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A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime. 26
0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Fernandez calls this the order of primeness of n.
a(A007097(n))=n, for any n >= 0. - Paul Tek, Nov 12 2013
When a nonoriented rooted tree is encoded as a Matula-Goebel number n, a(n) tells how many edges needs to be climbed up from the root of the tree until the first branching vertex (or the top of the tree, if n is one of the terms of A007097) is encountered. Please see illustrations at A061773. - Antti Karttunen, Jan 27 2014
Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - Allan C. Wechsler, Jan 09 2024
LINKS
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
FORMULA
a(n) = A049076(n)-1.
a(n) = if A049084(n) = 0 then 0 else a(A049084(n)) + 1. - Reinhard Zumkeller, Jul 14 2013
For all n, a(n) = A007814(A135141(n)) and a(A227413(n)) = A007814(n). Also a(A235489(n)) = a(n). - Antti Karttunen, Jan 27 2014
EXAMPLE
a(1) = 0 since 1 is not prime;
a(2) = a(prime(1)) = a(1) + 1 = 1 + 0 = 1;
a(3) = a(prime(2)) = a(2) + 1 = 1 + 1 = 2;
a(4) = 0 since 4 is not prime;
a(5) = a(prime(3)) = a(3) + 1 = 2 + 1 = 3;
a(6) = 0 since 6 is not prime;
a(7) = a(prime(4)) = a(4) + 1 = 0 + 1 = 1.
MAPLE
A078442 := proc(n)
if not isprime(n) then
0 ;
else
1+procname(numtheory[pi](n)) ;
end if;
end proc: # R. J. Mathar, Jul 07 2012
MATHEMATICA
a[n_] := a[n] = If[!PrimeQ[n], 0, 1+a[PrimePi[n]]]; Array[a, 105] (* Jean-François Alcover, Jan 26 2018 *)
PROG
(PARI) A078442(n)=for(i=0, n, isprime(n) | return(i); n=primepi(n)) \\ M. F. Hasler, Mar 09 2010
(Haskell)
a078442 n = fst $ until ((== 0) . snd)
(\(i, p) -> (i + 1, a049084 p)) (-2, a000040 n)
-- Reinhard Zumkeller, Jul 14 2013
CROSSREFS
A left inverse of A007097.
One less than A049076.
a(A000040(n)) = A049076(n).
Cf. A018252 (positions of zeros).
Cf. permutations A235489, A250247/A250248, A250249/A250250, A245821/A245822 that all preserve a(n).
Cf. also array A114537 (A138947) and permutations A135141/A227413, A246681.
Sequence in context: A322841 A291044 A113290 * A175663 A240672 A352288
KEYWORD
nonn
AUTHOR
Henry Bottomley, Dec 31 2002
STATUS
approved

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Last modified May 19 04:06 EDT 2024. Contains 372666 sequences. (Running on oeis4.)