A237038 -id:A237038 - cp's OEIS frontend

A002383 Primes of form k^2 + k + 1.

3, 7, 13, 31, 43, 73, 157, 211, 241, 307, 421, 463, 601, 757, 1123, 1483, 1723, 2551, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 20023, 20593, 21757, 22651, 23563

Offset: 1

N. J. A. Sloane

Also primes p such that 4p-3 is square. - Giovanni Teofilatto, Sep 07 2005
Also these primes are sums of 1 and some consecutive even numbers starting at 2; e.g., 31 = 1+2+4+6+8+10. - Labos Elemer, Apr 15 2003
Also primes of form n^2 - n + 1 (Prime central polygonal numbers, A002061). - Zak Seidov, Jan 26 2006
Also primes which are of the form TriangularNumber(n) + TriangularNumber(n+2): 7 = 1+6, 13 = 3+10, 31 = 10+21, 43 = 15+28, 73 = 28+45, ... - Vladimir Joseph Stephan Orlovsky, Apr 03 2009
It is not known whether there are infinitely many primes of the form n^2+n+1. See Rose reference. - Daniel Tisdale, Jun 27 2009
These numbers when >= 7 are prime repunits 111_n in a base n >= 2, so except for 3, they are all Brazilian primes belonging to A085104. (See Links "Les nombres brésiliens", Sections V.4 - V.5.) A002383 is generated by A002384 which lists the bases n of 111_n. A002383 = A053183 Union A185632. - Bernard Schott, Dec 22 2012
Conjecture: the set of these numbers, except 3, is the intersection of sets A085104 and A059055. See A225148. - Thomas Ordowski, May 02 2013
For a(n)>13, the fractional part of square root of a(n) starts with digit 5 (see A034101). - Charles Kusniec, Sep 06 2022

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 per l'intervallo compreso entro 121 milioni, e cioè per tutti i valori di n fino a 11000, Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6, v. 3 (1929), pages 193-218.
H. E. Rose, A Course in Number Theory, Clarendon Press, 1988, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10751
Cody S. Hansen and Pace P. Nielsen, Prime factors of phi3(x) of the same form, arXiv:2204.08971 [math.NT], 2022.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A002384, A088503, A110284, A085104.
Cf. A237037, A237038, A237039, A237040 (from semiprimes of form n^3 + 1).
See also A034101.

[ a: n in [1..100] | IsPrime(a) where a is n^2+n+1 ]; // Wesley Ivan Hurt, Jun 16 2014

select(isprime, [j^2+j+1$j=1..200])[];  # Alois P. Heinz, Apr 20 2022

Select[Table[n^2+n+1, {n,250}], PrimeQ] (* Harvey P. Dale, Mar 23 2012 *)

list(lim)=select(n->isprime(n),vector((sqrt(4*lim-3)-1)\2,k,k^2+k+1)) \\ Charles R Greathouse IV, Jul 25 2011

from sympy import isprime
print(list(filter(isprime, (n**2 + n + 1 for n in range(150))))) # Michael S. Branicky, Apr 20 2022

a(n) = A002384(n)^2 + A002384(n) + 1 = (A088503(n-1)^2 + 3)/4 = (A110284(n) + 3)/4. - Ray Chandler, Sep 07 2005

Extended by Ray Chandler, Sep 07 2005

A096173 Numbers k such that k^3+1 is an odd semiprime.

Original entry on oeis.org

2, 4, 6, 16, 18, 22, 28, 42, 58, 60, 70, 72, 78, 100, 102, 106, 112, 148, 156, 162, 190, 210, 232, 280, 310, 330, 352, 358, 382, 396, 448, 456, 490, 568, 606, 672, 756, 786, 820, 826, 828, 856, 858, 876, 928, 970, 982, 1008, 1012, 1030, 1068, 1092, 1108, 1150

Offset: 1

Hugo Pfoertner, Jun 20 2004

nonn

Numbers n such that n^3 + 1 is a semiprime, because then n^3 + 1 must be odd, since n^3 + 1 = (n+1)*(n^2 - n + 1) is a semiprime only if n+1 is odd. - Jonathan Sondow, Feb 02 2014

Obviously, n + 1 is always a prime number. Sequence is intersection of A006093 and A055494. - Altug Alkan, Dec 20 2015

			a(1)=2 because 2^3+1=9=3*3, a(13)=100: 100^3+1=1000001=101*9901.

R. J. Mathar, Table of n, a(n) for n = 1..2086

Cf. A001358; A081256: largest prime factor of k^3+1; A096174: (k^3+1)/(k+1) is prime; A046315, A237037, A237038, A237039, A237040.

Cf. A006093, A055494.

[n: n in [1..2*10^3] | IsPrime(n+1) and IsPrime(n^2-n+1)]; // Vincenzo Librandi, Dec 21 2015

select(n -> isprime(n+1) and isprime(n^2-n+1), [seq(2*i,i=1..1000)]); # Robert Israel, Dec 20 2015

Select[Range[1200], PrimeQ[#^2 - # + 1] && PrimeQ[# + 1] &] (* Jonathan Sondow, Feb 02 2014 *)

for(n=1, 1e5, if(bigomega(n^3+1)==2, print1(n, ", "))); \\ Altug Alkan, Dec 20 2015

a(n) = 2*A237037(n) = (A237040(n)-1)^(1/3). - Jonathan Sondow, Feb 02 2014

Corrected by Zak Seidov, Mar 08 2006

A237040 Semiprimes of the form k^3 + 1.

Original entry on oeis.org

9, 65, 217, 4097, 5833, 10649, 21953, 74089, 195113, 216001, 343001, 373249, 474553, 1000001, 1061209, 1191017, 1404929, 3241793, 3796417, 4251529, 6859001, 9261001, 12487169, 21952001, 29791001, 35937001, 43614209, 45882713, 55742969, 62099137, 89915393, 94818817, 117649001

Offset: 1

Jonathan Sondow, Feb 02 2014

nonn

k^3 + 1 is a term iff k + 1 and k^2 - k + 1 are both prime.
Is the sequence infinite? This is an analog of Landau's 4th problem, namely, are there infinitely many primes of the form k^2 + 1?
In other words: are there infinitely many primes p such that p^2 - 3*p + 3 is also prime? - Charles R Greathouse IV, Jul 02 2017

			9 = 3*3 = 2^3 + 1 is the first semiprime of the form n^3 + 1, so a(1) = 9.

Vincenzo Librandi, Table of n, a(n) for n = 1..1400
Eric Weisstein's World of Mathematics, Semiprime
Wikipedia, Semiprime
Wikipedia, Landau's problems

Cf. A001358, A002383, A002496, A046315, A081256, A096173, A096174, A237037, A237038, A237039.

Cf. A242262 (semiprimes of the form k^3 - 1).

IsSemiprime:= func; [s: n in [1..500] | IsSemiprime(s) where s is n^3 + 1]; // Vincenzo Librandi, Jul 02 2017

L = Select[Range[500], PrimeQ[# + 1] && PrimeQ[#^2 - # + 1] &]; L^3 + 1
Select[Range[50]^3 + 1, PrimeOmega[#] == 2 &] (* Zak Seidov, Jun 26 2017 *)

lista(nn) = for (n=1, nn, if (bigomega(sp=n^3+1) == 2, print1(sp, ", "));); \\ Michel Marcus, Jun 27 2017

list(lim)=my(v=List(),n,t); forprime(p=3,sqrtnint(lim\1-1,3)+1, if(isprime(t=p^2-3*p+3), listput(v,t*p))); Vec(v) \\ Charles R Greathouse IV, Jul 02 2017

a(n) = A096173(n)^3 + 1 = 8*A237037(n)^3 + 1.

A237037 Numbers k such that (2*k)^3 + 1 is a semiprime.

Original entry on oeis.org

1, 2, 3, 8, 9, 11, 14, 21, 29, 30, 35, 36, 39, 50, 51, 53, 56, 74, 78, 81, 95, 105, 116, 140, 155, 165, 176, 179, 191, 198, 224, 228, 245, 284, 303, 336, 378, 393, 410, 413, 414, 428, 429, 438, 464, 485, 491, 504, 506, 515, 534, 546, 554, 575, 596, 611, 638, 641, 648, 659, 680, 683, 711, 714, 725, 744, 765, 774, 791

Offset: 1

Jonathan Sondow, Feb 02 2014

nonn

Numbers k such that 2*k+1 and 4*k^2 - 2*k + 1 are both prime.

Same as k/2 such that k^3 + 1 is a semiprime, because then k must be even.

			(2*1)^3 + 1 = 9 = 3*3 is a semiprime, so a(1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime.
Wikipedia, Semiprime.

Cf. A001358, A046315, A081256, A096173, A096174, A237038, A237039, A237040.

Select[Range[800], PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
Select[Range[800],PrimeOmega[(2#)^3+1]==2&] (* Harvey P. Dale, Nov 28 2024 *)

a(n) = A096173(n)/2 = (1/2)*(A237040(n)-1)^(1/3).

A237039 Semiprimes of the form (2*p)^3 + 1, where p is prime.

Original entry on oeis.org

65, 217, 10649, 195113, 1191017, 45882713, 55742969, 946966169, 2106997769, 2289529433, 2548895897, 10735357817, 26946035993, 54958685609, 77199941513, 131561576057, 132495001193, 440016501017, 456888832409, 623273556089, 848202406697, 966188398457

Offset: 1

Jonathan Sondow, Feb 02 2014

nonn

			(2*2)^3 + 1 = 65 = 5*13 is a semiprime, so a(1) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime.
Wikipedia, Semiprime.

Cf. A001358, A046315, A081256, A096173, A096174, A237037, A237038, A237040.

L = Select[Range[5000], PrimeQ[#] && PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]; (2 L)^3 + 1
Select[Table[(2p)^3+1,{p,Prime[Range[1000]]}],PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 21 2021 *)

a(n) = (2*A237038(n))^3 + 1.

A239363 Numbers k such that phi(k+1) - phi(k) = k/2.

Original entry on oeis.org

2, 4, 16, 24, 216, 256, 4960, 10648, 65536, 195112, 1191016, 7544800, 11530464, 12705504, 45882712, 55742968, 179883264, 946966168, 2106997768, 2289529432, 2548895896, 3056745600, 3482123272, 7756683264, 10735357816, 15920801280, 26946035992, 46827274240

Offset: 1

Paolo P. Lava, Mar 17 2014

nonn

From Amiram Eldar, Nov 09 2024: (Start)
If p is a Fermat prime (A019434), then p-1 is a term.
If p is a term of A237038, then 8*p^3 is a term. (End)

			phi(217) = 180, phi(216) = 72, and 180-72 = 108, which is 216/2.

Giovanni Resta, Table of n, a(n) for n = 1..39 (terms < 10^12)

Cf. A000010, A019434, A237038.

with(numtheory); P:=proc(q) local n;
for n from 1 to q do if phi(n+1)-phi(n)=n/2 then print(n);
fi; od; end: P(10^9);

Select[Range[200000], EulerPhi[#+1] - EulerPhi[#] == #/2 &] (* Amiram Eldar, Nov 09 2024 *)

is(k) = eulerphi(k+1) - eulerphi(k) == k/2; \\ Amiram Eldar, Nov 09 2024

a(13)-a(27) from Giovanni Resta, Mar 17 2014

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A002383 Primes of form k^2 + k + 1.

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A096173 Numbers k such that k^3+1 is an odd semiprime.

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A237040 Semiprimes of the form k^3 + 1.

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A237037 Numbers k such that (2*k)^3 + 1 is a semiprime.

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A237039 Semiprimes of the form (2*p)^3 + 1, where p is prime.

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A239363 Numbers k such that phi(k+1) - phi(k) = k/2.

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