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.

A349357 a(n) = A349355(n) + A349356(n).

Original entry on oeis.org

2, 0, 0, 4, 0, 8, 0, 16, 4, 8, 0, 16, 0, 8, 8, 52, 0, 24, 0, 16, 8, 8, 0, 40, 4, 8, 24, 16, 0, 0, 0, 160, 8, 8, 8, 56, 0, 8, 8, 40, 0, 0, 0, 16, 24, 8, 0, 112, 4, 40, 8, 16, 0, 80, 8, 40, 8, 8, 0, 16, 0, 8, 24, 484, 8, 0, 0, 16, 8, 0, 0, 152, 0, 8, 40, 16, 8, 0, 0, 112, 112, 8, 0, 16, 8, 8, 8, 40, 0, 16, 8, 16
Offset: 1

Views

Author

Antti Karttunen, Nov 16 2021

Keywords

Links

Crossrefs

Cf. A003958, A003959, A063441, A097945, A349355, A349356.

Programs

  • Mathematica
    f1[p_, e_] := -2*(p - 1)^(e - 1); f2[p_, e_] := 2*(p + 1)^(e - 1); a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 16 2021 *)
  • PARI
    A349357(n) = (A349355(n)+A349356(n)); \\ Needs also code from A349355 and A349356.

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349355(d) * A349356(n/d). [As the sequences are Dirichlet inverses of each other]

A349173 Dirichlet convolution of A003415 with A003959, where A003415 is the arithmetic derivative and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 7, 1, 12, 1, 33, 10, 16, 1, 68, 1, 20, 18, 131, 1, 87, 1, 96, 22, 28, 1, 296, 16, 32, 67, 124, 1, 167, 1, 473, 30, 40, 26, 449, 1, 44, 34, 428, 1, 215, 1, 180, 147, 52, 1, 1128, 22, 171, 42, 208, 1, 510, 34, 560, 46, 64, 1, 881, 1, 68, 187, 1611, 38, 311, 1, 264, 54, 295, 1, 1871, 1, 80, 203, 292, 38, 359
Offset: 1

Views

Author

Antti Karttunen, Nov 09 2021

Keywords

Links

Crossrefs

Cf. A003415, A003959, A349129, A349133, A349170, A349171, A349172, A349356.

Programs

  • Mathematica
    f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a1[1] = 0; a1[n_] := n*Plus @@ (f1 @@@ FactorInteger[n]); a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := DivisorSum[n, a1[#] * a2[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 09 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A349173(n) = sumdiv(n,d,A003415(d)*A003959(n/d));

Formula

a(n) = Sum_{d|n} A003415(d) * A003959(n/d).
a(n) = Sum_{d|n} A349133(d) * A349356(n/d). - Antti Karttunen, Nov 16 2021
For all n >= 1, a(n) >= A349133(n).

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).

A349355 Dirichlet convolution of A003958 with A063441 (Dirichlet inverse of A003959), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

Original entry on oeis.org

1, -2, -2, -2, -2, 4, -2, -2, -4, 4, -2, 4, -2, 4, 4, -2, -2, 8, -2, 4, 4, 4, -2, 4, -8, 4, -8, 4, -2, -8, -2, -2, 4, 4, 4, 8, -2, 4, 4, 4, -2, -8, -2, 4, 8, 4, -2, 4, -12, 16, 4, 4, -2, 16, 4, 4, 4, 4, -2, -8, -2, 4, 8, -2, 4, -8, -2, 4, 4, -8, -2, 8, -2, 4, 16, 4, 4, -8, -2, 4, -16, 4, -2, -8, 4, 4, 4, 4, -2, -16
Offset: 1

Views

Author

Antti Karttunen, Nov 16 2021

Keywords

Comments

Multiplicative because both A003958 and A063441 are.
In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003959 with factor A003958. For example, convolving this with A003968 (the Möbius transform of A003959) produces A003966, the Möbius transform of A003958.

Links

Crossrefs

Cf. A003958, A003959, A003966, A003968, A063441, A349356 (Dirichlet inverse), A349357 (sum with it).
Cf. also A349382.

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
    A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A063441(n) = (moebius(n)*sigma(n)); \\ Also Dirichlet inverse of A003959.
A349355(n) = sumdiv(n,d,A003958(n/d)*A063441(d));

Formula

a(n) = Sum_{d|n} A003958(n/d) * A063441(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
Showing 1-5 of 5 results.

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