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.

A349122 Inverse Möbius transform of A349128, where A349128(n) = phi(A064989(n)), A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 5, 4, 4, 6, 7, 6, 11, 10, 6, 5, 13, 8, 17, 9, 10, 14, 19, 8, 9, 22, 8, 15, 23, 12, 29, 6, 14, 26, 15, 12, 31, 34, 22, 12, 37, 20, 41, 21, 12, 38, 43, 10, 25, 18, 26, 33, 47, 16, 21, 20, 34, 46, 53, 18, 59, 58, 20, 7, 33, 28, 61, 39, 38, 30, 67, 16, 71, 62, 18, 51, 35, 44, 73, 15, 16, 74, 79, 30, 39
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2021

Keywords

Comments

Multiplicative because A349128 is.

Links

Crossrefs

Cf. A000010, A003961, A064216, A064989, A151799, A285702, A349127, A349128, A349138.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p, -1]^e; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)
  • PARI
    A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i,1], 1, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1)))); };
A349122(n) = sumdiv(n,d,A349128(d));
  • Python
    from sympy import prevprime, factorint, prod
def f(p, e):
    return e+1 if p == 2 else prevprime(p)**e
def a(n):
    return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 15 2021

Formula

a(n) = Sum_{d|n} A349128(d).
For all n >= 1, a(A003961(n)) = n, a(2*n-1) = A064216(n).
From Sebastian Karlsson, Nov 15 2021: (Start)
a(2*n-1) = A064989(2*n-1).
Multiplicative with a(2^e) = e + 1 and a(p^e) = prevprime(p)^e for odd primes p. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/9) * Product_{p prime > 2} ((p^2-p)/(p^2-prevprime(p))) = 0.2942719052..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

A285702 a(n) = A000010(A064216(n)).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 10, 2, 12, 16, 4, 18, 6, 4, 22, 28, 6, 8, 30, 10, 36, 40, 4, 42, 20, 12, 46, 12, 16, 52, 58, 8, 20, 60, 18, 66, 70, 6, 24, 72, 8, 78, 24, 22, 82, 40, 28, 32, 88, 12, 96, 100, 8, 102, 106, 30, 108, 36, 20, 48, 42, 36, 18, 112, 40, 126, 64, 8, 130, 136, 42, 60, 44, 20, 138, 148, 24, 56, 150, 46, 72, 156, 12, 162, 110, 32, 166, 24, 52, 172, 178
Offset: 1

Views

Author

Antti Karttunen, Apr 26 2017

Keywords

Links

Crossrefs

Cf. A000010, A064216, A064989, A099774, A151799, A285703.
Odd bisection of the following sequences: A347115, A348045, A349127, A349128.

Programs

  • Mathematica
    Table[EulerPhi@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 91}] (* Michael De Vlieger, Apr 26 2017 *)
  • Scheme
    (define (A285702 n) (A000010 (A064216 n)))

Formula

a(n) = A000010(A064216(n)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.5366875995..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A349127 Möbius transform of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 4, 0, 2, 0, 6, 0, 10, 0, 2, 0, 12, 0, 16, 0, 4, 0, 18, 0, 6, 0, 4, 0, 22, 0, 28, 0, 6, 0, 8, 0, 30, 0, 10, 0, 36, 0, 40, 0, 4, 0, 42, 0, 20, 0, 12, 0, 46, 0, 12, 0, 16, 0, 52, 0, 58, 0, 8, 0, 20, 0, 60, 0, 18, 0, 66, 0, 70, 0, 6, 0, 24, 0, 72, 0, 8, 0, 78, 0, 24, 0, 22, 0, 82, 0, 40, 0, 28, 0, 32
Offset: 1

Views

Author

Antti Karttunen, Nov 13 2021

Keywords

Comments

The multiplicative definition of this sequence ("Möbius transform of prime shift towards lesser primes") differs from otherwise similarly defined A349128 (Euler phi applied to A064989) only in that here a(2^e) = 0, while A349128(2^e) = 1.
Compare the situation with A003961 ("prime shift towards larger primes"), where A003972(n) = A000010(A003961(n)) is also the Möbius transform of A003961.

Links

Crossrefs

Agrees with A347115, A348045 and A349128 on odd numbers.
Cf. A000004, A285702 (even and odd bisection).
Cf. A000010, A008683, A064989, A151799.
Cf. also A003961, A003972, A349125, A349136.

Programs

  • Mathematica
    f[p_, e_] := ((q = NextPrime[p, -1]) - 1)*q^(e - 1); a[1] = 1; a[n_] := If[EvenQ[n], 0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)
  • PARI
    A349127(n) = if(!(n%2),0, my(f = factor(n), q); prod(i=1, #f~, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1))));
  • PARI
    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); };
A349127(n) = if(n%2, eulerphi(A064989(n)), 0);
  • PARI
    A349127(n) = sumdiv(n,d,moebius(n/d)*A064989(d));

Formula

Multiplicative with a(2^e) = 0, and for odd primes p, a(p^e) = (q-1)*q^(e-1), where q = prevprime(p), where prevprime is A151799.
If n is odd, then a(n) = A000010(A064989(n)), and if n is even, then a(n) = 0.
a(n) = Sum_{d|n} A008683(d) * A064989(n/d).
For all n >= 1, a(2n-1) = A347115(2n-1) = A348045(2n-1) = A349128(2n-1) = A285702(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (16/Pi^4) / Product_{p prime > 2} (1+1/p-q(p)/p^2-q(p)/p^3) = 0.1341718..., where q(p) = prevprime(p) = A151799(p). - Amiram Eldar, Dec 24 2022
Showing 1-3 of 3 results.

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