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Moebius transform of A003961; a(n) = phi(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.
38

%I #38 Nov 18 2022 07:11:36

%S 1,2,4,6,6,8,10,18,20,12,12,24,16,20,24,54,18,40,22,36,40,24,28,72,42,

%T 32,100,60,30,48,36,162,48,36,60,120,40,44,64,108,42,80,46,72,120,56,

%U 52,216,110,84,72,96,58,200,72,180,88,60,60,144,66,72,200,486,96,96,70

%N Moebius transform of A003961; a(n) = phi(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

%H Antti Karttunen, <a href="/A003972/b003972.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Vincenzo Librandi)

%F Multiplicative with a(p^e) = (q-1)q^(e-1) where q = nextPrime(p). - _David W. Wilson_, Sep 01 2001

%F a(n) = A000010(A003961(n)) = A037225(A108228(n)) = A037225(A048673(n)-1). - _Antti Karttunen_, Aug 20 2020

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/Pi^2) * Product_{p prime} ((p^2-p)/(p^2 - nextPrime(p)) = 1.2547593344... . - _Amiram Eldar_, Nov 18 2022

%t b[1] = 1; b[p_?PrimeQ] := b[p] = Prime[ PrimePi[p] + 1]; b[n_] := b[n] = Times @@ (b[First[#]]^Last[#] &) /@ FactorInteger[n]; a[n_] := Sum[ MoebiusMu[n/d]*b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}] (* _Jean-François Alcover_, Jul 18 2013 *)

%o (PARI) A003972(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); eulerphi(factorback(f)); }; \\ _Antti Karttunen_, Aug 20 2020

%o (Python)

%o from math import prod

%o from sympy import nextprime, factorint

%o def A003972(n): return prod((q:=nextprime(p))**(e-1)*(q-1) for p, e in factorint(n).items()) # _Chai Wah Wu_, Jul 18 2022

%Y Cf. A000010, A003961, A037225, A048673, A108228.

%Y Cf. also A003973.

%K nonn,mult

%O 1,2

%A _Marc LeBrun_

%E More terms from _David W. Wilson_, Aug 29 2001

%E Secondary name added by _Antti Karttunen_, Aug 20 2020