A053182 -id:A053182 - cp's OEIS frontend

A065509 Primes p such that p^4 + p^3 + p^2 + p + 1 is prime.

2, 7, 13, 17, 23, 29, 43, 73, 79, 83, 127, 193, 227, 239, 263, 277, 337, 359, 373, 397, 439, 457, 479, 503, 557, 563, 617, 919, 967, 1009, 1129, 1187, 1249, 1297, 1327, 1429, 1493, 1553, 1579, 1657, 1663, 1979, 1987, 2069, 2243, 2383, 2617, 2663, 2789

Offset: 1

Vladeta Jovovic, Nov 26 2001

Primes in A049409. - Vincenzo Librandi, Aug 07 2010

The generated prime numbers are in A190527. - Bernard Schott, Dec 20 2012

			a(4) = 17 because 17 is prime and 17^4 + 17^3 + 17^2 + 17 + 1 = 88741 is prime.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith).

Cf. A053182.

[n: n in [0..10000]| IsPrime(n) and IsPrime(n^4+n^3+n^2+n+1)] // Vincenzo Librandi, Aug 07 2010

f[n_]:=1+n+n^2+n^3+n^4; lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst,p]], {n,6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 24 2009 *)
Select[Prime[Range[500]],PrimeQ[Total[#^Range[0,4]]]&] (* Harvey P. Dale, Apr 08 2017 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (isprime(p^4 + p^3 + p^2 + p + 1), write("b065509.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 20 2009

{A065509_vec(N,p=1)=vector(N,i,until(isprime((p^5-1)\(p-1)),p=nextprime(p+1));p)} \\ M. F. Hasler, Mar 03 2020

A065404 Squares of composite numbers k such that sigma(k) (sum of divisors of k, A000203) is a prime.

Original entry on oeis.org

16, 64, 729, 2401, 4096, 15625, 28561, 65536, 83521, 262144, 279841, 531441, 707281, 3418801, 4826809, 9765625, 24137569, 28398241, 38950081, 47458321, 244140625, 260144641, 887503681, 1073741824, 1387488001, 2655237841

Offset: 1

Labos Elemer, Nov 06 2001

nonn

			46 cases below 10^12; for M a Mersenne prime, (M+1)/2 is here: M=8191, 4096=(M+1)/2.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..100 from Harry J. Smith)

Cf. A062700, A000203 (sigma), A065403, A065405, A053182, A053183, A028982.

{ n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), write("b065404.txt", n++, " ", m^2); if (n==100, return)) ) } \\ Harry J. Smith, Oct 18 2009

sigma(a(n)) = A065403(n).

A163268 Primes p such that 1 + p + p^2 + p^3 + p^4 + p^5 + p^6 is prime.

Original entry on oeis.org

2, 3, 5, 13, 17, 31, 61, 73, 89, 149, 163, 251, 349, 353, 461, 523, 599, 647, 863, 941, 947, 1087, 1117, 1229, 1277, 1291, 1297, 1409, 1439, 1489, 1567, 1579, 1609, 1627, 1753, 1783, 1831, 2039, 2131, 2293, 2531, 2609, 2753, 2861, 3037, 3163, 3167, 3299

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2009

nonn

Primes in A100330. The generated prime numbers are exactly A194257. [Bernard Schott, Dec 21 2012]

Robert Israel, Table of n, a(n) for n = 1..10000 (first 102 terms from Zak Seidov)

Cf. A053182, A065509, A088550, A100330, A163268, A194257.

select(p -> isprime(p) and isprime(1+p+p^2+p^3+p^4+p^5+p^6), [2,seq(i,i=3..10000,2)]); # Robert Israel, May 05 2017

f[n_]:=1+n+n^2+n^3+n^4+n^5+n^6; lst={}; Do[p=Prime[n]; If[PrimeQ[f[p]], AppendTo[lst,p]], {n,7!}]; lst
Select[Prime[Range[500]],PrimeQ[Total[#^Range[0,6]]]&] (* Harvey P. Dale, Jul 13 2022 *)

n=0;forprime(p=2,10000,isprime((p^7-1)/(p-1))&&print(n++" "p))\\ Zak Seidov, Mar 09 2013

Edited (but not checked) by N. J. A. Sloane, Jul 25 2009

A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

Original entry on oeis.org

1, 6, 8, 12, 14, 15, 20, 21, 24, 27, 33, 38, 50, 54, 57, 62, 66, 69, 75, 77, 78, 80, 90, 99, 105, 110, 111, 117, 119, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 168, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278

Offset: 1

Bernard Schott, Dec 18 2012

nonn

All these numbers are in A002384 but not in A053182.
The generated prime numbers n^2 + n + 1 are in A185632.
All the generated numbers n^2 + n + 1 = 111_n are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
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, A002384, A053182, A053183, A085104, A125134, A185632.

Select[Range@ 280, And[! PrimeQ@ #, PrimeQ[#^2 + # + 1]] &] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = ! isprime(n) && isprime(n^2 + n + 1); \\ Michel Marcus, Sep 04 2013

A240971 Primes p such that (p^2 + p + 1)/3 is prime.

Original entry on oeis.org

7, 13, 19, 31, 43, 73, 97, 103, 127, 157, 199, 223, 241, 271, 409, 421, 661, 673, 727, 859, 883, 937, 1021, 1039, 1051, 1063, 1093, 1447, 1483, 1609, 1657, 1669, 1723, 1753, 1861, 1879, 1993, 2029, 2203, 2437, 2539, 2677, 2719, 2803, 2833, 2953, 3079, 3121

Offset: 1

Vincenzo Librandi, Aug 05 2014

nonn

Under Schinzel's hypothesis, there are infinitely many primes of this form.

p must be of form 6k+1 to give an integer. A053182 lists when p^2 + p + 1 is prime. - Jens Kruse Andersen, Aug 06 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..4900
Eric Weisstein's World of Mathematics, Schinzel's Hypothesis.

Cf. A053182.

[p: p in PrimesInInterval(3,3500)| IsPrime((p^2+p+1) div 3)];

select(n -> isprime(n) and isprime((n^2 + n + 1)/3), [seq(6*k+1,k=1..1000)]); # Robert Israel, Aug 05 2014

Select[Prime[Range[500]], PrimeQ[(#^2 + # + 1)/3] &]

forprime(p=1,10^4,s=(p^2+p+1)/3;if(floor(s)==s,if(isprime(s),print1(p,", ")))) \\ Derek Orr, Aug 05 2014

A259417 Even powers of the odd primes listed in increasing order.

Original entry on oeis.org

1, 9, 25, 49, 81, 121, 169, 289, 361, 529, 625, 729, 841, 961, 1369, 1681, 1849, 2209, 2401, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6561, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 14641, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649

Offset: 1

Hartmut F. W. Hoft, Jun 26 2015

nonn

Each of the following sequences, p^(q-1) with p >= 2 and q > 2 primes, except their respective first elements, powers of 2, is a subsequence:
A001248(p) = p^2,  A030514(p) = p^4,  A030516(p) = p^6,
A030629(p) = p^10, A030631(p) = p^12, A030635(p) = p^16,
A030637(p) = p^18, A137486(p) = p^22, A137492(p) = p^28,
A139571(p) = p^30, A139572(p) = p^36, A139573(p) = p^40,
A139574(p) = p^42, A139575(p) = p^46, A173533(p) = p^52,
A183062(p) = p^58, A183085(p) = p^60.
See also the link to the OEIS Wiki.
The sequences A053182(n)^2, A065509(n)^4, A163268(n)^6 and A240693(n)^10 are subsequences of this sequence.
The odd numbers in A023194 are a subsequence of this sequence.

			a(11) = 5^4 = 625 is followed by a(12) = 3^6 = 729 since no even power of an odd prime falls between them.

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..473
OEIS Wiki, Index entries for number of divisors.

a259417[bound_] := Module[{q, h, column = {}}, For[q = Prime[2], q^2 <= bound, q = NextPrime[q], For[h = 1, q^(2*h) <= bound, h++, AppendTo[column, q^(2*h)]]]; Prepend[Sort[column], 1]]
a259417[25000] (* data *)
With[{upto=25000},Select[Union[Flatten[Table[Prime[Range[2,Floor[ Sqrt[ upto]]]]^n,{n,0,Log[2,upto],2}]]],#<=upto&]] (* Harvey P. Dale, Nov 25 2017 *)

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=1} (P(2*k) - 1/2^(2*k)) = 1.21835996432366585110..., where P is the prime zeta function. - Amiram Eldar, Jul 10 2022

A096527 Number of permutations of divisors of n such that all sums of triple adjacent divisors are primes.

Original entry on oeis.org

0, 0, 0, 6, 0, 0, 0, 12, 6, 4, 0, 12, 0, 4, 4, 4, 0, 0, 0, 16, 12, 0, 0, 20, 6, 4, 12, 20, 0, 0, 0, 0, 4, 4, 24, 48, 0, 4, 12, 50, 0, 0, 0, 4, 12, 0, 0, 0, 0, 0, 0, 16, 0, 0, 24, 136, 12, 4, 0, 286, 0, 0, 96, 0, 24, 0, 0, 30, 0, 0, 0, 0, 0, 0, 32, 16, 4, 0, 0

Offset: 1

Reinhard Zumkeller, Jun 23 2004

nonn

a(A096530(n)) = 0, a(A096529(n)) > 0.

For square of terms of A053182(n), a(n) = 6. - Michel Marcus, May 08 2014

			Divisors of n=10 are {1,2,5,10}:
[1,2,10,5]->(1+2+10,2+5+10)=(13,17), [1,10,2,5]->(1+10+2,10+2+5)=(13,17)
[5,2,10,1]->(5+2+10,2+10+1)=(17,13) and
[5,10,2,1]->(5+10+2,10+2+1)=(17,13): therefore a(10)=4.

Cf. A096528, A000005.

isokperm(v, nbd, d) = {for (j=1, nbd-2, if (! isprime(d[v[j]] + d[v[j+1]] + d[v[j+2]]), return (0));); return (1);}
a(n) = {d = divisors(n); nbd = #d; if (nbd > 2, sum(i=1, nbd!, isokperm(numtoperm(nbd, i), nbd, d)));} \\ Michel Marcus, May 03 2014

More terms from Michel Marcus, May 03 2014

A096529 Numbers whose divisors can be permuted so that all sums of triple adjacent divisors are primes.

Original entry on oeis.org

4, 8, 9, 10, 12, 14, 15, 16, 20, 21, 24, 25, 26, 27, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 52, 55, 56, 57, 58, 60, 63, 65, 68, 75, 76, 77, 81, 84, 85, 86, 88, 92, 93, 99, 100, 104, 105, 111, 115, 117, 119, 123, 124, 125, 129, 132, 135, 136, 140, 143, 145, 147

Offset: 1

Reinhard Zumkeller, Jun 23 2004

nonn

Square of terms of A053182 are in this sequence. - Michel Marcus, May 08 2014
From Amiram Eldar, Nov 08 2024: (Start)
The possible values of the number of even divisors of even terms of this sequence is restricted by the number of odd divisors.
Let k be a term and d_odd(k) = A001227(k) and d_even(k) = A183063 be its number of odd divisors and number of even divisors, respectively. When k is even, in a valid permutation of its divisors there must be two even divisors between two odd divisors, at most 2 before the first odd divisor, and at most 2 after the last odd divisor.
Therefore, d_even(k) - 2*(d_odd(k) - 1) <= 4. Let d(k) = A000005(k) = d_odd(k) + d_even(k), and let e = A007814(k) and m = A000265(k). Then, k = 2^e * m, d(k) = (e+1) * d(m) = (e+1) * d_odd(k), so d_even(k) = e * d_odd(k), and |e-2| * d_odd(k) <= 2.
If m = 1, then d_odd(k) = 1 and e <= 4, so 16 = 2^4 is the largest power of 2 in this sequence.
If m = p is a prime, then d_odd(k) = 2 and e <= 3, and therefore only terms of the form 2*p, 4*p or 8*p are possible. 2*p is a term if and only if p is a term of A106067.
If m is composite, then d_odd(k) > 2 and e <= 2, and therefore k is not divisible by 8. (End)

			Divisors of 24 are {1,2,3,4,6,8,12,24}: [2,8,3,12,4,1,24,6] -> (2+8+3,8+3+12,3+12+4,12+4+1,4+1+24,1+24+6) = (13,23,19,17,29,31): therefore 24 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..74

Complement of A096530.
Cf. A053182, A096527.
Cf. A000005, A000265, A001227, A007814, A106067, A183063.

isok(p) = {my(n = #p); if(n < 3, return(0)); for(k = 1, n-2, if(!isprime(p[k]+p[k+1]+p[k+2]), return(0))); 1;}
is2(n) = {my(d = divisors(n)); forperm(d, p, if(isok(p), return(1))); 0;}
is1(k) = {my(e = valuation(k,2), o = k >> e); (e == 0) || (o == 1 && e <= 4) || (abs(e-2) * numdiv(o) <= 2);}
is(k) = is1(k) && is2(k); \\ Amiram Eldar, Nov 08 2024

A096527(a(n)) > 0.

a(30)-a(51) from Michel Marcus, May 03 2014

a(52) onwards from Amiram Eldar, Nov 08 2024

A136242 Numbers k among A008864 such that k^2 - k + 1 is prime.

Original entry on oeis.org

3, 4, 6, 18, 42, 60, 72, 90, 102, 132, 168, 174, 294, 384, 678, 702, 744, 762, 774, 828, 840, 858, 912, 1092, 1098, 1164, 1182, 1194, 1218, 1374, 1428, 1488, 1560, 1584, 1710, 1812, 1848, 1932, 1974, 2130, 2274, 2310, 2340, 2412, 2664, 2730, 2790, 2958

Offset: 1

Lekraj Beedassy, Dec 24 2007

See A053183 for the primes associated with a(n).

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

Cf. A008864, A053182, A053183, A136243.

Select[Prime[Range[500]] + 1, PrimeQ[#^2 - # + 1] &] (* Amiram Eldar, Apr 19 2024 *)

lista(pmax) = forprime(p=1, pmax, if(isprime(p^2+p+1), print1(p+1, ", "))); \\ Amiram Eldar, Apr 19 2024

a(n) = A053182(n) + 1.

A181149 a(n) = prime(n)^3 + prime(n)^2 + prime(n).

Original entry on oeis.org

14, 39, 155, 399, 1463, 2379, 5219, 7239, 12719, 25259, 30783, 52059, 70643, 81399, 106079, 151739, 208919, 230763, 305319, 363023, 394419, 499359, 578759, 712979, 922179, 1040603, 1103439, 1236599, 1307019

Offset: 1

Jani Melik, Jan 24 2011

a(n) is semiprime just when prime(n) is in A053182. - Charles R Greathouse IV, Apr 23 2022

			a(4)=399 because the 4th prime is 7, 7^3 = 343, 7^2 = 49, and 343 + 49 + 7 = 399.

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. p: A000040; p^2: A001248; p^3: A030078; p^2+p: A036690; p^3+p^2: A135178.

[p^3+p^2+p: p in PrimesUpTo(500)] // Vincenzo Librandi, Jan 26 2011

A181149 := n -> map (p -> p^(3)+p^(2)+p, ithprime(n)):
seq (A181149(n), n=1..40);

#^3+#^2+#&/@Prime[Range[30]] (* Harvey P. Dale, Aug 13 2013 *)

a(n,p=prime(n))=p^3+p^2+p \\ Charles R Greathouse IV, Apr 23 2022

a(n) = A135178(n) + A000040(n). - Elmo R. Oliveira, Mar 22 2023

cp's OEIS Frontend

A065509 Primes p such that p^4 + p^3 + p^2 + p + 1 is prime.

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A065404 Squares of composite numbers k such that sigma(k) (sum of divisors of k, A000203) is a prime.

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A163268 Primes p such that 1 + p + p^2 + p^3 + p^4 + p^5 + p^6 is prime.

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A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

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A240971 Primes p such that (p^2 + p + 1)/3 is prime.

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A259417 Even powers of the odd primes listed in increasing order.

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A096527 Number of permutations of divisors of n such that all sums of triple adjacent divisors are primes.

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A096529 Numbers whose divisors can be permuted so that all sums of triple adjacent divisors are primes.

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