jun 20 2014: (Start)
a(1) = 1; a(n) = -1, n > 1.
a(n) = 1 - 2*sign(n-1) = 1 - 2*A057427(n-1).
a(n) = (-1)^sign(1-n) = (-1)^A057427(1-n).
a(n) = 2*floor(1/n)-1 = 2*A063524(n)-1. (End)
Dirichlet g.f.: 2 - zeta(s). - Álvar Ibeas, Dec 30 2018
a(n) = Sum_{d|n} A033879(d)*A055615(n/d) = Sum_{d|n} A344587(d)*A346234(n/d). - Antti Karttunen, Nov 22 2024

Edited by Charles R Greathouse IV, Mar 18 2010

More terms from Antti Karttunen, Nov 22 2024

A378231 Deficiency of prime-shifted squares: a(n) = 2*A003961(n^2) - sigma(A003961(n^2)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 5, 19, 41, 41, 47, 109, 365, 469, 141, 155, 299, 271, 449, 683, 3281, 341, 1097, 505, 1041, 1927, 663, 811, 2567, 2001, 1211, 11719, 3509, 929, -921, 1331, 29525, 2777, 1545, 4277, 6749, 1639, 2333, 4933, 9141, 1805, 851, 2161, 5235, 16733, 3815, 2755, 22979, 13177, 6805, 6239, 9671, 3421, 27347, 6131, 31049

Offset: 1

Antti Karttunen, Nov 23 2024

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See comments in A377879 and in A337339.

Cf. A000203, A003961, A003973, A033879, A344587, A377879.

Cf. also A337339.

A003961(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)); };
A378231(n) = A344587(n^2);

a(n) = A344587(n^2) = A377879(A003961(n)).

A378216 Dirichlet inverse of A174725.

Original entry on oeis.org

1, 0, 0, -1, 0, -2, 0, -2, -1, -2, 0, -4, 0, -2, -2, -3, 0, -4, 0, -4, -2, -2, 0, -6, -1, -2, -2, -4, 0, -6, 0, -4, -2, -2, -2, -7, 0, -2, -2, -6, 0, -6, 0, -4, -4, -2, 0, -8, -1, -4, -2, -4, 0, -6, -2, -6, -2, -2, 0, -10, 0, -2, -4, -5, -2, -6, 0, -4, -2, -6, 0, -10, 0, -2, -4, -4, -2, -6, 0, -8, -3, -2, 0, -10

Offset: 1

Antti Karttunen, Nov 22 2024

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

Cf. A000005, A070824, A174725.

Cf. A023900, A033879, A344587, A378220.

PARI
```
A378216(n) = if(1==n,n,2-numdiv(n));
```

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA174725(n/d) * a(d).
For n > 1, a(n) = -A070824(n) = 2-A000005(n).
a(n) = Sum_{d|n} A023900(d) * A033879(n/d) = Sum_{d|n} A378220(d) * A344587(n/d).

A378978 Deficiency of even numbers: a(n) = 4n - sigma(2n).

Original entry on oeis.org

1, 1, 0, 1, 2, -4, 4, 1, -3, -2, 8, -12, 10, 0, -12, 1, 14, -19, 16, -10, -12, 4, 20, -28, 7, 6, -12, -8, 26, -48, 28, 1, -12, 10, -4, -51, 34, 12, -12, -26, 38, -56, 40, -4, -54, 16, 44, -60, 25, -17, -12, -2, 50, -64, 4, -24, -12, 22, 56, -120, 58, 24, -60, 1, 8, -72, 64, 2, -12, -56, 68, -115, 70, 30, -72, 4, 20

Offset: 1

Antti Karttunen, Dec 13 2024

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Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences related to sigma(n).

Even bisection of A033879.
Topmost row of A378979.
Cf. A000203, A033879, A062731.
Cf. also A344587, A377879.

a[n_]:= 4*n - DivisorSigma[1,2*n]; Array[a,77] (* Stefano Spezia, Dec 13 2024 *)

A033879(n) = (n+n-sigma(n));
A378978(n) = A033879(2*n);

a(n) = A033879(2*n) = 4*n - A062731(n).

cp's OEIS Frontend

A346246 Dirichlet inverse of A344587, 2*A003961(n) - sigma(A003961(n)).

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

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A153881 1 followed by -1, -1, -1, ... .

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A378231 Deficiency of prime-shifted squares: a(n) = 2*A003961(n^2) - sigma(A003961(n^2)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

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A378216 Dirichlet inverse of A174725.

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A378978 Deficiency of even numbers: a(n) = 4*n - sigma(2*n).

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A378978 Deficiency of even numbers: a(n) = 4n - sigma(2n).