A346247 -id:A346247 - 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).

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

sign

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).

A346236 Sum of A341530 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 4, 0, 70, 0, 8, 8, 9, 0, 134, 0, 2, 16, 4, 0, 52, 4, 4, -4, 44, 0, 368, 0, -3, 8, 4, 16, 1037, 0, 8, 8, 18, 0, 352, 0, 6, 460, 12, 0, 564, 16, -2, 8, 34, 0, -296, 8, 344, 16, 4, 0, 464, 0, 4, -8, 29, 8, 160, 0, 2, 24, 736, 0, -395, 0, 4, 460, 20, 16, 200, 0, -14, 21, 4, 0, 2152, 8, 8, 8, 740

Offset: 1

Antti Karttunen, Jul 11 2021

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Cf. A000203, A003961, A341530, A346235.

Cf. also A346247, A346250, A346255.

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(dA003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341530(n) = { my(t=A003961(n), s=sigma(t)); gcd((n*s), sigma(n)*t); };
v346235 = DirInverseCorrect(vector(up_to,n,A341530(n)));
A346235(n) = v346235[n];
A346236(n) = (A341530(n)+A346235(n));

a(n) = A341530(n) + A346235(n).

A346255 Sum of A336849 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 9, 0, 30, 0, 27, 25, 42, 0, -15, 0, 66, 70, 81, 0, -25, 0, 63, 110, 78, 0, 45, 49, 102, 125, -33, 0, -140, 0, 243, 130, 114, 154, 625, 0, 138, 170, 189, 0, -440, 0, 117, 175, 174, 0, -81, 121, 147, 190, -51, 0, 625, 182, 99, 230, 186, 0, 315, 0, 222, 275, 729, 238, -260, 0, 171, 290, -308, 0, 15, 0, 246, 245, -69, 286

Offset: 1

Antti Karttunen, Jul 19 2021

sign

Cf. A000203, A003961, A003973, A336849, A346254.

Cf. also A346236, A346247, 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
A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
v346254 = DirInverseCorrect(vector(up_to,n,A336849(n)));
A346254(n) = v346254[n];
A346255(n) = (A336849(n)+A346254(n));

a(n) = A336849(n) + A346254(n).

cp's OEIS Frontend

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

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A346250 Sum of -A252748 and its Dirichlet inverse.

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A346236 Sum of A341530 and its Dirichlet inverse.

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A346255 Sum of A336849 and its Dirichlet inverse.

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