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.

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

1,5

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences computed from indices in prime factorization.

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

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 *)

Prog

(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));

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.

Sequence in context: A053118 A175682 A326722 * A279228 A181481 A335872

Adjacent sequences: A349124 A349125 A349126 * A349128 A349129 A349130

Keyword

nonn,mult

Author

Antti Karttunen, Nov 13 2021

Status

approved