A002383 -id:A002383 - cp's OEIS frontend

A088548 Primes of the form k^4 + k^3 + k^2 + k + 1.

5, 31, 2801, 22621, 30941, 88741, 245411, 292561, 346201, 637421, 732541, 837931, 2625641, 3500201, 3835261, 6377551, 15018571, 16007041, 21700501, 28792661, 30397351, 35615581, 39449441, 48037081, 52822061, 78914411, 97039801, 147753211, 189004141, 195534851

Offset: 1

Cino Hilliard, Nov 17 2003

These numbers when >= 31 are primes repunits 11111_n in a base n >= 2, so except 5, they are all Brazilian primes belonging to A085104. (See Links "Les nombres brésiliens", § V.4 - § V.5.) A008858 is generated by the bases n present in A049409. - Bernard Schott, Dec 19 2012

			a(2) = 31 is prime and 31 = 2^4 + 2^3 + 2^2 + 2 + 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A002383, A049409, A085104, A088550.

[a: n in [0..200] | IsPrime(a) where a is n^4+n^3+n^2+n+1]; // Vincenzo Librandi, Jul 16 2012

lst={}; Do[a=1+n+n^2+n^3+n^4; If[PrimeQ[a], AppendTo[lst,a]], {n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 02 2009 *)
Select[Table[n^4+n^3+n^2+n+1, {n,0,2000}], PrimeQ] (* Vincenzo Librandi, Jul 16 2012 *)

polypn(n,p) = { for(x=1,n, if(p%2,y=2,y=1); for(m=1,p, y=y+x^m; ); if(isprime(y),print1(y",")); ) }

from sympy import isprime
print(list(filter(isprime, (k**4+k**3+k**2+k+1 for k in range(120))))) # Michael S. Branicky, May 31 2021

A000040 intersect A053699. - R. J. Mathar, Feb 07 2014

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.

A163418 Primes of the form ((p-1)/2)^2+((p+1)/2), where p is prime.

Original entry on oeis.org

3, 7, 13, 31, 43, 73, 211, 241, 421, 463, 1123, 1723, 2551, 2971, 4831, 5701, 6163, 8011, 8191, 9901, 11131, 12433, 14281, 17293, 19183, 20023, 23563, 24181, 28393, 30103, 31153, 35911, 37831, 43891, 46441, 53593, 60271, 77563, 83233, 86143

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 27 2009

Subsequence of A002383. - Charles R Greathouse IV, Aug 11 2009

			((3-1)/2)^2+((3+1)/2)=1+2=3;
((5-1)/2)^2+((5+1)/2)=4+3=7;
((7-1)/2)^2+((7+1)/2)=9+4=13,..

G. C. Greubel, Table of n, a(n) for n = 1..1500

Cf. A162652.

f[n_]:=((p-1)/2)^2+((p+1)/2); lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst
Select[((#-1)/2)^2+((#+1)/2)&/@Prime[Range[200]],PrimeQ] (* Harvey P. Dale, Aug 06 2012 *)

forprime(p=3,10^3,if(isprime(q=((p-1)/2)^2+(p+1)/2),print1(q,", "))) \\ Derek Orr, Apr 30 2015

A059055 Primes which can be written as (b^k+1)/(b+1) for positive integers b and k.

Original entry on oeis.org

3, 7, 11, 13, 31, 43, 61, 73, 157, 211, 241, 307, 421, 463, 521, 547, 601, 683, 757, 1123, 1483, 1723, 2551, 2731, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9091, 9901, 10303, 11131, 12211, 12433, 13421, 13807, 14281

Offset: 1

Henry Bottomley, Dec 21 2000

For (b^k+1)/(b+1) to be a prime, k must be an odd prime. 2=(0^0+1)/(0+1) has been excluded since neither b nor k would be positive.
From Bernard Schott, Apr 30 2021: (Start)
43 is the only known prime to have two such representations (examples).
The next two sequences realize a partition of this set: Brazilian primes of the form (c^q-1)/(c-1) (A002383 \ {3}) and primes that are not Brazilian (A343774). (End)

			43 is in the sequence since (2^7+1)/(2+1) = 129/3 = 43; indeed also (7^3+1)/(7+1) = 344/8 = 43.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 3880 terms from T. D. Noe)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

Cf. A002383, A059054.

Cf. A003424, A085104.

max = 89; maxdata = (1 + max^3)/(1 + max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 + m^i)/(1 + m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)

isok(p) = {if (isprime(p), for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1););););} \\ Michel Marcus, Apr 30 2021

A073337 Primes of the form 4k^2 - 10k + 7 with k positive.

Original entry on oeis.org

3, 13, 31, 241, 307, 463, 757, 1123, 1723, 3307, 3541, 4831, 5113, 5701, 6007, 8011, 9901, 10303, 11131, 12433, 13807, 14281, 17293, 20023, 20593, 21757, 23563, 24181, 26083, 28057, 30103, 35911, 41413, 43891, 46441, 53593, 60271, 78121, 82657, 86143, 95791, 108571, 123553, 127807, 136531, 145543, 147073, 156421

Offset: 1

Zak Seidov, Aug 25 2002

Primes of the form k^2 + k + 1 with k odd and positive. - Peter Munn, Jan 27 2018

Primes of the form A000217(2*k) + A000217(2*k+2). - J. M. Bergot, May 09 2018

			3 is a term because for k=2, 4*k^2 - 10*k + 7 = 3 a prime.
7 is not a term because 7 can only be obtained with k=0 or k=5/2.

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A000217, A054554, A073338, A168026.

Subset of A002383.

Filtered(List([2..300],n->4*n^2-10*n+7),IsPrime); # Muniru A Asiru, Apr 15 2018

[a: n in [1..400] | IsPrime(a) where a is 4*n^2 - 10*n + 7]; // Vincenzo Librandi, Dec 23 2019

select(isprime, [seq(4*n^2-10*n+7 ,n=2..300)]); # Muniru A Asiru, Apr 15 2018

Select[Table[4 n^2 - 10 n + 7, {n, 1, 200}], PrimeQ] (* Vincenzo Librandi, Dec 23 2019 *)

select(isprime,vector(300,k,4*k^2 - 10*k + 7)) \\ Joerg Arndt, Feb 28 2018

a(n) = A054554(A073338(n)). - Elmo R. Oliveira, Apr 20 2025

Edited by Dean Hickerson, Aug 28 2002
a(1)=7 inserted and typo in Mathematica code corrected by Vincenzo Librandi, Dec 09 2011
Incorrect term 7 removed by Joerg Arndt, Feb 28 2018

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

Original entry on oeis.org

3, 43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533

Offset: 1

Bernard Schott, Dec 18 2012

nonn

These are the primes associated with A182253.
All these numbers are in A002383 but not in A053183.
All the numbers n^2 + n + 1 = 111_n with n >= 2 are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A182253.

Cf. A085104, A125134.

Select[Table[If[PrimeQ[n],Nothing,n^2+n+1],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 02 2023 *)

lista(nn) = {for (n = 1, nn, if (! isprime(n) && isprime(p = n^2 + n + 1), print1(p, ", ");););} \\ Michel Marcus, Sep 04 2013

A081257 a(n) is the greatest prime factor of (n^3 - 1).

Original entry on oeis.org

7, 13, 7, 31, 43, 19, 73, 13, 37, 19, 157, 61, 211, 241, 13, 307, 17, 127, 421, 463, 13, 79, 601, 31, 37, 757, 271, 67, 29, 331, 151, 1123, 397, 97, 43, 67, 1483, 223, 547, 1723, 139, 631, 283, 109, 103, 61, 181, 43, 2551, 379, 919, 409, 2971, 79, 103, 3307, 163

Offset: 2

Jan Fricke, Mar 14 2003

The record values here (as well as those for A081256) appear to match the terms of A002383 for n > 1. - Bill McEachen, Jun 19 2023

			a(7)=19 because 7^3 - 1 = 342 = 2*3*3*19.

Jon E. Schoenfield, Table of n, a(n) for n = 2..10000 (first 999 terms from R. J. Mathar).

Cf. A081256, A081258, A002383.

Cf. A096175 (n^3-1 is an odd semiprime), A096176 ((n^3-1)/(n-1) is prime).

A081257 := proc(n)
        A006530( n^3-1) ;
end proc: # R. J. Mathar, Jul 18 2015

FactorInteger[#][[-1,1]]&/@(Range[2,60]^3-1) (* Harvey P. Dale, Oct 09 2017 *)

a(n)=my(f=factor(n^3-1)); f[#f~,1] \\ Charles R Greathouse IV, Mar 08 2017

from sympy import primefactors
def A081257(n): return max(primefactors(n-1)+primefactors(n*(n+1)+1)) # Chai Wah Wu, Oct 15 2022

a(n) = A006530(A068601(n)). - Michel Marcus, Jun 19 2023

More terms from Hugo Pfoertner, Jun 21 2004

A110284 Squares of the form 4p - 3, where p is a prime.

Original entry on oeis.org

9, 25, 49, 121, 169, 289, 625, 841, 961, 1225, 1681, 1849, 2401, 3025, 4489, 5929, 6889, 10201, 11881, 13225, 14161, 15625, 17689, 19321, 20449, 22801, 24025, 24649, 25921, 32041, 32761, 39601, 41209, 44521, 48841, 49729, 55225, 57121, 69169

Offset: 1

Giovanni Teofilatto, Sep 07 2005

nonn

Also: squares of the form 2*s-3, where s is a semiprime, A107317. - Franklin T. Adams-Watters, Jun 28 2010

Squares are less dense then primes and easy to generate so it's faster to check squares if they are of the required form than to check if primes are of the required form. - David A. Corneth, Oct 15 2018

David A. Corneth, Table of n, a(n) for n = 1..16289 (First 1316 terms by Marius A. Burtea, terms < 10^11)

Cf. A002383, A002384, A088503, A001358.

[4*p - 3: p in PrimesUpTo(10^5)|IsSquare (4*p - 3)]; // Vincenzo Librandi, Oct 17 2018

Select[ 4Prime[ Range[2000]] - 3, IntegerQ[ Sqrt[ # ]] &] (* Robert G. Wilson v, Sep 20 2005 *)

isok(n) = issquare(n) && (p=(n+3)/4) && (frac(p)==0) && isprime(p); \\ Michel Marcus, Oct 15 2018

upto(n) = my(res = List()); forstep(i = 3, sqrtint(n), 2, if(isprime((i^2+3)/4), listput(res, i^2))); res \\ David A. Corneth, Oct 15 2018

a(n) = 4*A002383(n) - 3 = A088503(n-1)^2.

Extended by Ray Chandler, Sep 07 2005

A285017 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1 where n is not prime.

Original entry on oeis.org

43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22621, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533

Offset: 1

Bernard Schott, Apr 08 2017

nonn

These numbers are Brazilian primes belonging to A085104.
Brazilian primes with n prime are A023195, except 3 which is not Brazilian.
A085104 = This sequence Union { A023195 \ number 3 }.
k + 1 is necessarily prime, but that's not sufficient: 1 + 10 + 100 = 111.
Most of these terms come from A185632 which are prime numbers 111_n with n no prime, the first other term is 22621 = 11111_12, the next one is 245411 = 11111_22.
Number of terms < 10^k: 0, 2, 9, 23, 64, 171, 477, 1310, 3573, 10098, ..., . - Robert G. Wilson v, Apr 15 2017

			157 = 12^2 + 12 + 1 = 111_12 is prime and 12 is composite.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1310 from Robert G. Wilson v)
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, April-June 2010, pages 30-38, included here with permission from the editors of Quadrature.

Cf. A002383, A023195, A053183, A053696, A085104, A088548, A088550, A185632, A190527, A194257.

N:= 40000: # to get all terms <= N
res:= NULL:
for k from 2 to ilog2(N) do if isprime(k) then
  for n from 2 do
    p:= (n^(k+1)-1)/(n-1);
    if p > N then break fi;
    if isprime(p) and not isprime(n) then res:= res, p fi
od fi od:
res:= {res}:
sort(convert(res,list)); # Robert Israel, Apr 14 2017

mx = 36000; g[n_] := Select[Drop[Accumulate@Table[n^ex, {ex, 0, Log[n, mx]}], 2], PrimeQ]; k = 1; lst = {}; While[k < Sqrt@mx, If[CompositeQ@k, AppendTo[lst, g@k]; lst = Sort@Flatten@lst]; k++]; lst (* Robert G. Wilson v, Apr 15 2017 *)

isok(n) = {if (isprime(n), forcomposite(b=2, n, d = digits(n, b); if ((#d > 2) && (vecmin(d) == 1) && (vecmax(d)== 1), return(1)););); return(0);} \\ Michel Marcus, Apr 09 2017

A285017_vec(n)={my(h=vector(n,i,1),y,c,z=4,L:list);L=List();forprime(x=3,,forcomposite(m=z,x-1,y=digits(x,m);if((y==h[1..#y])&&2<#y,listput(L,x);z=m;if(c++==n,return(Vec(L))))))} \\ R. J. Cano, Apr 18 2017

A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

Original entry on oeis.org

3, 7, 21, 43, 111, 157, 273, 343, 507, 813, 931, 1333, 1641, 1807, 2163, 2757, 3423, 3661, 4423, 4971, 5257, 6163, 6807, 7833, 9313, 10101, 10507, 11343, 11773, 12657, 16003, 17031, 18633, 19183, 22053, 22651, 24493, 26407, 27723, 29757, 31863

Offset: 1

Alexander Adamchuk, Aug 02 2006

nonn

Prime terms belong to A074268, which is a subset of A002383, A087126, A034915, A085104.

In every interval of prime(n)^2 integers, a(n) is the number that are not divisible by prime(n) plus the number that are divisible by prime(n)^2. - Peter Munn, Dec 12 2020

Cf. A002061, A074268, A002383, A087126, A034915, A068468, A085104, A083558.

Mathematica
```
Table[Prime[n]^2-Prime[n]+1,{n,1,100}]
```

a(n) = {my(p = prime(n)); p^2 - p + 1; } \\ Amiram Eldar, Nov 07 2022

a(n) = prime(n)^2 - prime(n) + 1.
a(n) = A036689(n)+1. - R. J. Mathar, Aug 13 2019
Product_{n>=1} (1 - 1/a(n)) = zeta(6)/(zeta(2)*zeta(3)) (A068468). - Amiram Eldar, Nov 07 2022

cp's OEIS Frontend

A088548 Primes of the form k^4 + k^3 + k^2 + k + 1.

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A237040 Semiprimes of the form k^3 + 1.

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A163418 Primes of the form ((p-1)/2)^2+((p+1)/2), where p is prime.

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A059055 Primes which can be written as (b^k+1)/(b+1) for positive integers b and k.

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A073337 Primes of the form 4*k^2 - 10*k + 7 with k positive.

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A185632 Primes of the form n^2 + n + 1 where n is nonprime.

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A081257 a(n) is the greatest prime factor of (n^3 - 1).

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A073337 Primes of the form 4k^2 - 10k + 7 with k positive.