A067720 -id:A067720 - cp's OEIS frontend

A070689 Numbers k such that k+1 and k^2+1 are primes.

1, 2, 4, 6, 10, 16, 36, 40, 66, 126, 130, 150, 156, 180, 210, 240, 250, 256, 270, 280, 306, 396, 400, 420, 430, 466, 490, 556, 570, 576, 646, 690, 700, 750, 760, 826, 906, 910, 936, 946, 966, 1060, 1096, 1150, 1276, 1290, 1306, 1320, 1366, 1566, 1570

Offset: 1

Robert G. Wilson v, May 13 2002

nonn

For any n > 1 in this sequence, (n+1)*(n^2+1) has the same nonzero digits as its prime factors in base n. - Ely Golden, Dec 12 2016

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Cf. A067720.

Select[ Range[2000], PrimeQ[ # + 1] && PrimeQ[ #^2 + 1] & ]
Select[Prime[Range[250]],PrimeQ[(#-1)^2+1]&]-1 (* Harvey P. Dale, Feb 10 2022 *)

list(lim)=my(v=List()); forprime(p=2,lim+1, if(isprime(1+(p-1)^2), listput(v,p-1))); Vec(v) \\ Charles R Greathouse IV, Dec 13 2016

A127435 Primes p such that (p-1)^2 + 1 is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 37, 41, 67, 127, 131, 151, 157, 181, 211, 241, 251, 257, 271, 281, 307, 397, 401, 421, 431, 467, 491, 557, 571, 577, 647, 691, 701, 751, 761, 827, 907, 911, 937, 947, 967, 1061, 1097, 1151, 1277, 1291, 1307, 1321, 1367, 1567, 1571, 1861

Offset: 1

Lekraj Beedassy, Jan 14 2007

nonn

Consists of 3 and a subsequence of A045349.

These are the primes of the form A067720(k)+1. - Michel Marcus, Nov 21 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..2289

For the associated primes, see A127436.

Cf. A045349, A067720.

[p: p in PrimesUpTo(2000) | IsPrime((p^2-2*p+2))]; // Vincenzo Librandi, Jul 20 2025

Select[Prime@Range[300], PrimeQ[(# - 1)^2 + 1] &] (* Ray Chandler, Jan 23 2007 *)

listp(nn) = {forprime(p=2, nn, if (isprime((p-1)^2 + 1), print1(p, ", ")););} \\ Michel Marcus, Jun 08 2016

a(n) = sqrt(A127436(n)-1) + 1.

Corrected and extended by Ray Chandler, Jan 23 2007

A333169 a(n) = phi(n^2 + 1), where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 4, 4, 16, 12, 36, 20, 48, 40, 100, 60, 112, 64, 196, 112, 256, 112, 240, 180, 400, 192, 384, 208, 576, 312, 676, 288, 624, 420, 832, 432, 800, 432, 1056, 612, 1296, 544, 1088, 760, 1600, 812, 1408, 720, 1776, 1012, 2016, 768, 1840, 1200, 2400, 1300, 2160

Offset: 0

Amiram Eldar, Mar 09 2020

nonn

			a(0) = phi(0^2 + 1) = phi(1) = 1.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000010, A002522, A067720, A193432, A193433, A333167.

Mathematica
```
Table[EulerPhi[k^2 + 1], {k, 0, 100}]
```

a(n) = eulerphi(n^2+1); \\ Michel Marcus, Mar 10 2020

a(n) = A000010(A002522(n)).

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A070689 Numbers k such that k+1 and k^2+1 are primes.

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A127435 Primes p such that (p-1)^2 + 1 is prime.

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A333169 a(n) = phi(n^2 + 1), where phi is the Euler totient function (A000010).

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