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 1-6 of 6 results.

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).

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).

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

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A323911, A344587, A346246.
Cf. also A346250.

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];
A346247(n) = (A344587(n)+A346246(n));

Formula

a(n) = A344587(n) + A346246(n).
a(n) = A323911(A003961(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

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A003973, A336849, A346254.
Cf. also A346236, A346247, A346250.

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];
A346255(n) = (A336849(n)+A346254(n));

Formula

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

Views

Author

Antti Karttunen, Jul 30 2021

Keywords

Links

Crossrefs

Cf. A000040, A000720, A062234, A250469, A252748, A346476, A346477.
Cf. also A323911, A346250, A346480.

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(dA346476(n) = (n+n-A250469(n));
v346477 = DirInverseCorrect(vector(up_to,n,A346476(n)));
A346477(n) = v346477[n];
A346478(n) = (A346476(n)+A346477(n));

Formula

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

A347099 a(1) = 2; and for n > 1, a(n) = A336853(n) + A347098(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 9, 4, 4, 0, 32, 0, 8, 8, 49, 0, 56, 0, 36, 16, 4, 0, 153, 4, 8, 56, 66, 0, 96, 0, 207, 8, 4, 16, 295, 0, 8, 16, 187, 0, 168, 0, 48, 120, 12, 0, 553, 16, 80, 8, 78, 0, 444, 8, 323, 16, 4, 0, 480, 0, 12, 216, 745, 16, 144, 0, 60, 24, 200, 0, 1016, 0, 8, 152, 90, 16, 216, 0, 723, 472, 4, 0, 786, 8, 8, 8, 289
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Sum of {the pointwise sum of A336853 and A063524 (1, 0, 0, 0, ...)} and its Dirichlet inverse.
The first negative term is a(720) = -6306.

Links

Crossrefs

Cf. A001223, A001248, A003961, A063524, A336853, A347098.
Cf. also A346250, A347097.

Programs

  • PARI
    up_to = 16384;
A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
Aux347098(n) = if(1==n,n,A336853(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA347098(n) = v347098[n];
A347099(n) = if(1==n,2,A336853(n)+A347098(n));

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A336853(d) * A347098(n/d).
For all n >= 1, a(A001248(n)) = A001223(n)^2.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.