A259145 -id:A259145 - cp's OEIS frontend

A259172 Numbers in A259145 that are neither prime nor semiprime.

561, 595, 1105, 1235, 1245, 1495, 1547, 1885, 2405, 2555, 2717, 2849, 3115, 3495, 3655, 3657, 3689, 3815, 4521, 4795, 4945, 5035, 5385, 5395, 5453, 5457, 5709, 5865, 6083, 6141, 6251, 6285, 6365, 6391, 6501, 6695, 6755, 6969, 7021, 7887, 8113, 8255, 8355

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Regarding the distribution: Let K be the union of primes and semiprimes in A259145. Let S be the set of other terms. The growth rate of the cardinality of S with respect to the cardinality of K is significantly slower. For instance, if we take the first 50000 terms of A259145, about 32.5 percent are contained in S. If we take the first 350000 terms, about 38.2 percent are contained in S.
a(n) that are in A002997 (Carmichael numbers) for a(n) <= 10^6 are 561, 1105, 8911, 10585, 29341, 825265.
a(n) that are in A051015 (Zeisel numbers) for a(n) <= 3*10^6 are 1885, 353977, 2953711.

Carlos Eduardo Olivieri, Table of n, a(n) for n = 1..8906
Eric Naslund, Integers with a predetermined prime factorization, arXiv:1203.2363 [math.NT], 2012.

Cf. A001221, A001358, A002997, A007053, A051015, A125527, A259145.

Subsequence of A000469, A033942, A050384 (conjuctered).

Select[Range[25000], PrimeQ[#^2 - EulerPhi[#]] && PrimeNu[#] > 2 &]

A001221(a(n)) > 2.

A000005(a(n)) = 2^k, k >= 3.

A258435 Primes of form x^2 - phi(x) in increasing order.

Original entry on oeis.org

3, 7, 43, 157, 1069, 1201, 4177, 4423, 5869, 6163, 8209, 17581, 19183, 22651, 26407, 37057, 48649, 60793, 61837, 82129, 89137, 102829, 113233, 115981, 121453, 141793, 143263, 190573, 208393, 230929, 283609, 292141, 303097, 314401, 337069

Offset: 1

Carlos Eduardo Olivieri, May 30 2015

			a(1) = 3, because  2^2 - 1 = 3, and 1^2 - 1 = 0 is not a prime.
a(2) = 7, since 3^2 = 9, phi(3) = 2, so 9-2 = 7 (prime).
a(3) = 43, since 7^2 = 49, phi(7) = 6, so 49-6 = 43 (prime).
a(6) = 1201, since 35^2 = 1225, phi(35) = 24, so 1225-24 = 1201 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subset of A258434.
For phi see A000010.
A074268 is a subsequence. - Michel Marcus, Jun 19 2015
Cf. A259145.

[a: n in [1..1000] | IsPrime(a) where a is n^2-EulerPhi(n) ]; // Vincenzo Librandi, Jun 03 2015

lst = Table[n^2 - EulerPhi[n], {n, 1000}]; Select[lst, PrimeQ]
Select[Table[n^2 - EulerPhi[n], {n, 1000}], PrimeQ] (* Vincenzo Librandi, Jun 03 2015 *)

lista(nn) = {for (n=1, nn, if (isprime(p=n^2 -eulerphi(n)), print1(p, ", ")););} \\ Michel Marcus, Jul 08 2015

More terms from Vincenzo Librandi, Jun 03 2015

Edited by Wolfdieter Lang, Jun 16 2015

A275059 Numbers n such that A000010(n) + n^2 is a prime.

Original entry on oeis.org

1, 2, 3, 5, 11, 13, 15, 19, 31, 33, 35, 41, 51, 53, 59, 65, 83, 89, 91, 101, 103, 115, 131, 141, 149, 161, 163, 181, 185, 187, 191, 193, 199, 217, 221, 233, 241, 263, 281, 287, 295, 303, 329, 331, 349, 373, 401, 415, 419, 431, 433, 445, 449, 461, 463, 469, 473, 499, 517

Offset: 1

Vincenzo Librandi, Jul 27 2016

For n >= 2, a(n) is odd and squarefree. - Robert Israel, Jul 29 2016

			5 is a term because A000010(5) + 5^2 = 29 is a prime number.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A259145.

[n: n in [1..800] | IsPrime(n^2+EulerPhi(n))];

select(t -> isprime(numtheory:-phi(t) + t^2), [1,2,seq(n,n=3..1000,2)]); # Robert Israel, Jul 29 2016

Select[Range[2000], PrimeQ[#^2 + EulerPhi[#]] &]

isok(n) = isprime(eulerphi(n) + n^2); \\ Altug Alkan, Jul 27 2016

cp's OEIS Frontend

A259172 Numbers in A259145 that are neither prime nor semiprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A258435 Primes of form x^2 - phi(x) in increasing order.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A275059 Numbers n such that A000010(n) + n^2 is a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI