A349127 - OEIS

%I #35 Dec 24 2022 03:50:46

%S 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,

%T 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,

%U 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

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

%C 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.

%C Compare the situation with A003961 ("prime shift towards larger primes"), where A003972(n) = A000010(A003961(n)) is also the Möbius transform of A003961.

%H Antti Karttunen, <a href="/A349127/b349127.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%F 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.

%F If n is odd, then a(n) = A000010(A064989(n)), and if n is even, then a(n) = 0.

%F a(n) = Sum_{d|n} A008683(d) * A064989(n/d).

%F For all n >= 1, a(2n-1) = A347115(2n-1) = A348045(2n-1) = A349128(2n-1) = A285702(n).

%F 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

%t 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 *)

%o (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))));

%o (PARI)

%o 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); };

%o A349127(n) = if(n%2, eulerphi(A064989(n)), 0);

%o (PARI)

%o A349127(n) = sumdiv(n,d,moebius(n/d)*A064989(d));

%Y Agrees with A347115, A348045 and A349128 on odd numbers.

%Y Cf. A000004, A285702 (even and odd bisection).

%Y Cf. A000010, A008683, A064989, A151799.

%Y Cf. also A003961, A003972, A349125, A349136.

%K nonn,mult

%O 1,5

%A _Antti Karttunen_, Nov 13 2021