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 1-3 of 3 results.

A342417 Dirichlet inverse of Čiurlionis sequence, A342002.

Original entry on oeis.org

1, -1, -5, -1, -7, 9, -7, -5, -6, 1, -41, 5, -9, 3, 33, -1, -47, -6, -11, 19, 27, 63, -53, 41, 36, 1, 136, 7, -59, 7, -9, 25, 369, 77, 43, 16, -59, -49, -157, -9, -317, -19, -73, -13, -10, -21, -359, -53, -38, -52, 139, -105, -401, -348, 473, -23, -263, -51, -443, -201, -11, 5, 292, 23, 65, -893, -69, -5, 253, 31
Offset: 1

Views

Author

Antti Karttunen, Mar 13 2021

Keywords

Links

Crossrefs

Cf. A342002, A342419.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
Cf. also A359427.

Programs

  • PARI
    up_to = 11550;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
v342417 = DirInverseCorrect(vector(up_to,n,A342002(n)));
A342417(n) = v342417[n];

A359603 Dirichlet inverse of function f(n) = 1+(A003415(n)*A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, -4, -7, -21, -19, 30, -11, 51, -132, -164, -91, -11, -51, -588, -935, -5904, -451, -1402, -251, -5979, -7347, -13898, -2251, -25507, -12140, -27718, -99060, -174307, -11251, 11610, -15, 52653, 685, 2410, -1095, 24800, -71, -198, -2647, 53673, -631, 61020, -351, 94173, -20052, -21368, -3151, 207838
Offset: 1

Views

Author

Antti Karttunen, Jan 11 2023

Keywords

Links

Crossrefs

Cf. A003415, A276086, A358669, A359590 (parity of terms), A359604 [= a(n) mod 60].
Cf. also A359427, A359589.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A358669(n) = (A003415(n)*A276086(n));
memoA359603 = Map();
A359603(n) = if(1==n,1,my(v); if(mapisdefined(memoA359603,n,&v), v, v = -sumdiv(n,d,if(dA358669(n/d))*A359603(d),0)); mapput(memoA359603,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA358669(n/d)) * a(d).

A359428 Sum of A358764 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 16, 9, 36, 0, 16, 0, 20, 54, 20, 0, 30, 0, 120, 30, 180, 0, 20, 81, 100, 63, 200, 0, 228, 0, 544, 270, 900, 90, 446, 0, 500, 150, 820, 0, 1660, 0, 3000, 477, 4500, 0, 2760, 25, 2554, 1350, 5000, 0, 7788, 810, 14900, 750, 22500, 0, 936, 0, 28, 2265, -12, 450, 3324, 0, 168, 6750
Offset: 1

Views

Author

Antti Karttunen, Jan 02 2023

Keywords

Links

Crossrefs

Cf. A060681, A276086, A358764, A359427.
Cf. A053850 (positions of odd terms), A353569 (parity of terms).
Cf. also A342419.

Programs

Formula

a(n) = A358764(n) + A359427(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A358764(d) * A359427(n/d).
Showing 1-3 of 3 results.

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

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