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 1-3 of 3 results.

A348970 a(n) = A003959(n) - A129283(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 7, 1, 1, 0, 8, 0, 1, 1, 33, 0, 9, 0, 10, 1, 1, 0, 40, 1, 1, 10, 12, 0, 11, 0, 131, 1, 1, 1, 48, 0, 1, 1, 54, 0, 13, 0, 16, 12, 1, 0, 164, 1, 13, 1, 18, 0, 57, 1, 68, 1, 1, 0, 64, 0, 1, 14, 473, 1, 17, 0, 22, 1, 15, 0, 204, 0, 1, 14, 24, 1, 19, 0, 230, 67, 1, 0, 80, 1, 1, 1, 96, 0, 75, 1, 28
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

There are no negative terms. We prove this by induction over the prime factorization of n, showing that A348507(n) >= A003415(n) for all values of n >= 1. At n=1, both sequences have value 0, and at the primes both sequences obtain the value 1, so the base cases hold. We know that A348507(n)-(n/p) = (p+1)*A348507(n/p) for all prime factors p of n (see comment in A348507). With the arithmetic derivative we obtain respectively that A003415(n) = A003415(p*(n/p)) = A003415(p)*(n/p) + p*A003415(n/p) = (n/p) + p*A003415(n/p), for any prime factor p of n. Now A348507(p*(n/p)) >= A003415(p*(n/p)) iff A348507(p*(n/p)) - (n/p) >= A003415(p*(n/p)) - (n/p), that is, iff (p+1)*A348507(n/p) >= p*A003415(n/p), which indeed follows by the induction hypothesis, which assumes that A348507(x) >= A003415(x) for all proper divisors x of n.

Links

Crossrefs

Cf. A003415, A003959, A129283, A211991, A348029, A348507, A348971, A348972, A348973, A348974.
Cf. A008578 (positions of zeros), A001358 (positions of ones).

Programs

  • Mathematica
    d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n - d[n]; Array[a, 100] (* Amiram Eldar, Nov 05 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); };
A348970(n) = (A003959(n) - (n+A003415(n)));

Formula

a(n) = A003959(n) - A129283(n) = A003959(n) - (n+A003415(n)).
a(n) = A348029(n) - A211991(n).
a(n) = A348507(n) - A003415(n).
For all n >= 1, a(A001358(n)) = 1.

A348973 Numerator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 8, 1, 11, 1, 20, 15, 17, 1, 7, 1, 23, 23, 16, 1, 13, 1, 22, 31, 35, 1, 17, 35, 41, 27, 5, 1, 61, 1, 112, 47, 53, 47, 2, 1, 59, 55, 2, 1, 83, 1, 23, 7, 71, 1, 40, 63, 95, 71, 6, 1, 45, 71, 37, 79, 89, 1, 19, 1, 95, 57, 256, 83, 127, 1, 70, 95, 43, 1, 19, 1, 113, 65, 13, 95, 149, 1, 128, 189, 125, 1, 13, 107
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Comments

It is known that A129283(n) <= A003959(n) for all n (see A348970 for a proof), which implies that each ratio a(n)/A348974(n) is at most 1: 1/1, 1/1, 1/1, 8/9, 1/1, 11/12, 1/1, 20/27, 15/16, 17/18, 1/1, 7/9, 1/1, 23/24, 23/24, 16/27, 1/1, 13/16, 1/1, 22/27, 31/32, 35/36, 1/1, 17/27, 35/36, 41/42, 27/32, 5/6, 1/1, 61/72, 1/1, 112/243, etc.

Links

Crossrefs

Cf. A003415, A003959, A129283, A348970, A348972, A348974 (denominators).
Cf. also A345059.

Programs

  • Mathematica
    f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Numerator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 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); };
A348973(n) = { my(u=n+A003415(n)); (u/gcd(A003959(n),u)); };

Formula

a(n) = A129283(n) / A348972(n) = A129283(n) / gcd(A003959(n), A129283(n)).

A348974 Denominator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 9, 1, 12, 1, 27, 16, 18, 1, 9, 1, 24, 24, 27, 1, 16, 1, 27, 32, 36, 1, 27, 36, 42, 32, 6, 1, 72, 1, 243, 48, 54, 48, 3, 1, 60, 56, 3, 1, 96, 1, 27, 8, 72, 1, 81, 64, 108, 72, 7, 1, 64, 72, 54, 80, 90, 1, 27, 1, 96, 64, 729, 84, 144, 1, 81, 96, 48, 1, 36, 1, 114, 72, 15, 96, 168, 1, 243, 256, 126, 1, 18, 108
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Links

Crossrefs

Cf. A003415, A003959, A129283, A348970, A348972, A348973 (numerators).
Cf. also A343227.

Programs

  • Mathematica
    f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Denominator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 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); };
A348974(n) = { my(s=A003959(n)); (s/gcd(s,(n+A003415(n)))); };

Formula

a(n) = A003959(n) / A348972(n) = A003959(n) / gcd(A003959(n), A129283(n)).
Showing 1-3 of 3 results.

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