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A070820
Difference between n-th prime and the value of commutator[phi,gpf] = commutator[A000010, A006530] at the same prime argument.
1
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
1,1
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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.
EXAMPLE
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.
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}]
PROG
(PARI) a(n)=if(n>1, my(f=factor(prime(n)-1)[, 1]); f[#f]+1, 2)
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, May 10 2002
STATUS
approved