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A291049
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Primes of the form 2^r * 17^s + 1.
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2
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2, 3, 5, 17, 137, 257, 65537, 157217, 295937, 557057, 1336337, 96550277, 1212153857, 2281701377, 5473632257, 395469930497, 1401249857537, 2637646790657, 4964982194177, 28572702478337, 1271035441709057, 38280596832649217, 1872540629620228097, 6634884445436379137
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OFFSET
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1,1
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COMMENTS
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Primes of the forms a^r * b^s + 1 where (a, b) = (2,1), (2,3), (2,5), (2,7), (2,11) and (2,13) are A092506, A005109, A077497, A077498, A077499 and A173236.
Fermat prime exponents r are 0, 1, 2, 4, 8, 16.
For n > 2, all terms are congruent to 5 (mod 6).
Also, these are prime numbers p for which (p*34^p)/(p-1) is an integer.
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LINKS
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EXAMPLE
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With n = 1, a(1) = 2^0 * 17^0 + 1 = 2.
With n = 5, a(5) = 2^3 * 17^1 + 1 = 137.
list of (r,s): (0,0), (1,0), (2,0), (4,0), (3,1), (8,0), (16,0), (5,3), (10,2), (15,1), (4,4), (2,6).
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MAPLE
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N:= 10^20: # to get all terms <= N+1
S:= NULL:
for r from 0 to ilog2(N) do
for s from 0 to floor(log[17](N/2^r)) do
p:= 2^r*17^s +1;
if isprime(p) then
S:= S, p
fi
od od:
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MATHEMATICA
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With[{nn = 10^19, q = 17}, Select[Sort@ Flatten@ Table[2^i*q^j + 1, {i, 0, Log[2, nn]}, {j, 0, Log[q, nn/2^i]}], PrimeQ]] (* Michael De Vlieger, Sep 18 2017, after Robert G. Wilson v at A005109 *)
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PROG
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(GAP)
K:=26*10^7+1;; # to get all terms <= K.
A:=Filtered(Filtered([1, 3..K], i-> i mod 6=5), IsPrime);; I:=[17];;
B:=List(A, i->Elements(Factors(i-1)));;
C:=List([0..Length(I)], j->List(Combinations(I, j), i->Concatenation([2], i)));;
A291049:=Concatenation([2, 3], List(Set(Flat(List([1..Length(C)], i->List([1..Length(C[i])], j->Positions(B, C[i][j]))))), i->A[i]));
(PARI) lista(nn) = my(t, v=List([])); for(r=0, logint(nn, 2), t=2^r; for(s=0, logint(nn\t, 17), if(isprime(t+1), listput(v, t+1)); t*=17)); Vec(vecsort(v)) \\ Jinyuan Wang, Jun 26 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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