A070820 Difference between n-th prime and the value of commutator[phi,gpf] = commutator[A000010, A006530] at the same prime argument.

2, 3, 3, 4, 6, 4, 3, 4, 12, 8, 6, 4, 6, 8, 24, 14, 30, 6, 12, 8, 4, 14, 42, 12, 4, 6, 18, 54, 4, 8, 8, 14, 18, 24, 38, 6, 14, 4, 84, 44, 90, 6, 20, 4, 8, 12, 8, 38, 114, 20, 30, 18, 6, 6, 3, 132, 68, 6, 24, 8, 48, 74, 18, 32, 14, 80, 12, 8, 174, 30, 12, 180, 62, 32, 8, 192, 98, 12, 6, 18

Offset: 1

Author

Labos Elemer, May 10 2002

Keywords

• nonn

Examples

			For n=100: prime(100) = 541, phi(541) = 540, gpf(540) = 5, gpf(541) = 541, phi(541) = 540, A070819(100) = 535, a(100) = A070820(100) = 541 - 535.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000040, A006530, A070812, A070813, A070819.

Programs

Mathematica

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] f[x_] := EulerPhi[pf[x]]-pf[EulerPhi[x]] Table[Prime[w]-f[Prime[w]], {w, 2, 128}]

PARI

a(n)=if(n>1, my(f=factor(prime(n)-1)[, 1]); f[#f]+1, 2)

Formula

a(n) = A000040(n) - A070812(A000040(n)) = prime(n) - A070819(n); odd differences (=3) appear only at Fermat primes, i.e., at prime indices of n = {2, 3, 7, 55, 6543}. a(1) = 2 - 0 by convention.