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 1-7 of 7 results.

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]

A323910 Dirichlet inverse of the deficiency of n, A033879.

Original entry on oeis.org

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2
Offset: 1

Views

Author

Antti Karttunen, Feb 12 2019

Keywords

Links

Crossrefs

Cf. A033879, A323911, A323912, A359549 (parity of terms).
Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

Programs

  • Mathematica
    b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)
  • PARI
    up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A344587 Deficiency of prime-shifted n: a(n) = 2*A003961(n) - sigma(A003961(n)).

Original entry on oeis.org

1, 2, 4, 5, 6, 6, 10, 14, 19, 10, 12, 12, 16, 18, 22, 41, 18, 26, 22, 22, 38, 22, 28, 30, 41, 30, 94, 42, 30, 18, 36, 122, 46, 34, 58, 47, 40, 42, 62, 58, 42, 42, 46, 52, 102, 54, 52, 84, 109, 66, 70, 72, 58, 126, 70, 114, 86, 58, 60, 6, 66, 70, 178, 365, 94, 54, 70, 82, 110, 78, 72, 110, 78, 78, 148, 102, 118, 78
Offset: 1

Views

Author

Antti Karttunen, May 28 2021

Keywords

Comments

First negative value occurs as a(120) = -30.
Questions: Which subsets of natural numbers generate the "cut sigmoid" graph(s) that cross the X-axis in the (lowermost) scatter plot?

Links

Crossrefs

Cf. A000203, A003961, A003973, A033879, A153881, A336851, A337386 (positions of terms <= 0), A346246 (Dirichlet inverse), A349387, A378216, A378231 [= a(n^2)].
Inverse Möbius transform of A337544.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };

Formula

a(n) = A033879(A003961(n)) = 2*A003961(n) - A003973(n).
a(n) = Sum_{d|n} A337544(d).
From Antti Karttunen, Nov 23 2024: (Start)
a(n) = Sum_{d|n} A003961(d)*A153881(n/d) = A003961(n) - A336851(n).
a(n) = Sum_{d|n} A033879(d)*A349387(n/d).
a(n) = Sum_{d|n} A003972(d)*A378216(n/d).
(End)

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).

A349388 Dirichlet convolution of A000027 with A346234 (Dirichlet inverse of A003961), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, -1, -2, -2, -2, 2, -4, -4, -6, 2, -2, 4, -4, 4, 4, -8, -2, 6, -4, 4, 8, 2, -6, 8, -10, 4, -18, 8, -2, -4, -6, -16, 4, 2, 8, 12, -4, 4, 8, 8, -2, -8, -4, 4, 12, 6, -6, 16, -28, 10, 4, 8, -6, 18, 4, 16, 8, 2, -2, -8, -6, 6, 24, -32, 8, -4, -4, 4, 12, -8, -2, 24, -6, 4, 20, 8, 8, -8, -4, 16, -54, 2, -6, -16, 4, 4, 4
Offset: 1

Views

Author

Antti Karttunen, Nov 17 2021

Keywords

Comments

Multiplicative because A000027 and A346234 are.

Links

Crossrefs

Cf. A000027, A000040, A001223, A003961, A151800, A346234, A349387 (Dirichlet inverse), A349389 (sum with it), A378607 (inverse Möbius transform).
Cf. also A347238.

Programs

  • Mathematica
    f[p_, e_] := p^e - NextPrime[p] * p^(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); };
A346234(n) = (moebius(n)*A003961(n));
A349388(n) = sumdiv(n,d,d*A346234(n/d));

Formula

a(n) = Sum_{d|n} d * A346234(n/d).
For all n >= 1, a(A000040(n)) = -A001223(n).
Multiplicative with a(p^e) = p^e - nextprime(p) * p^(e-1), where nextprime function is A151800. - Amiram Eldar, Nov 18 2021

A349571 Dirichlet convolution of A048673 with A055615 (Dirichlet inverse of n).

Original entry on oeis.org

1, 0, 0, 1, -1, 2, -1, 4, 4, 3, -4, 4, -4, 5, 6, 13, -7, 6, -7, 5, 10, 6, -8, 10, 5, 8, 24, 9, -13, -2, -12, 40, 12, 9, 16, 16, -16, 11, 16, 11, -19, -2, -19, 8, 14, 14, -20, 28, 19, 9, 18, 12, -23, 26, 22, 21, 22, 15, -28, 2, -27, 18, 26, 121, 28, -8, -31, 11, 28, -8, -34, 46, -33, 20, 18, 15, 34, -8, -37, 29, 124
Offset: 1

Views

Author

Antti Karttunen, Nov 23 2021

Keywords

Comments

Also Dirichlet convolution of A349385 with A349387.

Links

Crossrefs

Cf. A048673, A055615, A349385, A349387, A349572 (Dirichlet inverse).
Cf. also A349398, A349573.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)
  • PARI
    A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A055615(n) = (n*moebius(n));
A349571(n) = sumdiv(n,d,A048673(n/d)*A055615(d));

Formula

a(n) = Sum_{d|n} A048673(n/d) * A055615(d).
a(n) = Sum_{d|n} A349385(n/d) * A349387(d).

A378606 Dirichlet convolution of A046692 and A003961, where A046692 is the Dirichlet inverse of sigma, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 3, 6, 8, 0, 1, 2, 3, 0, 1, 18, 1, 0, 3, 2, 3, 0, 5, 6, 12, 0, 40, 6, 1, 0, 5, 54, 1, 0, 3, 16, 3, 0, 3, 6, 1, 0, 3, 2, 8, 0, 5, 18, 40, 0, 1, 6, 5, 0, 1, 18, 3, 0, 1, 2, 5, 0, 24, 162, 3, 0, 3, 2, 5, 0, 1, 48, 5, 0, 12, 6, 3, 0, 3, 18, 200, 0, 5, 6, 1, 0, 1, 6, 7, 0, 9, 10, 5, 0, 3, 54, 3, 0, 8, 24
Offset: 1

Views

Author

Antti Karttunen, Dec 11 2024

Keywords

Links

Crossrefs

Cf. A003961, A008683, A016825 (positions of 0's), A046692, A151800, A349387 (inverse Möbius transform), A378607 (Dirichlet inverse).

Programs

  • Mathematica
    f[p_, e_] := Module[{q = NextPrime[p]}, If[e == 1, q - p - 1, q^e - (p + 1)*q^(e - 1) + p*q^(e - 2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 11 2024 *)
  • PARI
    A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A046692(n) = { my(f=factor(n)~); prod(i=1, #f, if(1==f[2,i], -(f[1,i]+1), if(2==f[2,i], f[1,i], 0))); };
A378606(n) = sumdiv(n,d,A046692(d)*A003961(n/d));

Formula

a(n) = Sum_{d|n} A046692(d)*A003961(n/d).
a(n) = Sum_{d|n} A008683(d)*A349387(n/d).
Multiplicative with a(p^e) = q(p)^e - (p+1) * q(p)^(e-1) + p * q(p)^(e-2) if e >= 2, and q(p) - p - 1 if e = 1, where q(p) = A151800(p) is the prime next to p. - Amiram Eldar, Dec 11 2024
Showing 1-7 of 7 results.

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