A003972 Moebius transform of A003961; a(n) = phi(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

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, 32, 100, 60, 30, 48, 36, 162, 48, 36, 60, 120, 40, 44, 64, 108, 42, 80, 46, 72, 120, 56, 52, 216, 110, 84, 72, 96, 58, 200, 72, 180, 88, 60, 60, 144, 66, 72, 200, 486, 96, 96, 70

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Vincenzo Librandi)

Formula

Multiplicative with a(p^e) = (q-1)q^(e-1) where q = nextPrime(p). - David W. Wilson, Sep 01 2001

a(n) = A000010(A003961(n)) = A037225(A108228(n)) = A037225(A048673(n)-1). - Antti Karttunen, Aug 20 2020

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

Mathematica

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 *)

Prog

(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
(Python)
from math import prod
from sympy import nextprime, factorint
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

Crossrefs

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

Cf. also A003973.

Sequence in context: A085271 A347376 A260812 * A023853 A056526 A049066

Adjacent sequences: A003969 A003970 A003971 * A003973 A003974 A003975

Keyword

nonn,mult

Author

Marc LeBrun

Extensions

More terms from David W. Wilson, Aug 29 2001

Secondary name added by Antti Karttunen, Aug 20 2020

Status

approved