%I #11 Sep 21 2015 20:09:10

%S 7,11,29,37,41,67,79,97,137,191,211,277,379,631,709,821,947,967,991,

%T 1129,1327,1597,1831,2017,2081,2267,2347,2557,2683,2851,2927,3571,

%U 3917,4561,4657,4951,5051,5779,6217,6329,6763,8273,8647,8779,9181,9871,10093

%N Primes in the array A136431 that are not Fibonacci numbers.

%C A generalization of prime Fibonacci numbers (A005478) are the prime hyperfibonacci numbers (primes in A136431). Referring to the array A(k,n) = Apply partial sum operator k times to Fibonacci numbers, we see that every prime occurs in the n=2 column (as it contains every positive integer).

%C So excluding n=2 and k=0 (A005478) we have the nontrivially prime hyperfibonacci numbers which are not Fibonacci numbers.

%C Note that this sequence does not indicate multiplicity (e.g., 7 occurs twice in the valid part of the table).

%C Continuing the table of primes in the examples, from a computation by _Joshua Zucker_, we have:

%C k=1: {7, ...} no more through n = 1000.

%C k=2: {7, 79, 514201, 14930317, 956722025983, 5527939700884681 4660046610375530219, ...}

%C k=3: {11, 97, 17519, next value has 60 digits, ...}

%C k=4: {41, 10093, 16703, 3520457, 591286703533, 6557470285501, 19740274219868101499, ...}

%C k=5: {709, 8273, 14323, 466004661037329684,1 298611126818977061133263, ...}

%C k=6: {29, 2683, 23945893, 1835540197, 4052735290427, 27777884012083, ...}

%C k=7: {37, 967, 2267, 127921, 226007, 62048869, 1131463777, 7540113804271826929, ...}

%C k=8: {27777538280521, 1409869790947669143312035590804646728957, ...}

%C k=9: {1033628323428189498226451492123369099, next value has 60 digits, ...}

%C k=10: {67, 5972304273877744135569337875802249660927, ...}

%C k=11: {79, 4478413, 19008291293, 61305228407581679, ...}

%C k=12: {6763, 1982269, 37886753582095837, 2791715456569622316696636389, ...}.

%F Primes in the hyperfibonacci number array of A136431, excluding the n=2 column (which contains every positive integer).

%e k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 1000.

%e k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79, A001924(25) = 514201, ...}

%e k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97, A014162(16) = 17519}, no more through n = 30.

%e k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093, A014166(14) = 16703}

%e k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273, A053739(11) = 14323}, no more through n = 27.

%e k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683, 23945893(24) = 23945893}, no more through n = 27.

%e k=7: primes in A053296 = {A053296(3) = 37, A053296(6) = 967, A053296(7) = 2267, A053296(12) = 127921, A053296(13) = 226007}, no more through n = 27.

%p A136431 := proc(k,n) local x ; coeftayl(x/(1-x-x^2)/(1-x)^k,x=0,n) ; end: A136338 := proc(amax) local a,k,n,a136431; a := [] ; for k from 1 do if A136431(k,3) > amax then break ; fi ; for n from 3 do a136431 := A136431(k,n) ; if a136431 > amax then break ; fi ; if isprime(a136431) and not a136431 in a then a := [op(a),a136431] ; fi ; od: od: sort(a) ; end: A136338(20000) ; # _R. J. Mathar_, Apr 21 2008

%o (PARI) partsumfib(N,s=[],P=[])={ for( n=1+#s,N, s=concat(s,n+1); forstep( i=n,1,-1, isprime( s[i]+= if( i>1, s[i-1], fibonacci(n+2) ) ) & P=setunion(P,[s[i]]) ); print(s); );vecsort(eval(P))} \\ _M. F. Hasler_

%Y Cf. A000040, A005478, A136431, A137176.

%Y Cf. A136431.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Apr 12 2008

%E Revised definition from _N. J. A. Sloane_, May 09 2008

%E More terms from _R. J. Mathar_, Apr 21 2008