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.

A349383 a(n) = A349381(n) + A349382(n).

Original entry on oeis.org

2, 0, 0, 4, 0, 12, 0, 16, 9, 16, 0, 24, 0, 24, 24, 52, 0, 42, 0, 32, 36, 24, 0, 60, 16, 24, 63, 48, 0, 0, 0, 160, 36, 24, 48, 102, 0, 24, 36, 80, 0, 0, 0, 48, 84, 40, 0, 168, 36, 80, 36, 48, 0, 174, 48, 120, 36, 32, 0, 48, 0, 32, 126, 484, 48, 0, 0, 48, 60, 0, 0, 282, 0, 40, 120, 48, 72, 0, 0, 224, 351, 24, 0, 72, 48
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Links

Crossrefs

Cf. A003961, A064989, A349381, A349382.
Cf. also A349357.

Programs

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349381(d)*A349382(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).

A349356 Dirichlet convolution of A003959 with A097945 (Dirichlet inverse of A003958), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 18, 8, 4, 2, 12, 2, 4, 4, 54, 2, 16, 2, 12, 4, 4, 2, 36, 12, 4, 32, 12, 2, 8, 2, 162, 4, 4, 4, 48, 2, 4, 4, 36, 2, 8, 2, 12, 16, 4, 2, 108, 16, 24, 4, 12, 2, 64, 4, 36, 4, 4, 2, 24, 2, 4, 16, 486, 4, 8, 2, 12, 4, 8, 2, 144, 2, 4, 24, 12, 4, 8, 2, 108, 128, 4, 2, 24, 4, 4, 4, 36, 2, 32, 4, 12, 4
Offset: 1

Views

Author

Antti Karttunen, Nov 16 2021

Keywords

Comments

In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003958 with factor A003959. For example, convolving this with A349133 produces A349173.

Links

Crossrefs

Cf. A003958, A003959, A097945, A349355 (Dirichlet inverse), A349357 (sum with it).
Cf. also A349133, A349173, A349381.

Programs

  • Mathematica
    f[p_, e_] := 2*(p + 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A097945(n) = (moebius(n)*eulerphi(n)); \\ Also Dirichlet inverse of A003958.
A349356(n) = sumdiv(n,d,A003959(n/d)*A097945(d));

Formula

a(n) = Sum_{d|n} A003959(n/d) * A097945(d).
Multiplicative with a(p^e) = 2*(p+1)^(e-1). - Amiram Eldar, Nov 16 2021

A349382 Dirichlet convolution of A064989 with A346234 (Dirichlet inverse of A003961), where A003961 and A064989 are fully multiplicative sequences that shift the prime factorization of n one step towards larger and smaller primes respectively.

Original entry on oeis.org

1, -2, -3, -2, -4, 6, -6, -2, -6, 8, -6, 6, -6, 12, 12, -2, -6, 12, -6, 8, 18, 12, -10, 6, -12, 12, -12, 12, -8, -24, -8, -2, 18, 12, 24, 12, -10, 12, 18, 8, -6, -36, -6, 12, 24, 20, -10, 6, -30, 24, 18, 12, -12, 24, 24, 12, 18, 16, -8, -24, -8, 16, 36, -2, 24, -36, -10, 12, 30, -48, -6, 12, -8, 20, 36, 12, 36, -36
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Multiplicative because both A064989 and A346234 are.

Links

Crossrefs

Cf. A003961, A064989, A151799, A151800, A346234, A349381 (Dirichlet inverse), A349383 (sum with it).
Cf. also A349355, A349356.

Programs

  • Mathematica
    f[p_, e_] := If[p == 2, -2, NextPrime[p, -1]^e - NextPrime[p]*NextPrime[p, -1]^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A346234(n) = (moebius(n)*A003961(n));
A349382(n) = sumdiv(n,d,A064989(n/d)*A346234(d));

Formula

a(n) = Sum_{d|n} A064989(n/d) * A346234(d).
a(n) = A349383(n) - A349381(n).
Multiplicative with a(p^e) = -2 if p = 2, and prevprime(p)^e - nextprime(p) * prevprime(p)^(e-1) otherwise, where prevprime function is A151799 and nextprime function is A151800. - Amiram Eldar, Nov 17 2021

A349631 Dirichlet convolution of A003961 with A346479, which is Dirichlet inverse of A250469.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 6, 0, -6, 0, 12, 0, -6, 0, 18, 0, 24, 0, 24, 0, -24, 0, 0, 0, -24, 60, 36, 0, 48, 0, 42, -20, -42, 0, -12, 0, -42, -10, 12, 0, 72, 0, 60, 60, -48, 0, -24, 0, 42, -30, 72, 0, -84, 0, 12, -30, -78, 0, -120, 0, -72, 120, 126, 0, 180, 0, 96, -30, 132, 0, -48, 0, -96, 60, 108, 0, 174, 0, -84, 120
Offset: 1

Views

Author

Antti Karttunen, Nov 27 2021

Keywords

Comments

Note that for n = 2..36, a(n) = -A349632(n).
Dirichlet convolution of this sequence with A347376 is A003972.

Links

Crossrefs

Cf. A003961, A250469, A346479, A349632 (Dirichlet inverse).
Cf. also A003972, A347376, A349381.
Cf. also arrays A083221, A246278, A249821, A249822 and permutations A250245, A250246.

Programs

  • PARI
    up_to = 20000;
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A250469(n) = if(1==n,n,my(spn = nextprime(1+A020639(n)), c = A078898(n), k = 0); while(c, k++; if((1==k)||(A020639(k)>=spn),c -= 1)); (k*spn));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA250469(n)));
A346479(n) = v346479[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A349631(n) = sumdiv(n,d,A003961(d)*A346479(n/d));

Formula

a(n) = Sum_{d|n} A003961(d) * A346479(n/d).
Showing 1-5 of 5 results.

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