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.

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]

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

A349632 Dirichlet convolution of A250469 with A346234, which is Dirichlet inverse of A003961.

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, -90, 0, -180, 0, -96, 30, -132, 0, 48, 0, 96, -60, -108, 0, -174, 0, 12, -120
Offset: 1

Views

Author

Antti Karttunen, Nov 27 2021

Keywords

Comments

Note that for n = 2..36, a(n) = -A349631(n).
Dirichlet convolution of this sequence with A003972 is A347376.

Links

Crossrefs

Cf. A003961, A250469, A346234, A349631 (Dirichlet inverse).
Cf. also A003972, A347376, A349382.
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));
A346234(n) = (moebius(n)*A003961(n));
A349632(n) = sumdiv(n,d,A250469(n/d)*A346234(d));

Formula

a(n) = Sum_{d|n} A250469(d) * A346234(n/d).
Showing 1-4 of 4 results.

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