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.

A349389 a(n) = A349387(n) + A349388(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 5, 4, 4, 0, 10, 0, 8, 8, 19, 0, 16, 0, 10, 16, 4, 0, 26, 4, 8, 32, 20, 0, 0, 0, 65, 8, 4, 16, 42, 0, 8, 16, 26, 0, 0, 0, 10, 32, 12, 0, 70, 16, 24, 8, 20, 0, 68, 8, 52, 16, 4, 0, 4, 0, 12, 64, 211, 16, 0, 0, 10, 24, 0, 0, 114, 0, 8, 48, 20, 16, 0, 0, 70, 196, 4, 0, 8, 8, 8, 8, 26, 0, 8, 32, 30
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Links

Crossrefs

Cf. A349387, A349388.
Cf. also A349383.

Programs

  • Mathematica
    f1[p_,e_] := (q = NextPrime[p])^e - p * q^(e-1); f2[p_, e_] := p^e - NextPrime[p] * p^(e-1); a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    A349389(n) = (A349387(n) + A349388(n)); \\ Needs also code from A349387 and A349388.

Formula

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A349387(d) * A349388(n/d). [As the sequences are Dirichlet inverses of each other]

A349387 Dirichlet convolution of A003961 with A055615 (Dirichlet inverse of n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 4, 9, 10, 2, 2, 6, 4, 4, 4, 27, 2, 10, 4, 6, 8, 2, 6, 18, 14, 4, 50, 12, 2, 4, 6, 81, 4, 2, 8, 30, 4, 4, 8, 18, 2, 8, 4, 6, 20, 6, 6, 54, 44, 14, 4, 12, 6, 50, 4, 36, 8, 2, 2, 12, 6, 6, 40, 243, 8, 4, 4, 6, 12, 8, 2, 90, 6, 4, 28, 12, 8, 8, 4, 54, 250, 2, 6, 24, 4, 4, 4, 18, 8, 20, 16, 18, 12, 6
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Multiplicative because A003961 and A055615 are.
Convolving this with A000010 gives A003972, and convolving this with A000203 gives A003973.
Multiplicative with a(p^e) = nextprime(p)^e - p * nextprime(p)^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021

Links

Crossrefs

Cf. A000040, A001223, A003961, A055615, A151800, A349388 (Dirichlet inverse), A349389 (sum with it), A378606 (Möbius transform).
Cf. also A000010, A000203, A003972, A003973, A347236, A349381.

Programs

  • Mathematica
    f[p_,e_] := (q = NextPrime[p])^e - p * q^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A055615(n) = (n*moebius(n));
A349387(n) = sumdiv(n,d,A003961(n/d)*A055615(d));

Formula

a(n) = Sum_{d|n} A003961(n/d) * A055615(d).
For all n >= 1, a(A000040(n)) = A001223(n).

A349572 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A048673.

Original entry on oeis.org

1, 0, 0, -1, 1, -2, 1, -4, -4, -3, 4, -4, 4, -5, -6, -12, 7, -6, 7, -7, -10, -6, 8, -6, -4, -8, -24, -11, 13, -2, 12, -32, -12, -9, -14, -4, 16, -11, -16, -13, 19, -2, 19, -16, -22, -14, 20, -4, -18, -15, -18, -20, 23, -10, -14, -19, -22, -15, 28, 14, 27, -18, -34, -80, -20, -8, 31, -25, -28, -8, 34, 14, 33, -20
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Comments

Also Dirichlet convolution of A349384 with A349388.

Links

Crossrefs

Cf. A000027, A048673, A323893, A349384, A349388, A349571 (Dirichlet inverse).
Cf. also A349397.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
A349572(n) = sumdiv(n,d,d*A323893(n/d));

Formula

a(n) = Sum_{d|n} d * A323893(n/d).
a(n) = Sum_{d|n} A349384(d) * A349388(n/d).

A378607 Dirichlet convolution of sigma and the Dirichlet inverse of A003961 (A346234).

Original entry on oeis.org

1, 0, -1, -2, -1, 0, -3, -6, -7, 0, -1, 2, -3, 0, 1, -14, -1, 0, -3, 2, 3, 0, -5, 6, -11, 0, -25, 6, -1, 0, -5, -30, 1, 0, 3, 14, -3, 0, 3, 6, -1, 0, -3, 2, 7, 0, -5, 14, -31, 0, 1, 6, -5, 0, 1, 18, 3, 0, -1, -2, -5, 0, 21, -62, 3, 0, -3, 2, 5, 0, -1, 42, -5, 0, 11, 6, 3, 0, -3, 14, -79, 0, -5, -6, 1, 0, 1, 6, -7, 0, 9, 10
Offset: 1

Views

Author

Antti Karttunen, Dec 11 2024

Keywords

Links

Crossrefs

Cf. A000203, A003961, A016825, A151800, A346234, A378606 (Dirichlet inverse).
Inverse Möbius transform of A349388.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - NextPrime[p]*(p^e - 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 12 2025 *)
  • PARI
    A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A378607(n) = sumdiv(n,d,sigma(d)*A346234(n/d));

Formula

a(n) = Sum_{d|n} A000203(d)*A346234(n/d).
a(n) = Sum_{d|n} A349388(d).
Multiplicative with a(p^e) = (p^(e+1) - nextprime(p)*(p^e-1) - 1)/(p-1), where nextprime(p) = A151800(p). - Amiram Eldar, Jan 12 2025
Showing 1-4 of 4 results.

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