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

1,1

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, ...}.

Links

Table of n, a(n) for n=1..47.

Formula

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

Example

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.

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

Prog

(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

Crossrefs

Cf. A000040, A005478, A136431, A137176.

Cf. A136431.

Sequence in context: A356467 A076304 A122560 * A193867 A110572 A023254

Adjacent sequences: A136335 A136336 A136337 * A136339 A136340 A136341

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 12 2008

Extensions

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

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

Status

approved