cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Comments

Dirichlet inverse of the deficiency of prime shifted n.

Links

Crossrefs

Cf. A000203, A003961, A003973, A323910, A344587, A346247, A346251 (positions of zeros).
Cf. also A346235, A346248, A346254.

Programs

  • PARI
    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];

Formula

a(n) = A323910(A003961(n)).
a(n) = A346247(n) - A344587(n).

A346248 Dirichlet inverse of -A252748, 2*n - A003961(n).

Original entry on oeis.org

1, -1, -1, 2, -3, 5, -3, 8, 8, 7, -9, 10, -9, 11, 11, 32, -15, 16, -15, 6, 19, 13, -17, 48, 8, 17, 56, 18, -27, -3, -25, 128, 17, 19, 25, 104, -33, 23, 25, 32, -39, 9, -39, -6, 24, 29, -41, 224, 32, 16, 23, 6, -47, 144, 35, 88, 31, 31, -57, 78, -55, 37, 72, 512, 43, -33, -63, -18, 41, -13, -69, 512, -67, 41, 40, -6, 43
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Comments

Zeros occur at n = 352, 26840, 34816, 3787168, ...

Links

Crossrefs

Cf. A003961, A252748, A346250.
Cf. also A323910, A346235, A346246, A346254.

Programs

  • PARI
    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
v346248 = DirInverseCorrect(vector(up_to,n,(n+n)-A003961(n)));
A346248(n) = v346248[n];

Formula

a(n) = A346250(n) + A252748(n).

A346254 Dirichlet inverse of A336849.

Original entry on oeis.org

1, -3, -5, 0, -7, 25, -11, 0, 0, 21, -13, -30, -17, 55, 35, 0, -19, -100, -23, 0, 55, 39, -29, 36, 0, 85, 0, -66, -31, -175, -37, 0, 65, 57, 77, 400, -41, 115, 85, 0, -43, -495, -47, 108, 0, 145, -53, -216, 0, 98, 171, -68, -59, 500, 169, 0, 115, 93, -61, 210, -67, 111, 0, 0, 119, -325, -71, 0, 261, -385, -73, -120, -79, 205, 0, -138
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A003973, A336849, A346255, A346256 (positions of zeros).
Cf. also A346235, A346246, A346248.

Programs

  • PARI
    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];

Formula

a(n) = A346255(n) - A336849(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

Views

Author

Antti Karttunen, Jul 11 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A341530, A346235.
Cf. also A346247, A346250, A346255.

Programs

  • PARI
    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));

Formula

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

A347096 a(1) = 1; a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d), where A341512(n) = sigma(n)*A003961(n) - n*sigma(A003961(n)).

Original entry on oeis.org

1, -1, -2, -10, -2, -32, -4, -64, -42, -54, -2, -214, -4, -112, -112, -316, -2, -469, -4, -412, -232, -168, -6, -792, -90, -262, -612, -860, -2, -1208, -6, -1216, -340, -354, -320, -1655, -4, -484, -532, -1760, -2, -2528, -4, -1438, -1850, -732, -6, 160, -364, -1863, -712, -2210, -6, -4596, -384, -3696, -976, -942, -2
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Dirichlet inverse of the pointwise sum of A341512 and A063524 (1, 0, 0, 0, ...).

Links

Crossrefs

Cf. A000203, A001223, A003961, A063524, A341512, A347097.
Cf. also A346235, A346239.

Programs

  • PARI
    up_to = 16384;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341512(n) = { my(u=A003961(n)); ((sigma(n)*u) - (n*sigma(u))); };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA341512(n));
v347096 = DirInverseCorrect(vector(up_to,n,Aux347096(n)));
A347096(n) = v347096[n];

Formula

a(1) = 1; and for n > 1, a(n) = -Sum_{d|n, d < n} A341512(n/d) * a(d).
For all n >= 1, a(A000040(n)) = -A001223(n).
Showing 1-5 of 5 results.

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