A260678 -id:A260678 - cp's OEIS frontend

257, 227, 199, 173, 149, 127, 107, 89, 73, 59, 47, 37, 29, 23, 19, 17, 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 289, 323, 359, 397, 437, 479, 523, 569, 617, 667, 719, 773, 829, 887, 947, 1009, 1073, 1139, 1207, 1277, 1349, 1423, 1499, 1577, 1657

Offset: 1

M. F. Hasler, Nov 15 2015

Motivated by the fact that the first 32 terms of this sequence are primes. This has an explanation through Heegener numbers, similar to Euler's prime-generating polynomial (cf. A002837 and related crossrefs).
See also A007635 for the primes in this sequence, A260678 for indices k for which a(k) is composite.
Sequence provides all numbers m for which 4*m - 67 is a square. - Bruno Berselli, Nov 16 2015

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007635 (primes in this sequence = primes of the form n^2 + n + 17).
Cf. A002837 (n^2 - n + 41 is prime), A005846 (primes of form n^2 + n + 41), A007634 (n^2 + n + 41 is composite), A097823 (n^2 + n + 41 is not squarefree).
Cf. A260678.

[n+(17-n)^2: n in [1..70]]; // Vincenzo Librandi, Nov 16 2015

Table[n + (17 - n)^2, {n, 70}] (* Vincenzo Librandi, Nov 16 2015 *)
LinearRecurrence[{3,-3,1},{257,227,199},60] (* Harvey P. Dale, May 12 2019 *)

PARI
```
for(n=1,99,print1(n+(17-n)^2,","))
```

G.f.: x*(257 - 544*x + 289*x^2)/(1 - x)^3.
From Elmo R. Oliveira, Feb 11 2025: (Start)
E.g.f.: exp(x)*(x^2 - 32*x + 289) - 289.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

cp's OEIS Frontend

A260679 a(n) = n + (17 - n)^2.

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