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.

A378749 Dirichlet inverse of A378747, where A378747(n) = A048673(n) - A001065(n).

Original entry on oeis.org

1, -1, -2, -1, -3, 2, -5, -4, -5, 3, -6, -1, -8, 3, 3, -13, -9, -3, -11, -1, 3, 6, -14, -4, -10, 6, -22, -3, -15, -5, -18, -37, 6, 9, 4, -7, -20, 9, 6, -4, -21, -7, -23, -1, -1, 10, -26, -10, -28, -2, 9, -3, -29, -18, 7, -4, 9, 15, -30, 3, -33, 14, 1, -94, 7, -8, -35, -1, 10, -8, -36, 4, -39, 18, 2, -3, 7, -10, -41
Offset: 1

Views

Author

Antti Karttunen, Dec 09 2024

Keywords

Links

Crossrefs

Cf. A001065, A048673, A378747.
Cf. also A378750.

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A378747(n) = (A048673(n)-(sigma(n)-n));
memoA378749 = Map();
A378749(n) = if(1==n,1,my(v); if(mapisdefined(memoA378749,n,&v), v, v = -sumdiv(n,d,if(dA378747(n/d)*A378749(d),0)); mapput(memoA378749,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378747(n/d) * a(d).

A378748 Möbius transform of A378747.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 4, 5, 7, 0, 5, 4, 7, 2, 5, 19, 8, 8, 10, 6, 11, 0, 13, 20, 16, 2, 41, 14, 14, 2, 17, 65, 11, 0, 19, 36, 19, 2, 17, 30, 20, 10, 22, 12, 39, 4, 25, 76, 48, 12, 17, 20, 28, 64, 21, 58, 23, 0, 29, 28, 32, 4, 73, 211, 31, 2, 34, 18, 31, 14, 35, 132, 38, 2, 49, 26, 43, 10, 40, 114, 223, 0, 43, 60, 33, 2
Offset: 1

Views

Author

Antti Karttunen, Dec 09 2024

Keywords

Comments

No negative terms.

Links

Crossrefs

Cf. A000010, A003961, A003972, A008683, A048673, A051953, A378521, A378747.
Positions of 0's is given by {2} U A108605.

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A378747(n) = (A048673(n)-(sigma(n)-n));
A378748(n) = sumdiv(n,d,moebius(d)*A378747(n/d));

Formula

a(n) = Sum_{d|n} A008683(d)*A378747(n/d).
a(n) = A378521(n) - A051953(n).
For n > 1, a(n) = A000010(n) + A000010(A003961(n))/2 - n.

A378751 a(n) = A000203(n) - A048673(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 2, 1, 0, 7, 5, 5, 5, 7, 6, -10, 8, 1, 8, 10, 4, 16, 9, -8, 6, 16, -23, 6, 14, 19, 13, -59, 15, 25, 9, -22, 17, 25, 13, -5, 20, 13, 20, 25, -10, 28, 21, -79, -4, 19, 24, 21, 24, -68, 26, -29, 22, 43, 29, 10, 28, 40, -34, -238, 24, 46, 32, 40, 23, 28, 35, -143, 34, 52, 1, 36, 24, 40, 38, -98, -192
Offset: 1

Views

Author

Antti Karttunen, Dec 09 2024

Keywords

Links

Crossrefs

Cf. A000203, A048673, A377984, A378747.
Cf. A337378 (positions of negative terms), A337379 (of terms >= 0).

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A378751(n) = (sigma(n)-A048673(n));

Formula

a(n) = n - A378747(n).
a(n) = (1/2) * (A377984(n)-1).
Showing 1-3 of 3 results.

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