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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.

%I A078442 #63 Jul 18 2024 14:26:34
%S A078442 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,
%T A078442 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,
%U A078442 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
%N A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.
%C A078442 Fernandez calls this the order of primeness of n.
%C A078442 a(A007097(n))=n, for any n >= 0. - _Paul Tek_, Nov 12 2013
%C A078442 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
%C A078442 Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - _Allan C. Wechsler_, Jan 09 2024
%H A078442 Reinhard Zumkeller, <a href="/A078442/b078442.txt">Table of n, a(n) for n = 1..10000</a>
%H A078442 N. Fernandez, <a href="http://www.borve.org/primeness/FOP.html">An order of primeness, F(p)</a>
%H A078442 N. Fernandez, <a href="/A006450/a006450.html">An order of primeness</a> [cached copy, included with permission of the author]
%H A078442 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A078442 a(n) = A049076(n)-1.
%F A078442 a(n) = if A049084(n) = 0 then 0 else a(A049084(n)) + 1. - _Reinhard Zumkeller_, Jul 14 2013
%F A078442 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
%e A078442 a(1) = 0 since 1 is not prime;
%e A078442 a(2) = a(prime(1)) = a(1) + 1 = 1 + 0 = 1;
%e A078442 a(3) = a(prime(2)) = a(2) + 1 = 1 + 1 = 2;
%e A078442 a(4) = 0 since 4 is not prime;
%e A078442 a(5) = a(prime(3)) = a(3) + 1 = 2 + 1 = 3;
%e A078442 a(6) = 0 since 6 is not prime;
%e A078442 a(7) = a(prime(4)) = a(4) + 1 = 0 + 1 = 1.
%p A078442 A078442 := proc(n)
%p A078442     if not isprime(n) then
%p A078442         0 ;
%p A078442     else
%p A078442         1+procname(numtheory[pi](n)) ;
%p A078442     end if;
%p A078442 end proc: # _R. J. Mathar_, Jul 07 2012
%t A078442 a[n_] := a[n] = If[!PrimeQ[n], 0, 1+a[PrimePi[n]]]; Array[a, 105] (* _Jean-François Alcover_, Jan 26 2018 *)
%o A078442 (PARI) A078442(n)=for(i=0,n, isprime(n) || return(i); n=primepi(n)) \\ _M. F. Hasler_, Mar 09 2010
%o A078442 (Haskell)
%o A078442 a078442 n = fst $ until ((== 0) . snd)
%o A078442                         (\(i, p) -> (i + 1, a049084 p)) (-2, a000040 n)
%o A078442 -- _Reinhard Zumkeller_, Jul 14 2013
%Y A078442 A left inverse of A007097.
%Y A078442 One less than A049076.
%Y A078442 a(A000040(n)) = A049076(n).
%Y A078442 Cf. A373338 (mod 2), A018252 (positions of zeros).
%Y A078442 Cf. A000720, A049084, A061773.
%Y A078442 Cf. permutations A235489, A250247/A250248, A250249/A250250, A245821/A245822 that all preserve a(n).
%Y A078442 Cf. also array A114537 (A138947) and permutations A135141/A227413, A246681.
%K A078442 nonn
%O A078442 1,3
%A A078442 _Henry Bottomley_, Dec 31 2002