|
|
A136338
|
|
Primes in the array A136431 that are not Fibonacci numbers.
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
Primes in the hyperfibonacci number array of A136431, excluding the n=2 column (which contains every positive integer).
|
|
EXAMPLE
|
k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683, 23945893(24) = 23945893}, 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
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|