A065509 -id:A065509 - cp's OEIS frontend

A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.

31, 2801, 30941, 88741, 292561, 732541, 3500201, 28792661, 39449441, 48037081, 262209281, 1394714501, 2666986681, 3276517921, 4802611441, 5908670381, 12936304421, 16656709681, 19408913261, 24903325661, 37226181521, 43713558101, 52753304641, 64141071121, 96427561501, 100648118041

Offset: 1

Bernard Schott, Dec 20 2012

nonn

These primes are generated by exactly A065509, cf. 2nd formula.
These numbers are repunit primes 11111_p, so they are Brazilian primes (A085104).
When p^4 + p^3 + p^2 + p + 1 = sigma(p^4) is prime, then it equals A193574(p^4), so that this sequence is a subsequence of A193574; by definition it is also a subsequence of A053699 and A131992. - Hartmut F. W. Hoft, May 05 2017

			a(3) = 30941 = 11111_13 = 13^4 + 13^3 + 13^2 + 13^1 + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Cf. A049409 (n^4 + ... + 1 is prime), A065509 (primes among these n), A193574.
Subsequence of A088548 (primes n^4 + ... + 1) and A085104 ("Brazilian" primes, of the form 1 + n + n^2 + ... + n^k).
Intersection of A000040 (primes) and A131992 (p^4 + ... + 1), subsequence of A053699 (n^4 + ... + 1).

[p: p in PrimesUpTo(600) | IsPrime(p) where p is p^4 +p^3+p^2+p+1]; // Vincenzo Librandi, May 06 2017

a190527[n_] := Select[Map[(Prime[#]^5-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a190527[100] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[#^4 + #^3 + #^2 + # + 1 &/@Prime[Range[100]], PrimeQ] (* Vincenzo Librandi, May 06 2017 *)

[q|p<-primes(100),ispseudoprime(q=(p^5-1)\(p-1))]
A190527_vec(N)=[(p^5-1)\(p-1)|p<-A065509_vec(N)] \\ M. F. Hasler, Mar 03 2020

a(n) = A193574(A065509(n)^4). - Hartmut F. W. Hoft, May 08 2017

a(n) = A053699(A065509(n)) = A000203(A065509(n)^4). - M. F. Hasler, Mar 03 2020

a(7) corrected and a(18)-a(26) added by Hartmut F. W. Hoft, May 05 2017

Edited by M. F. Hasler, Mar 06 2020

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

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

A193366 Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.

Original entry on oeis.org

5, 22621, 245411, 346201, 637421, 837931, 2625641, 3835261, 6377551, 15018571, 16007041, 21700501, 30397351, 35615581, 52822061, 78914411, 97039801, 147753211, 189004141, 195534851, 209102521, 223364311, 279086341, 324842131, 421106401, 445120421, 566124791, 693025471, 727832821, 745720141, 880331261, 943280801, 987082981, 1544755411, 1740422941

Offset: 1

Jonathan Vos Post, Dec 20 2012

Note that there are no primes of the form n^3 + n^2 + n + 1 = (n+1)*(n^2+1) as irreducible components over Z.
From Bernard Schott, May 15 2017: (Start)
These are the primes associated with A286094.
A088548 = A190527 Union {This sequence}.
All the numbers of this sequence n^4 + n^3 + n^2 + n + 1 = 11111_n with n > 1 are Brazilian numbers, so belong to A125134 and A085104. (End)

			a(1) = 1^4 + 1^3 + 1^2 + 1 + 1 = 5.
a(2) = 12^4 + 12^3 + 12^2 + 12 + 1 = 22621.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38.

Subsequence of A088548.
Cf. A000040, A018252, A185632, A192321.
Cf. A049409, A053699, A065509, A085104, A088548, A125134, A190527, A286094.

for n from 1 to 150 do p(n):= 1+n+n^2+n^3+n^4;
if tau(n)>2 and isprime(p(n)) then print(n,p(n)) else fi od: # Bernard Schott, May 15 2017

Select[Map[Total[#^Range[0, 4]] &, Select[Range@ 204, ! PrimeQ@ # &]], PrimeQ] (* Michael De Vlieger, May 15 2017 *)

print1(5);forcomposite(n=4,1e3,if(isprime(t=n^4+n^3+n^2+n+1),print1(", "t))) \\ Charles R Greathouse IV, Mar 25 2013

{n^4 + n^3 + n^2 + n + 1 where n is in A018252}.

A286094 Nonprime numbers n such that n^4 + n^3 + n^2 + n + 1 is prime.

Original entry on oeis.org

1, 12, 22, 24, 28, 30, 40, 44, 50, 62, 63, 68, 74, 77, 85, 94, 99, 110, 117, 118, 120, 122, 129, 134, 143, 145, 154, 162, 164, 165, 172, 175

Offset: 1

Bernard Schott, May 02 2017

nonn

A065509 Union {this sequence} = A049409.

The corresponding prime numbers n^4 + n^3 + n^2 + n + 1 = 11111_n are in A193366; these Brazilian primes, except 5 which is not Brazilian, belong to A085104 and A285017.

			12 is in the sequence because 12^4 + 12^3 + 12^2 + 12 + 1 = 11111_12 = 22621, which is prime.

R. J. Cano, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.

Cf. A049409, A065509, A085104, A088548, A125134, A190527, A193366, A285017.

Select[Range@ 414, And[! PrimeQ@ #, PrimeQ[Total[#^Range[0, 4]]]] &] (* Michael De Vlieger, May 03 2017 *)

isok(n)=if(n==1,5,if(ispseudoprime(n), 0, isprime(fromdigits([1, 1, 1, 1, 1], n))));
getfirstterms(n)={my(L:list, c:small); L=List(); c=0; forstep(k=1, +oo, 1, if(isok(k), listput(L, k); if(c++==n, break))); return(Vec(L))} \\ R. J. Cano, May 09 2017

A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).

Original entry on oeis.org

1, 6, 12, 28, 30, 56, 117, 120, 132, 140, 182, 306, 380, 496, 552, 672, 775, 870, 992, 1080, 1287, 1406, 1428, 1680, 1722, 1892, 2016, 2184, 2256, 2480, 2793, 2862, 3276, 3540, 3640, 3782, 3960, 4060, 4556, 4560, 4650, 5112, 5382, 5402, 5460, 6120, 6320, 6552

Offset: 1

Marius A. Burtea, Dec 06 2020

nonn

All terms are nonprimes.
The sequence includes all numbers of the form p*(p + 1) with p prime. Indeed: sigma(p*(p + 1)) = sigma(p)*sigma(p + 1) = (p + 1)*sigma(p + 1). So A036690 is a subsequence. Thus, the sequence is infinite.
Let k >= 1. If p and q = 1 + p + ... + p^(2*k) are prime numbers, then m = p^(2*k)*q is a term. Indeed, sigma(m) = sigma(p^(2*k)*q) = sigma(p^(2*k))*sigma(q) = q*sigma(q).
p is in: A053182 (k = 1), A065509 (k = 2), A163268 (k = 3), and A240693 (k = 5).
For k = 4 there are no prime p because 1 + p + p^2 + p^3 + p^4 + p^5 + p^6 + p^7 + p^8 = (p^6 + p^3 + 1)*(p^2 + p + 1).
If m = 2^(p - 1)*(2^p - 1), p >= 1, (see A006516), then sigma(m) = sigma(2^(p - 1)*(2^p - 1)) = sigma(2^(p - 1))*sigma(2^p - 1) = (2^p - 1)*sigma(2^p - 1), so m is a term.
Thus, A006516(n) and A000396(n), for n >= 1, are terms.

			sigma(6) = 12 = 3*4 = 3*sigma(3), so 6 is a term.
sigma(12) = 28 = 4*7 = 4*sigma(4), so 12 is a term.
sigma(30) = 72 = 6*12 = 6*sigma(6), so 30 is a term.
sigma(56) = 120 = 8*15 = 8*sigma(8), so 56 is a term.
sigma(117) = 182 = 13*14 = 13*sigma(13), so 117 is a term.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A000203, A000396, A006516, A036690, A053182, A064987, A065509, A163268, A327165, A327599.

Magma
```
s:=func; [n:n in [1..6600]|s(n)];
```

q[n_] := Module[{d = Divisors[n], s}, s = Plus @@ d; AnyTrue[d, #*DivisorSigma[1, #] == s &]]; Select[Range[7000], q] (* Amiram Eldar, Dec 06 2020 *)

isok(k) = my(sk=sigma(k)); fordiv(k, d, if (d*sigma(d) == sk, return(1))); \\ Michel Marcus, Dec 06 2020

cp's OEIS Frontend

A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is 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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A259417 Even powers of the odd primes listed in increasing order.

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A193366 Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.

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A286094 Nonprime numbers n such that n^4 + n^3 + n^2 + n + 1 is prime.

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A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).

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