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 1-5 of 5 results.

A354351 Dirichlet inverse of A108951, primorial inflation of n.

Original entry on oeis.org

1, -2, -6, 0, -30, 12, -210, 0, 0, 60, -2310, 0, -30030, 420, 180, 0, -510510, 0, -9699690, 0, 1260, 4620, -223092870, 0, 0, 60060, 0, 0, -6469693230, -360, -200560490130, 0, 13860, 1021020, 6300, 0, -7420738134810, 19399380, 180180, 0, -304250263527210, -2520, -13082761331670030, 0, 0, 446185740, -614889782588491410
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2022

Keywords

Comments

Multiplicative with a(p^e) = 0 if e > 1, and -A034386(p) otherwise.

Links

Crossrefs

Cf. A002110, A008683, A013929 (positions of 0's), A034386, A108951, A354352.
Cf. also A347379, A354186, A354349, A354359, A354365, A354366.

Programs

Formula

a(n) = A008683(n) * A108951(n).
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A108951(n/d) * a(d).
a(n) = A354352(n) - A108951(n).

A354349 Dirichlet inverse of A181819, prime shadow of n.

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -2, -1, 1, 4, -2, -2, -2, 4, 4, 2, -2, -2, -2, -2, 4, 4, -2, 2, 1, 4, -1, -2, -2, -8, -2, -3, 4, 4, 4, 1, -2, 4, 4, 2, -2, -8, -2, -2, -2, 4, -2, -4, 1, -2, 4, -2, -2, 2, 4, 2, 4, 4, -2, 4, -2, 4, -2, 7, 4, -8, -2, -2, 4, -8, -2, -1, -2, 4, -2, -2, 4, -8, -2, -4, 2, 4, -2, 4, 4, 4, 4, 2, -2, 4, 4
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2022

Keywords

Comments

Multiplicative because A181819 is.

Links

Crossrefs

Cf. A181819.
Cf. also A354186, A354351, A354359.

Programs

  • PARI
    A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
memoA354349 = Map();
A354349(n) = if(1==n,1,my(v); if(mapisdefined(memoA354349,n,&v), v, v = -sumdiv(n,d,if(dA181819(n/d)*A354349(d),0)); mapput(memoA354349,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A181819(n/d) * a(d).

A354866 Dirichlet inverse of A122111.

Original entry on oeis.org

1, -2, -4, 1, -8, 10, -16, -1, 7, 20, -32, -10, -64, 40, 46, 2, -128, -27, -256, -20, 92, 80, -512, 14, 37, 160, -17, -40, -1024, -150, -2048, -3, 184, 320, 202, 53, -4096, 640, 368, 28, -8192, -300, -16384, -80, -146, 1280, -32768, -26, 175, -129, 736, -160, -65536, 85, 404, 56, 1472, 2560, -131072, 242, -262144
Offset: 1

Views

Author

Antti Karttunen, Jun 09 2022

Keywords

Links

Crossrefs

Cf. A122111, A354867, A354868 (parity), A354869 (positions of odd terms).
Cf. also A354186, A354349, A354351, A354359, A354826.

Programs

  • PARI
    A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
memoA354866 = Map();
A354866(n) = if(1==n,1,my(v); if(mapisdefined(memoA354866,n,&v), v, v = -sumdiv(n,d,if(dA122111(n/d)*A354866(d),0)); mapput(memoA354866,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA122111(n/d) * a(d).
a(n) = A354867(n) - A122111(n).

A354358 Möbius transform of A124859.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 24, 4, 1, 1, 4, 1, 1, 1, 180, 1, 4, 1, 4, 1, 1, 1, 24, 4, 1, 24, 4, 1, 1, 1, 2100, 1, 1, 1, 16, 1, 1, 1, 24, 1, 1, 1, 4, 4, 1, 1, 180, 4, 4, 1, 4, 1, 24, 1, 24, 1, 1, 1, 4, 1, 1, 4, 27720, 1, 1, 1, 4, 1, 1, 1, 96, 1, 1, 4, 4, 1, 1, 1, 180, 180, 1, 1, 4, 1, 1, 1, 24, 1, 4, 1, 4, 1, 1, 1, 2100
Offset: 1

Views

Author

Antti Karttunen, Jun 05 2022

Keywords

Comments

Multiplicative because A124859 is.

Links

Crossrefs

Cf. A002110, A008683, A124859.
Cf. also A347379, A354359.

Programs

  • Mathematica
    primorial[n_] := Product[Prime[i], {i, 1, n}]; primorial[0] = 1; f[p_, e_] := primorial[e] - primorial[e-1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 07 2023 *)
  • PARI
    A124859(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1] = prod(j=1, f[k, 2], prime(j)); f[k, 2] = 1); factorback(f); }; \\ From A124859
A354358(n) = sumdiv(n,d,moebius(n/d)*A124859(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A124859(d).
Multiplicative with a(p^e) = primorial(e) - primorial(e-1). - Sebastian Karlsson, Jul 30 2022

A354826 Dirichlet inverse of A238745.

Original entry on oeis.org

1, -2, -2, 0, -2, 5, -2, 0, 0, 5, -2, -2, -2, 5, 5, 0, -2, -2, -2, -2, 5, 5, -2, 0, 0, 5, 0, -2, -2, -17, -2, 0, 5, 5, 5, 8, -2, 5, 5, 0, -2, -17, -2, -2, -2, 5, -2, 0, 0, -2, 5, -2, -2, 0, 5, 0, 5, 5, -2, 16, -2, 5, -2, 0, 5, -17, -2, -2, 5, -17, -2, -8, -2, 5, -2, -2, 5, -17, -2, 0, 0, 5, -2, 16, 5, 5, 5, 0, -2, 16
Offset: 1

Views

Author

Antti Karttunen, Jun 09 2022

Keywords

Links

Crossrefs

Cf. A238745.
Cf. also A354349, A354359.

Programs

  • PARI
    A124859(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1] = prod(j=1, f[k, 2], prime(j)); f[k, 2] = 1); factorback(f); }; \\ From A124859
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A238745(n) = A181819(A124859(n));
memoA354826 = Map();
A354826(n) = if(1==n,1,my(v); if(mapisdefined(memoA354826,n,&v), v, v = -sumdiv(n,d,if(dA238745(n/d)*A354826(d),0)); mapput(memoA354826,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA238745(n/d) * a(d).
Showing 1-5 of 5 results.

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