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.

A349381 Dirichlet convolution of A003961 with A349125 (Dirichlet inverse of A064989), 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, 6, 4, 6, 6, 18, 15, 8, 6, 18, 6, 12, 12, 54, 6, 30, 6, 24, 18, 12, 10, 54, 28, 12, 75, 36, 8, 24, 8, 162, 18, 12, 24, 90, 10, 12, 18, 72, 6, 36, 6, 36, 60, 20, 10, 162, 66, 56, 18, 36, 12, 150, 24, 108, 18, 16, 8, 72, 8, 16, 90, 486, 24, 36, 10, 36, 30, 48, 6, 270, 8, 20, 84, 36, 36, 36, 10, 216, 375, 12
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Multiplicative because both A003961 and A349125 are.
Convolving this with A349127 gives A003972.

Links

Crossrefs

Cf. A003961, A064989, A349125, A349382 (Dirichlet inverse), A349383 (sum with it).
Cf. also A003972, A349127, and A349355, A349356 and A349384, A349385, and A349387.

Programs

  • 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); };
A349125(n) = (moebius(n)*A064989(n));
A349381(n) = sumdiv(n,d,A003961(n/d)*A349125(d));

Formula

a(n) = Sum_{d|n} A003961(n/d) * A349125(d).
a(n) = A349383(n) - A349382(n).

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

A349386 a(n) = A349384(n) + A349385(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 3, 4, 6, 0, 2, 0, 10, 12, 7, 0, 0, 0, 3, 20, 12, 0, -4, 9, 16, 16, 5, 0, -36, 0, 15, 24, 18, 30, -10, 0, 22, 32, -6, 0, -60, 0, 6, 0, 28, 0, -20, 25, -3, 36, 8, 0, -20, 36, -10, 44, 30, 0, -48, 0, 36, 0, 31, 48, -72, 0, 9, 56, -90, 0, -32, 0, 40, -6, 11, 60, -96, 0, -30, 52, 42, 0, -80, 54, 46, 60
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Links

Crossrefs

Cf. A003961, A048673, A349384, A349385.
Cf. also A349383.

Programs

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349384(d) * A349385(n/d). [As the sequences are Dirichlet inverses of each other]

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]
Showing 1-4 of 4 results.

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