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 1-6 of 6 results.

A347238 Dirichlet inverse of A347236.

Original entry on oeis.org

1, -1, -2, -6, -2, 2, -4, 0, -15, 2, -2, 12, -4, 4, 4, 0, -2, 15, -4, 12, 8, 2, -6, 0, -35, 4, 0, 24, -2, -4, -6, 0, 4, 2, 8, 90, -4, 4, 8, 0, -2, -8, -4, 12, 30, 6, -6, 0, -77, 35, 4, 24, -6, 0, 4, 0, 8, 2, -2, -24, -6, 6, 60, 0, 8, -4, -4, 12, 12, -8, -2, 0, -6, 4, 70, 24, 8, -8, -4, 0, 0, 2, -6, -48, 4, 4, 4, 0, -8
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Multiplicative because A347236 is.
It seems that A046099 gives the positions of zeros.
This follows from the formula for a(p^e). - Sebastian Karlsson, Sep 01 2021

Links

Crossrefs

Cf. A000040, A001223, A003961, A046099, A061019, A347236, A347239.
Cf. A151800.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));
v347238 = DirInverseCorrect(vector(up_to,n,A347236(n)));
A347238(n) = v347238[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A347236(n/d) * a(d).
a(n) = A347239(n) - A347236(n).
For all n >= 1, a(A000040(n)) = -A001223(n).
Multiplicative with a(p^e) = p - A151800(p) if e = 1, -p*A151800(p) if e = 2 and 0 if e > 2. - Sebastian Karlsson, Sep 01 2021

A347237 Möbius transform of A347236.

Original entry on oeis.org

1, 0, 1, 6, 1, 0, 3, 6, 17, 0, 1, 6, 3, 0, 1, 42, 1, 0, 3, 6, 3, 0, 5, 6, 37, 0, 49, 18, 1, 0, 5, 78, 1, 0, 3, 102, 3, 0, 3, 6, 1, 0, 3, 6, 17, 0, 5, 42, 89, 0, 1, 18, 5, 0, 1, 18, 3, 0, 1, 6, 5, 0, 51, 330, 3, 0, 3, 6, 5, 0, 1, 102, 5, 0, 37, 18, 3, 0, 3, 42, 353, 0, 5, 18, 1, 0, 1, 6, 7, 0, 9, 30, 5, 0, 3, 78, 3, 0, 17
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Dirichlet convolution of A003972 (prime shifted phi) with A061019.
Dirichlet convolution of A003961 with A158523.
Multiplicative because A003972 and A061019 are, and also because A347236 is.
From Antti Karttunen, Aug 25 2021: (Start)
All terms are nonnegative because sequence is multiplicative and a(p^k) >= 0 for all primes p and k >= 0.
Proof: For any prime p, sequence a(p^k), k>=0, is obtained as an ordinary convolution of sequences (-p)^k and the first differences of q^k, where q = A151800(p). (E.g., for powers of 2, the sequences convolved are A122803 and A025192, giving A102901.) This convolution is an alternating sum, with the terms 1*(q-1)*q^(k-1), -(p)*(q-1)*q^(k-2), (p^2)*(q-1)*q^(k-3), -(p^3)*(q-1)*q^(k-4), ..., (p^(k-1))*(q-1), -(p^k), for odd k, with sum of each consecutive pair being nonnegative because q >= p+1, while with an even exponent k, the leftover term p^k at the end is also positive, thus the whole sum is nonnegative also in that case.
(End)

Links

Crossrefs

Cf. A000040, A001223, A003961, A003972, A008683, A008836, A016825 (positions of zeros), A061019, A102901, A120612, A158523, A347236.
Cf. also A347137.
Cf. A151800.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));
\\ Or alternatively as:
A158523(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((-1)^e)*(p+1)*(p^(e-1))); };
A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A347236(d).
a(n) = Sum_{d|n} A003972(n/d) * A061019(d).
a(n) = Sum_{d|n} A003961(n/d) * A158523(d).
For all n >= 1, a(A000040(n)) = A001223(n) - 1.
For all n >= 0, a(2^n) = A102901(n).
For all n >= 0, a(3^n) = A120612(n).
Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A347239 Sum of A347236 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 13, 4, 4, 0, 26, 0, 8, 8, 55, 0, 34, 0, 26, 16, 4, 0, 26, 4, 8, 68, 52, 0, 0, 0, 133, 8, 4, 16, 223, 0, 8, 16, 26, 0, 0, 0, 26, 68, 12, 0, 110, 16, 74, 8, 52, 0, 68, 8, 52, 16, 4, 0, 4, 0, 12, 136, 463, 16, 0, 0, 26, 24, 0, 0, 247, 0, 8, 148, 52, 16, 0, 0, 110, 421, 4, 0, 8, 8, 8, 8, 26, 0, 8, 32
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

It seems that A030059 gives the positions of all zeros.

Links

Crossrefs

Cf. A000290, A001223, A001248, A003961, A061019, A030059, A030229, A347236, A347238.

Programs

  • PARI
    up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));
v347238 = DirInverseCorrect(vector(up_to,n,A347236(n)));
A347238(n) = v347238[n];
A347239(n) = (A347236(n)+A347238(n));

Formula

a(n) = A347236(n) + A347238(n).
a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A347236(d) * A347238(n/d).
For all n >= 1, a(A030059(n)) = 0 and a(A030229(n)) = 2*A347236(A030229(n)).
For all n >= 1, a(A001248(n)) = A000290(A001223(n)).

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).

A347136 a(n) = Sum_{d|n} d * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes.

Original entry on oeis.org

1, 5, 8, 19, 12, 40, 18, 65, 49, 60, 24, 152, 30, 90, 96, 211, 36, 245, 42, 228, 144, 120, 52, 520, 109, 150, 272, 342, 60, 480, 68, 665, 192, 180, 216, 931, 78, 210, 240, 780, 84, 720, 90, 456, 588, 260, 100, 1688, 247, 545, 288, 570, 112, 1360, 288, 1170, 336, 300, 120, 1824, 128, 340, 882, 2059, 360, 960, 138
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Dirichlet convolution of the identity function (A000027) with the prime shifted identity (A003961). Multiplicative because both A000027 and A003961 are.
Dirichlet convolution of Euler phi (A000010) with the prime shifted sigma (A003973).
Dirichlet convolution of sigma (A000203) with the prime shifted phi (A003972).
Inverse Möbius transform of A347137.

Links

Crossrefs

Cf. A003961, A003972, A003973, A151800, A347121, A347137 (Möbius transform).
Cf. also A038040, A336845, A336475, A347236.

Programs

  • Mathematica
    f[p_, e_] := ((np = NextPrime[p])^(e + 1) - p^(e + 1))/(np - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347136(n) = sumdiv(n,d,d*A003961(n/d));

Formula

a(n) = Sum_{d|n} d * A003961(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A003973(d).
a(n) = Sum_{d|n} A000203(n/d) * A003972(d).
a(n) = Sum_{d|n} A347137(d).
For all primes p, a(p) = p + A003961(p).
a(n) = A347121(n) + 2*n.
Multiplicative with a(p^e) = (A151800(p)^(e+1) - p^(e+1))/(A151800(p)-p). - Amiram Eldar, Aug 24 2021

A347121 a(n) = A347136(n) - 2*n.

Original entry on oeis.org

-1, 1, 2, 11, 2, 28, 4, 49, 31, 40, 2, 128, 4, 62, 66, 179, 2, 209, 4, 188, 102, 76, 6, 472, 59, 98, 218, 286, 2, 420, 6, 601, 126, 112, 146, 859, 4, 134, 162, 700, 2, 636, 4, 368, 498, 168, 6, 1592, 149, 445, 186, 466, 6, 1252, 178, 1058, 222, 184, 2, 1704, 6, 216, 756, 1931, 230, 828, 4, 548, 278, 940, 2, 3041, 6
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Links

Crossrefs

Cf. A000040, A001223, A003961, A347136, A347122 (Möbius transform).
Cf. also A341512, A346239, A347236.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347121(n) = (sumdiv(n,d,d*A003961(n/d))-(2*n));

Formula

a(n) = A347136(n) - 2*n.
a(A000040(n)) = A001223(n).
Showing 1-6 of 6 results.

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