A229854 -id:A229854 - cp's OEIS frontend

A229853 a(n) = 384*n + 1.

1, 385, 769, 1153, 1537, 1921, 2305, 2689, 3073, 3457, 3841, 4225, 4609, 4993, 5377, 5761, 6145, 6529, 6913, 7297, 7681, 8065, 8449, 8833, 9217, 9601, 9985, 10369, 10753, 11137, 11521, 11905, 12289, 12673, 13057, 13441, 13825, 14209, 14593, 14977, 15361, 15745

Offset: 0

Arkadiusz Wesolowski, Oct 01 2013

Every composite Fermat number has a divisor of the form 384*n + 1, 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, A094358, A229854, A229855, A229856.

Magma
```
[384*n+1 : n in [0..40]];
```
Maple
```
seq(384*n+1, n=0..40);
```
Mathematica
```
Table[384*n + 1, {n, 0, 40}]
```
PARI
```
for(n=0, 40, print1(384*n+1, ", "));
```

G.f.: (1 + 383*x)/(1 - x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: exp(x)*(1 + 384*x).
a(n) = 2*a(n-1) - a(n-2). (End)

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

Original entry on oeis.org

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)

A229850 Number of prime factors congruent to 1 mod 3 that divide the Fermat number 2^(2^n) + 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 3, 2

Offset: 0

Arkadiusz Wesolowski, Oct 01 2013

a(n) < A046052(n) because all Fermat numbers greater than 3 are equal to 2 (mod 3).
a(n) = 1 if A046052(n) = 2.
If A046052(n) = 3, then a(n) = 0 or 2.
a(n) <= A228846(n) - n - 1 for n = 0 to 11.

M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 61-63, 65-66.

Wilfrid Keller, Fermat factoring status
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A002476, A023394, A046052, A229854.

cp's OEIS Frontend

A229853 a(n) = 384*n + 1.

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A229850 Number of prime factors congruent to 1 mod 3 that divide the Fermat number 2^(2^n) + 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs