A229853 -id:A229853 - cp's OEIS frontend

A229855 a(n) = 384*n + 257.

257, 641, 1025, 1409, 1793, 2177, 2561, 2945, 3329, 3713, 4097, 4481, 4865, 5249, 5633, 6017, 6401, 6785, 7169, 7553, 7937, 8321, 8705, 9089, 9473, 9857, 10241, 10625, 11009, 11393, 11777, 12161, 12545, 12929, 13313, 13697, 14081, 14465, 14849, 15233, 15617, 16001

Offset: 0

Arkadiusz Wesolowski, Oct 01 2013

Every composite Fermat number has at least two divisors of the form 384*n + 257, n > 0.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
Wikipedia, Fermat number.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000215, A016789, A094358, A229853, A229854, A229856.

Magma
```
[384*n+257 : n in [0..40]];
```
Maple
```
seq(384*n+257, n=0..40);
```
Mathematica
```
Table[384*n + 257, {n, 0, 40}]
```
PARI
```
for(n=0, 40, print1(384*n+257, ", "));
```

G.f.: (257 + 127*x)/(1 - x)^2.
a(n) = 128*A016789(n) + 1.
From Elmo R. Oliveira, Dec 08 2024: (Start)
E.g.f.: exp(x)*(257 + 384*x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)

A229856 Primes of the form 384*k + 257.

Original entry on oeis.org

257, 641, 1409, 3329, 4481, 7937, 9473, 9857, 11393, 11777, 12161, 13313, 13697, 14081, 15233, 16001, 17921, 19073, 19457, 19841, 21377, 23297, 25601, 28289, 30593, 30977, 35201, 35969, 36353, 37889, 38273, 39041, 40193, 40577, 40961, 41729, 43649, 44417

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

nonn

Every Fermat number greater than 257 has a prime factor of the form 384*k + 257, k > 0.

Wikipedia, Fermat number

Subsequence of A107181 (primes of the form 8x^2+9y^2).

Cf. A000215, A023394, A094358, A229853, A229855, A229856.

[384*n+257 : n in [0..115] | IsPrime(384*n+257)];

Select[Table[384*n + 257, {n, 0, 115}], PrimeQ]

A229854 Primes of the form 384*k + 1.

Original entry on oeis.org

769, 1153, 2689, 3457, 4993, 6529, 7297, 7681, 9601, 10369, 10753, 12289, 13441, 14593, 15361, 18049, 18433, 20353, 21121, 22273, 23041, 26113, 26497, 26881, 29569, 31489, 31873, 32257, 33409, 36097, 37633, 39937, 43777, 45697, 49537, 49921, 52609, 53377

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

nonn

Not every composite Fermat number has a prime factor of the form 384*k + 1.

Wikipedia, Fermat number

Subsequence of A002476 (primes of form 6m + 1).

Cf. A000215, A023394, A094358, A229850, A229853, A229855, A229856.

[384*n+1 : n in [1..139] | IsPrime(384*n+1)];

Select[Table[384*n + 1, {n, 139}], PrimeQ]

A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

Original entry on oeis.org

274177, 319489, 6700417, 825753601, 1214251009, 6487031809, 646730219521, 6597069766657, 25409026523137, 31065037602817, 46179488366593, 151413703311361, 231292694251438081, 1529992420282859521, 2170072644496392193, 3603109844542291969

Offset: 1

Arkadiusz Wesolowski, Feb 07 2022

nonn

Subsequence of A014752.

			a(1) = 503^2 + 27*28^2 = 274177 is a prime factor of 2^(2^6) + 1;
a(2) = 383^2 + 27*80^2 = 319489 is a prime factor of 2^(2^11) + 1;
a(3) = 887^2 + 27*468^2 = 6700417 is a prime factor of 2^(2^5) + 1;
a(4) = 27017^2 + 27*1884^2 = 825753601 is a prime factor of 2^(2^16) + 1;
a(5) = 2561^2 + 27*6688^2 = 1214251009 is a prime factor of 2^(2^15) + 1;

Allan Cunningham, Haupt-exponents of 2, The Quarterly Journal of Pure and Applied Mathematics, Vol. 37 (1906), pp. 122-145.

Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.

Cf. A002476, A014752, A023394, A204620, A229850, A229853.

isok(p) = if(p%6==1 && isprime(p), my(z=znorder(Mod(2, p))); z>>valuation(z, 2)==1, return(0));

A002476 INTERSECT A023394.

cp's OEIS Frontend

A229855 a(n) = 384*n + 257.

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A229856 Primes of the form 384*k + 257.

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A229854 Primes of the form 384*k + 1.

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A351332 Primes congruent to 1 (mod 3) that divide some Fermat number.

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