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 1-4 of 4 results.

A346242 Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

Original entry on oeis.org

1, -1, -3, 0, -1, 5, -1, 0, 6, -3, -1, -2, -1, 1, -9, 0, -1, -16, -1, 4, 3, 1, -1, 0, -24, 1, -12, 0, -1, 43, -1, 0, 3, 1, -5, 14, -1, 1, 3, 0, -1, -11, -1, 0, 54, 1, -1, 0, -6, 32, 3, 0, -1, 44, -3, -6, 3, 1, -1, -50, -1, 1, -24, 0, 1, -5, -1, 0, 3, -15, -1, -4, -1, 1, 96, 0, -5, -5, -1, 0, 24, 1, -1, 8, -3, 1, 3, 0, -1
Offset: 1

Views

Author

Antti Karttunen, Jul 13 2021

Keywords

Links

Crossrefs

Cf. A276086, A324198, A346243.
Cf. A008966 (parity of terms), A005117 (positions of odd terms), A013929 (of even terms), A045344 (of -1's, at least a subset of them), A354810 (of 0's), A354811 (of 1's), A354812 (of 2's), A354813 (of 3's), A354814 (of 4's), A354822 (of -2's).
Cf. also A354347, A354348, A354823, A354824.

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(dA324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
v346242 = DirInverseCorrect(vector(up_to,n,A324198(n)));
A346242(n) = v346242[n];

Formula

a(n) = A346243(n) - A324198(n).
From Antti Karttunen, Jun 09 2022: (Start)
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA324198(n/d) * a(d).
For all n >= 1, A000035(a(n)) = A008966(n).
For all n >= 1, a(A045344(n)) = -1.
(End)

A354347 Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, -1, 0, 1, -1, 1, -1, 1, 1, -9, -1, -2, -1, 1, -3, 1, -1, 1, -4, -3, 0, 1, -1, -1, -1, 21, 1, 1, -1, -6, -1, 1, 1, 3, -1, 7, -1, -1, 0, -3, -1, 23, 0, 4, -3, 7, -1, 2, 1, 3, 1, 1, -1, -1, -1, 1, 8, 15, -1, -1, -1, 1, 1, 3, -1, 14, -1, 1, -46, -7, -1, 7, -1, 5, 0, 1, -1, 3, 1, -3, 1, -131
Offset: 1

Views

Author

Antti Karttunen, Jun 07 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A345000.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms), A354815 (positions of 0's), A354816 (of -1's).
Cf. also A346242, A354348, A354823, A354824.

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); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
memoA354347 = Map();
A354347(n) = if(1==n,1,my(v); if(mapisdefined(memoA354347,n,&v), v, v = -sumdiv(n,d,if(dA345000(n/d)*A354347(d),0)); mapput(memoA354347,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA345000(n/d) * a(d).
For all n >= 1, A000035(a(n)) = A353627(n).

A354823 Dirichlet inverse of A351083, where A351083(n) = gcd(n, A327860(n)), and A327860 is the arithmetic derivative of the primorial base exp-function.

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -7, -5, 0, 1, -1, 1, -1, 13, -3, -1, -1, -2, -1, -7, 13, 1, -1, 9, -24, 1, 0, 7, -1, 7, -1, 33, 1, -15, 9, -6, -1, 1, -11, 27, -1, -25, -1, -1, 4, 1, -1, 7, 48, 24, 1, -1, -1, 2, -3, 59, 1, 1, -1, 19, -1, 1, -12, 23, 1, -1, -1, 33, 1, -23, -1, -2, -1, 1, 52, 1, 7, 23, -1, -67, 0, 1, -1, -25
Offset: 1

Views

Author

Antti Karttunen, Jun 09 2022

Keywords

Links

Crossrefs

Cf. A327860, A351083.
Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).
Cf. also A346242, A354347, A354348, A354824.

Programs

  • PARI
    A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A351083(n) = gcd(n, A327860(n));
memoA354823 = Map();
A354823(n) = if(1==n,1,my(v); if(mapisdefined(memoA354823,n,&v), v, v = -sumdiv(n,d,if(dA351083(n/d)*A354823(d),0)); mapput(memoA354823,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA351083(n/d) * a(d).

A355692 Dirichlet inverse of A355442, gcd(A003961(n), A276086(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

Original entry on oeis.org

1, -3, -1, 0, -1, 1, -1, 24, -4, 3, -1, 16, -1, 3, -3, -72, -1, 6, -1, 6, -3, 3, -1, -68, 0, 3, -116, 0, -1, 21, -1, 24, 1, 3, -5, 72, -1, 3, -3, -120, -1, 23, -1, 6, -158, 3, -1, 28, 0, -18, -3, 0, -1, 632, -5, -24, -3, 3, -1, -54, -1, 3, 16, 504, -5, -1, -1, 6, -3, 15, -1, -400, -1, 3, -236, 0, 1, 23, -1, 474, 136
Offset: 1

Views

Author

Antti Karttunen, Jul 18 2022

Keywords

Links

Crossrefs

Cf. A003961, A276086.
Cf. also A346242, A354347, A354348, A354823, A354824.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
memoA355692 = Map();
A355692(n) = if(1==n,1,my(v); if(mapisdefined(memoA355692,n,&v), v, v = -sumdiv(n,d,if(dA355442(n/d)*A355692(d),0)); mapput(memoA355692,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA355442(n/d) * a(d).
Showing 1-4 of 4 results.

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