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

Author

Antti Karttunen, Nov 13 2021

Keywords

• nonn
• mult

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.

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