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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.

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A336840 Inverse Möbius transform of A048673.

Original entry on oeis.org

1, 3, 4, 8, 5, 14, 7, 22, 17, 18, 8, 42, 10, 26, 26, 63, 11, 65, 13, 55, 38, 30, 16, 124, 30, 38, 80, 81, 17, 100, 20, 185, 44, 42, 50, 206, 22, 50, 56, 164, 23, 148, 25, 94, 127, 62, 28, 368, 68, 117, 62, 120, 31, 316, 58, 244, 74, 66, 32, 318, 35, 78, 189, 550, 74, 172, 37, 133, 92, 196, 38, 626, 41, 86, 174, 159
Offset: 1

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Comments

Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).
The local minima occur on primes p, where p/2 < a(p) <= (p+1).

Links

Crossrefs

Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336840(n) = sumdiv(n,d,A048673(d));
  • PARI
    A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };

Formula

a(n) = Sum_{d|n} A048673(d).
a(n) = (1/2) * (A000005(n) + A003973(n)).
a(n) = A113415(A003961(n)). - Antti Karttunen, Jun 01 2022
a(n) = A349371(A003961(n)) = A364063(A048673(n)). - Antti Karttunen, Nov 30 2024

A323893 Dirichlet inverse of A048673, where A048673(n) = (A003961(n)+1) / 2, and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

1, -2, -3, -1, -4, 4, -6, -2, -4, 5, -7, 3, -9, 7, 6, -4, -10, 8, -12, 4, 8, 8, -15, 8, -9, 10, -12, 6, -16, 5, -19, -8, 9, 11, 9, 8, -21, 13, 11, 11, -22, 11, -24, 7, 16, 16, -27, 20, -25, 18, 12, 9, -30, 32, 10, 17, 14, 17, -31, 6, -34, 20, 24, -16, 12, 14, -36, 10, 17, 20, -37, 16, -40, 22, 27, 12, 12, 20, -42, 28, -36, 23, -45, 12, 13
Offset: 1

Views

Author

Antti Karttunen, Feb 08 2019

Keywords

Links

Crossrefs

Cf. A003961, A048673, A323894, A349134, A378520 (Möbius transform).

Programs

  • PARI
    up_to = 20000;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v323893 = DirInverse(vector(up_to,n,A048673(n)));
A323893(n) = v323893[n];
  • PARI
    memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v))); \\ Antti Karttunen, Nov 30 2024

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA048673(n/d) * a(d).
a(n) = A349134(A003961(n)). - Antti Karttunen, Nov 30 2024
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