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.

A346249 Möbius transform of A337549.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 3, 8, 10, 1, 1, 8, 3, 3, 7, 28, 1, 10, 3, 12, 15, 1, 5, 28, 16, 3, 62, 24, 1, 7, 5, 92, 13, 1, 21, 52, 3, 3, 21, 44, 1, 15, 3, 24, 56, 5, 5, 92, 58, 16, 19, 36, 5, 62, 15, 84, 27, 1, 1, 44, 5, 5, 108, 292, 27, 13, 3, 36, 37, 21, 1, 168, 5, 3, 68, 48, 39, 21, 3, 148, 346, 1, 5, 84, 21, 3, 31, 92, 7, 56
Offset: 1

Views

Author

Antti Karttunen, Jul 14 2021

Keywords

Links

Crossrefs

Cf. A000010, A000040, A001223, A003961, A003972, A008683, A337549.
Sequences A001359, A029710, A031924 give the subsets of positions of 1's, 3's and 5's in this sequence.

Programs

  • PARI
    A003972(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); eulerphi(factorback(f)); };
A337549(n) = (A003972(n) - n);
A346249(n) = sumdiv(n,d,moebius(n/d)*A337549(d));

Formula

a(n) = A008683(n/d) * A337549(d).
a(A000040(n)) = A001223(n)-1. - Antti Karttunen, Dec 06 2021

A286385 a(n) = A003961(n) - A000203(n).

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 3, 12, 12, 3, 1, 17, 3, 9, 11, 50, 1, 36, 3, 21, 23, 3, 5, 75, 18, 9, 85, 43, 1, 33, 5, 180, 17, 3, 29, 134, 3, 9, 29, 99, 1, 69, 3, 33, 97, 15, 5, 281, 64, 54, 23, 55, 5, 255, 19, 177, 35, 3, 1, 147, 5, 15, 171, 602, 35, 51, 3, 45, 49, 87, 1, 480, 5, 9, 121, 67, 47, 87, 3, 381, 504, 3, 5, 271, 25, 9, 35, 171, 7, 291, 75, 93, 57, 15, 41, 963
Offset: 1

Views

Author

Antti Karttunen, May 09 2017

Keywords

Comments

Are all terms nonnegative? This question is equivalent to the question posed in A285705.
From Antti Karttunen, Aug 05 2020: (Start)
The answer to the above question is yes. Because both A000203 and A003961 are multiplicative sequences, it suffices to prove that for any prime p, and e >= 1, q^e >= sigma(p^e) = ((p^(1+e))-1) / (p-1), where q = A151800(p), i.e., the next larger prime after p. If p is a lesser twin prime, then q = p+2 (and this difference can't be less than 2, apart from case p=2), and it is easy to see that (n+2)^e > ((n^(e+1)) - 1) / (n-1), for all n >= 2, e >= 1.
See comments in A326042.
(End)
This is the inverse Möbius transform of A337549, from which it is even easier to see that all terms are nonnegative. - Antti Karttunen, Sep 22 2020

Links

Crossrefs

Cf. A000203, A001359, A003961, A001065, A031924, A033879, A048673, A151800, A285705, A326042, A336702, A336851, A336852, A337549 (Möbius transform).
Cf. A326057 [= gcd(a(n), A252748(n))].

Programs

  • Mathematica
    Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] - DivisorSigma[1, #] &, 96] (* Michael De Vlieger, Oct 05 2020 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A286385(n) = (A003961(n) - sigma(n));
for(n=1, 16384, write("b286385.txt", n, " ", A286385(n)));
  • Python
    from sympy import factorint, nextprime, divisor_sigma as D
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return 2*a048673(n) - D(n) - 1 # Indranil Ghosh, May 12 2017
  • Scheme
    (define (A286385 n) (- (A003961 n) (A000203 n)))

Formula

a(n) = A285705(A048673(n)) - 1 = 2*A048673(n) - A000203(n) - 1.
a(n) = A336852(n) - A336851(n). - Antti Karttunen, Aug 05 2020
a(n) = Sum_{d|n} A337549(d). - Antti Karttunen, Sep 22 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) - Pi^2/12 = 1.24152934..., where q(p) = nextprime(p) (A151800). - Amiram Eldar, Dec 21 2023

A347100 a(n) = phi(A003961(n)) - phi(n), where A003961 is the prime shift towards larger primes, and phi is Euler totient function.

Original entry on oeis.org

0, 1, 2, 4, 2, 6, 4, 14, 14, 8, 2, 20, 4, 14, 16, 46, 2, 34, 4, 28, 28, 14, 6, 64, 22, 20, 82, 48, 2, 40, 6, 146, 28, 20, 36, 108, 4, 26, 40, 92, 2, 68, 4, 52, 96, 34, 6, 200, 68, 64, 40, 72, 6, 182, 32, 156, 52, 32, 2, 128, 6, 42, 164, 454, 48, 76, 4, 76, 68, 96, 2, 336, 6, 44, 128, 96, 60, 104, 4, 292, 446, 44, 6
Offset: 1

Views

Author

Antti Karttunen, Aug 19 2021

Keywords

Comments

Möbius transform of A336853.

Links

Crossrefs

Möbius transform of A336853.
Cf. A000010, A000040, A001223, A003961, A003972, A008683, A051953, A337549.
Cf. also A346249, A347098.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; a[n_] := EulerPhi[Times @@ f @@@ FactorInteger[n]] - EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)
  • PARI
    A347100(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (eulerphi(factorback(f))-eulerphi(n)); };
  • PARI
    A336853(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)-n); };
A347100(n) = sumdiv(n,d,moebius(n/d)*A336853(d));

Formula

a(n) = A003972(n) - A000010(n).
a(n) = A337549(n) + A051953(n).
a(n) = Sum_{d|n} A008683(n/d) * A336853(d).
For all n >= 1, a(A000040(n)) = A001223(n).

A349754 a(n) = phi(A003961(n)) - 2*phi(n), where A003961 is fully multiplicative with a(p) = nextprime(p), and phi is Euler totient function.

Original entry on oeis.org

-1, 0, 0, 2, -2, 4, -2, 10, 8, 4, -8, 16, -8, 8, 8, 38, -14, 28, -14, 20, 16, 4, -16, 56, 2, 8, 64, 36, -26, 32, -24, 130, 8, 4, 12, 96, -32, 8, 16, 76, -38, 56, -38, 32, 72, 12, -40, 184, 26, 44, 8, 48, -46, 164, -8, 132, 16, 4, -56, 112, -54, 12, 128, 422, 0, 56, -62, 44, 24, 72, -68, 312, -66, 8, 88, 60, 0, 80
Offset: 1

Views

Author

Antti Karttunen, Dec 01 2021

Keywords

Comments

Möbius transform of A252748.

Links

Crossrefs

Cf. A000010, A003961, A003972, A008683, A083254, A252748, A337549, A347100.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := EulerPhi[s[n]] - 2*EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Dec 04 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349754(n) = (eulerphi(A003961(n))-2*eulerphi(n));

Formula

a(n) = A003972(n) - 2*A000010(n) = A337549(n) - A083254(n).
a(n) = A347100(n) - A000010(n).
a(n) = Sum_{d|n} A008683(n/d) * A252748(d).
Showing 1-4 of 4 results.

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