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.

A324641 Sum of the Doudna sequence and its Dirichlet inverse: a(n) = A005940(n) + A324640(n).

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 8, 9, 20, 0, 12, 0, 36, 30, 16, 0, 10, 0, 20, 54, 60, 0, 24, 25, 100, 15, 36, 0, 48, 0, 32, 90, 44, 90, 28, 0, 84, 150, 40, 0, 32, 0, 60, 97, 180, 0, 48, 81, 146, 66, 100, 0, 270, 150, 72, 126, 500, 0, 108, 0, 324, 93, 64, 250, -128, 0, 44, 270, -48, 0, 56, 0, 220, 339, 84, 270, -48, 0, 80, 391, 308, 0, 140, 110
Offset: 1

Views

Author

Antti Karttunen, Mar 11 2019

Keywords

Links

Programs

  • PARI
    up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
v324640 = DirInverse(vector(up_to,n,A005940(n)));
A324640(n) = v324640[n];
A324641(n) = (A005940(n)+A324640(n));

Formula

a(n) = A005940(n) + A324640(n).

A364574 Dirichlet inverse of A005941.

Original entry on oeis.org

1, -2, -3, 0, -5, 6, -9, 0, 2, 10, -17, 0, -33, 18, 19, 0, -65, -4, -129, 0, 35, 34, -257, 0, 12, 66, 0, 0, -513, -38, -1025, 0, 67, 130, 69, 0, -2049, 258, 131, 0, -4097, -70, -8193, 0, -22, 514, -16385, 0, 56, -24, 259, 0, -32769, 0, 133, 0, 515, 1026, -65537, 0, -131073, 2050, -42, 0, 261, -134, -262145, 0, 1027
Offset: 1

Views

Author

Antti Karttunen, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A005941, A364575.
Cf. also A324640.

Programs

  • PARI
    A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
memoA364574 = Map();
A364574(n) = if(1==n,1,my(v); if(mapisdefined(memoA364574,n,&v), v, v = -sumdiv(n,d,if(dA005941(n/d)*A364574(d),0)); mapput(memoA364574,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA005941(n/d) * a(d).
a(p) = -A005941(p) for all primes p.

A364575 a(n) = A364574(A005940(1+n)), where A364574 is the Dirichlet inverse of A005941 [the inverse permutation of A005940].

Original entry on oeis.org

1, -2, -3, 0, -5, 6, 2, 0, -9, 10, 19, 0, 12, -4, 0, 0, -17, 18, 35, 0, 69, -38, -22, 0, 56, -24, -64, 0, -24, 0, 0, 0, -33, 34, 67, 0, 133, -70, -42, 0, 265, -138, -339, 0, -276, 44, 8, 0, 240, -112, -288, 0, -640, 128, 124, 0, -336, 48, 176, 0, 48, 0, 0, 0, -65, 66, 131, 0, 261, -134, -82, 0, 521, -266, -659, 0
Offset: 0

Views

Author

Antti Karttunen, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A005940, A005941, A085405 (reduced modulo 2), A364574.
Cf. also A324052, A324640 (scatter plots).

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A005941(n) = { my(f=factor(n), p, p2=1, res=0); for(i=1, #f~, p = 1 << (primepi(f[i, 1])-1); res += (p * p2 * (2^(f[i, 2])-1)); p2 <<= f[i, 2]); (1+res) }; \\ (After David A. Corneth's program for A156552)
memoA364574 = Map();
A364574(n) = if(1==n,1,my(v); if(mapisdefined(memoA364574,n,&v), v, v = -sumdiv(n,d,if(dA005941(n/d)*A364574(d),0)); mapput(memoA364574,n,v); (v)));
A364575(n) = A364574(A005940(1+n));

Formula

a(n) = A364574(A005940(1+n)).
Showing 1-3 of 3 results.

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