A136338 Primes in the array A136431 that are not Fibonacci numbers.

7, 11, 29, 37, 41, 67, 79, 97, 137, 191, 211, 277, 379, 631, 709, 821, 947, 967, 991, 1129, 1327, 1597, 1831, 2017, 2081, 2267, 2347, 2557, 2683, 2851, 2927, 3571, 3917, 4561, 4657, 4951, 5051, 5779, 6217, 6329, 6763, 8273, 8647, 8779, 9181, 9871, 10093

Offset: 1

Author

Jonathan Vos Post, Apr 12 2008

Keywords

• easy
• nonn

Comments

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).

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

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

Continuing the table of primes in the examples, from a computation by Joshua Zucker, we have:

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

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

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

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

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

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

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

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

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

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

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

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

Examples

			k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 1000.
k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79, A001924(25) = 514201, ...}
k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97, A014162(16) = 17519}, no more through n = 30.
k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093, A014166(14) = 16703}
k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273, A053739(11) = 14323}, no more through n = 27.
k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683, 23945893(24) = 23945893}, no more through n = 27.
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.
		

Crossrefs

Cf. A000040, A005478, A136431, A137176.

Cf. A136431.

Programs

Maple

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

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

Formula

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

Extensions

Revised definition from N. J. A. Sloane, May 09 2008

More terms from R. J. Mathar, Apr 21 2008