A053182 -id:A053182 - cp's OEIS frontend

A253849 Numbers k such that 2^sigma(k) - 1 is a prime.

2, 4, 9, 16, 25, 64

Offset: 1

Jaroslav Krizek, Jan 16 2015

Also numbers n such that sigma(n) is in A000043, i.e., p = 2^sigma(n) - 1 is a Mersenne prime (A000668). The sequence of corresponding primes p reads: 7, 127, 8191, 2147483647, 2147483647, 170141183460469231731687303715884105727, ..., see A253851.
Subsequence of A023194 (numbers n such that sigma(n) is a prime), see there for an explanation why all terms except the first one are squares.
The sequence of values of sigma(a(n)) is 3, 7, 13, 31, 31, 127, ... and each term of this sequence must be a prime from the sequence of Mersenne exponents (A000043). See A253850.
Sequence differs from A023194 because A023194(7) = 289 but if a(7) exists, it must be a number n such that sigma(n) > A000043(43) = 30402457.
a(n) must be an even power of a prime. If it is the square of an odd prime, then this prime must be in A053182. If a(n) is an even power of 2, a(n)=2^(2k), then sigma(a(n))=2^(2k+1)-1. Thus, 2k+1 must be a double Mersenne prime exponent, i.e., such that the corresponding Mersenne prime is again a Mersenne exponent, cf. A103901. Only 4 such primes are known, and a(6)=2^6 (k=3) corresponds to the largest known prime of this type, 2^(2k+1)-1 = 127. - M. F. Hasler, Jan 21 2015

			4 is in the sequence because 2^sigma(4)-1 = 2^7-1 = 127 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851.

[n: n in[1..10000] | IsPrime((2^SumOfDivisors(n)) - 1)];

a253849[n_] := Select[Range@ n, PrimeQ[2^DivisorSigma[1, #] - 1] &]; a253849[20000] (* Michael De Vlieger, Jan 19 2015 *)

A261810 n and (2n^2 + 2n - 1) are primes.

Original entry on oeis.org

2, 3, 5, 11, 23, 59, 71, 113, 131, 137, 149, 179, 227, 257, 263, 269, 293, 317, 347, 353, 401, 419, 443, 449, 467, 557, 653, 659, 677, 683, 743, 773, 809, 839, 857, 881, 911, 929, 947, 977, 1019, 1049, 1277, 1301, 1319, 1433, 1571, 1697, 1847, 1871, 1901, 1913

Offset: 1

Jaroslav Krizek, Sep 01 2015

nonn

Primes p such that (number of divisors of p * sum of divisors of p * product of divisors of p - 1) is also a prime.
Primes p such that (A000005(p) * A000203(p) * A007955(p) - 1) is also a prime.
See similar sequences of type primes p such that x is also a prime for some x wherein tau(p) = A000005(p) = number of divisors of p, sigma(p) = A000203(p) = sum of divisors of p and pod(p) = A007955(p) = product of divisors of p:
A001359 (for x = tau(p) + sigma(p) - 1 and x = tau(p) + pod(p)),
A005382 (for x = tau(p) * pod(p) - 1),
A005384 (for x = sigma(p) + pod(p), x = tau(p) * sigma(p) - 1 and x = tau(p) * pod(p) + 1),
A023200 (for x = tau(p) + sigma(p) + 1),
A023204 (for x = tau(p) + sigma(p) + pod(p) and x = tau(p) * sigma(p) + 1),
A053182 (for x = sigma(p) * pod(p) + 1),
A053184 (for x = sigma(p) * pod(p) - 1),
A158526 (for x = tau(p) * sigma(p) * pod(p) + 1).
For n >= 3, a(n) == 5 mod 6. - Robert Israel, Sep 02 2015

			3 and 2*3^2 + 2*3 - 1 = 23 are primes.

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

Cf. A000005, A000203, A007955.

Cf. A001359, A005382, A005384, A023200, A023204, A053182, A053184, A158526.

[n: n in[1..10000] | IsPrime(n) and IsPrime(2*n*n + 2*n - 1)];

select(t -> isprime(t) and isprime(2*t^2 + 2*t-1), [2,3,seq(6*i-1,i=1..1000)]); # Robert Israel, Sep 02 2015

Select[Prime[Range[300]], PrimeQ[2 #^2 + 2 # - 1] &] (* Vincenzo Librandi, Sep 02 2015 *)

is(n)=isprime(n)&&isprime(2*n^2 + 2*n - 1) \\ Anders Hellström, Sep 01 2015

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

A341659 Primes p such that p^3 - 1 has 8 divisors.

Original entry on oeis.org

59, 167, 383, 839, 1487, 4259, 5087, 6047, 6599, 6719, 8543, 8963, 9743, 12227, 12647, 13163, 14087, 14867, 18947, 20123, 22643, 23099, 23159, 24083, 24239, 24659, 25583, 27107, 27299, 30203, 30803, 32507, 34319, 37463, 37799, 38603, 41879, 42839, 44519, 44687

Offset: 1

Jon E. Schoenfield, Feb 26 2021

nonn

Intersection of A005385 (Safe primes p: (p-1)/2 is also prime) and A053182 (Primes p such that p^2 + p + 1 is prime).
For each term p, p^3 - 1  = (p-1)*(p^2 + p + 1) is a number of the form 2*q*r (where q and r are distinct primes): p-1 = 2*q and p^2 + p + 1 = r.
Conjecture: sequence is infinite.

			     p =                    factorization
  n  a(n)    p^3 - 1         of (p^3 - 1)
  -  ----  ------------  -------------------
  1    59        205378  2 *   29 *     3541
  2   167       4657462  2 *   83 *    28057
  3   383      56181886  2 *  191 *   147073
  4   839     590589718  2 *  419 *   704761
  5  1487    3288008302  2 *  743 *  2212657
  6  4259   77254345978  2 * 2129 * 18143341
  7  5087  131639193502  2 * 2543 * 25882657
  8  6047  221115865822  2 * 3023 * 36572257
  9  6599  287365339798  2 * 3299 * 43553401

David A. Corneth, Table of n, a(n) for n = 1..6183

Cf. A000005, A000040, A005385, A053182, A309906.

Select[Range[50000], PrimeQ[#] && DivisorSigma[0, #^3 - 1] == 8 &] (* Amiram Eldar, Feb 26 2021 *)
Select[Prime[Range[5000]],DivisorSigma[0,#^3-1]==8&] (* Harvey P. Dale, Apr 17 2025 *)

isok(p) = isprime(p) && (numdiv(p^3-1) == 8); \\ Michel Marcus, Feb 26 2021

A237613 Numbers k such that tau(sigma(tau(k))) = sigma(tau(sigma(k))), where tau is A000005 and sigma is A000203.

Original entry on oeis.org

1, 4, 9, 25, 81, 289, 1681, 3481, 5041, 7921, 10201, 17161, 27889, 29929, 85849, 146689, 331776, 458329, 491401, 552049, 579121, 597529, 683929, 703921, 734449, 786432, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089, 1885129, 2036329, 2211169

Offset: 1

Paolo P. Lava, Feb 10 2014

nonn

The squares of the terms of A053182 are a subset of this sequence. In fact, in general, if p is prime we have tau(p)=2 and tau(p^2)=3. Therefore tau(p^2)=3 -> sigma(3)=4 -> tau(4)=tau(2^2)=3 and if p belongs to A053182 we also have that sigma(p^2)=p^2+p+1 (prime) -> tau(p^2+p+1)=2 -> sigma(2)=3.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000005, A000203, A053182.

[k:k in [1..2300000]| #Divisors(SumOfDivisors(#Divisors(k))) eq SumOfDivisors(#Divisors(SumOfDivisors(k)))]; // Marius A. Burtea, Aug 17 2019

with(numtheory); P:=proc(q) local n;
for n from 1 to q do
  if tau(sigma(tau(n)))=sigma(tau(sigma(n))) then print(n); fi;
od; end: P(10^6);

s = {}; Do[If[DivisorSigma[1, DivisorSigma[0, DivisorSigma[1, n]]] == DivisorSigma[0, DivisorSigma[1, DivisorSigma[0, n]]], AppendTo[s, n]], {n, 1, 2500000}]; s (* Amiram Eldar, Aug 17 2019 *)
With[{ds=DivisorSigma},Select[Range[2220000],ds[0,ds[1,ds[0,#]]]==ds[1,ds[0,ds[1,#]]]&]] (* Harvey P. Dale, Nov 04 2024 *)

s=[]; for(n=1, 2500000, if(sigma(sigma(sigma(n, 0)), 0) == sigma(sigma(sigma(n), 0)), s=concat(s, n))); s \\ Colin Barker, Feb 10 2014

A243471 Primes p such that p^6 - p^5 + 1 is prime.

Original entry on oeis.org

3, 31, 73, 181, 367, 373, 523, 631, 733, 1021, 1039, 1171, 1489, 1723, 1777, 2203, 2557, 2683, 3121, 3187, 3319, 4441, 4591, 4621, 4801, 4957, 5113, 5167, 5323, 5431, 5659, 5839, 5851, 5857, 6883, 7057, 7129, 7297, 7309, 7477, 7993, 8017, 8209, 8221, 8689, 8821

Offset: 1

K. D. Bajpai, Jun 05 2014

nonn

			31 appears in the sequence because it is prime and 31^6 - 31^5  + 1 = 858874531 is also prime.
73 appears in the sequence because it is prime and 73^6 - 73^5  + 1 = 149261154697 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5457

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

A243471 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5+1; if isprime (b) then RETURN (a); fi; end: seq(A243471 (), n=1..2000);

c=0; Do[k=Prime[n]; If[PrimeQ[k^6-k^5+1], c++; Print[c," ",k]], {n,1,200000}];

A243472 Primes p such that p^6 - p^5 - 1 is prime.

Original entry on oeis.org

2, 31, 101, 151, 181, 199, 229, 277, 307, 317, 379, 439, 479, 491, 647, 691, 797, 911, 997, 1039, 1051, 1181, 1291, 1367, 1381, 1471, 1511, 1549, 1657, 1709, 1847, 1867, 1987, 2081, 2099, 2111, 2207, 2467, 2621, 2707, 3041, 3221, 3259, 3541, 3571, 3581, 3769

Offset: 1

K. D. Bajpai, Jun 05 2014

nonn

			31 appears in the sequence because it is prime and 31^6 - 31^5 - 1 = 858874529 is also prime.
101 appears in the sequence because it is prime and 101^6 - 101^5  - 1 = 1051010050099 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..4699

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

A243472 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5-1; if isprime (b) then RETURN (a); fi; end: seq(A243472 (), n=1..2000);

c = 0;  Do[k=Prime[n]; If[PrimeQ[k^6-k^5-1], c++; Print[c," ",k]], {n,1,200000}];
Select[Prime[Range[600]],PrimeQ[#^6-#^5-1]&] (* Harvey P. Dale, Jan 21 2015 *)

s=[]; forprime(p=2, 4000, if(isprime(p^6-p^5-1), s=concat(s, p))); s \\ Colin Barker, Jun 06 2014

A344448 Square array read by antidiagonals upwards: T(n,k) for integer k >= 0 is the n-th prime p such that p^(2*3^k) + p^(3^k) + 1 is prime.

Original entry on oeis.org

2, 3, 2, 5, 3, 2, 17, 11, 11, 191, 41, 191, 263, 311, 4457, 59, 269, 557, 557, 5867, 3803, 71, 383, 761, 659, 7001, 13859, 1889, 89, 509, 797, 887, 7019, 22961, 16829, 17, 101, 809, 863, 1607, 7541, 31223, 62549, 69677, 113921, 131, 827, 977, 2309, 8609, 44351, 67103, 102647, 176459, 24071

Offset: 1

Martin Becker, May 19 2021

T(n,k)^(3^k), for all n >= 1, k >= 0, arranged by increasing values, is A342690. It is conjectured that all columns are infinite. If 3^k was replaced by k in the definition, all additional columns would be empty, as x^(2*k) + x^k + 1 is reducible if k has prime factors other than 3. For checking the property, Pocklington-Lehmer type primality tests seem particularly effective, as n-1 always has a large smooth factor p^(3^k), cf. the paper of Brillhart, Lehmer and Selfridge (1975), Theorem 5.

This array describes the essence of A342690 and A342691 in much more terse form. T(1, 8) = 113921 matches the 33177-digit value q = 113921^3^8 in A342690 and the 66353-digit prime q^2+q+1 in A342691.

			Array begins:
===============================================================
n\k |   0    1    2    3     4     5      6      7      8     9
----+----------------------------------------------------------
  1 |   2    2    2  191  4457  3803   1889     17 113921 24071
  2 |   3    3   11  311  5867 13859  16829  69677 176459 ...
  3 |   5   11  263  557  7001 22961  62549 102647 ...
  4 |  17  191  557  659  7019 31223  67103 164963 ...
  5 |  41  269  761  887  7541 44351 181931 170669 ...
  6 |  59  383  797 1607  8609 45737 188333 207923 ...
  7 |  71  509  863 2309  8627 61751 205433 235679 ...
  8 |  89  809  977 2621 21773 63377 210407 342833 ...
  9 | 101  827 1091 2687 22871 79481 219761 459209 ...

J. Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2^m+-1, Math. Compl. 29 (1975) 620-647.

The first column T(n,0) is A053182(n). The second column T(n,1) is A066100(n).

Cf. A342690, A342691.

N=5; K=2; m=matrix(N, K+1); for(k=0, K, i=0; forprime(p=2, , q=p^3^k;if(isprime(q^2+q+1, 1), i+=1; m[i,k+1]=p; if(i==N, break)))); m

A065510 Primes p such that p^4 - p^3 + p^2 - p + 1 is prime.

Original entry on oeis.org

2, 3, 5, 11, 37, 43, 47, 71, 131, 157, 223, 251, 257, 307, 487, 641, 1087, 1093, 1187, 1291, 1433, 1567, 1621, 1637, 1831, 1873, 1901, 2017, 2111, 2143, 2293, 2333, 2473, 2621, 2663, 2683, 2707, 2777, 2843, 2903, 3257, 3413, 3463, 3613, 3617, 3761, 3793

Offset: 1

Vladeta Jovovic, Nov 26 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A053182.

Select[Prime[Range[600]],PrimeQ[#^4-#^3+#^2-#+1]&] (* Harvey P. Dale, May 20 2019 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (isprime(p^4 - p^3 + p^2 - p + 1), write("b065510.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 20 2009

A137460 Prime numbers k such that k^2 +- (k+1) are primes.

Original entry on oeis.org

3, 5, 17, 71, 101, 131, 677, 839, 857, 1091, 1217, 2129, 2309, 2339, 2957, 3137, 3449, 3989, 4409, 6569, 6719, 6761, 7229, 8501, 8627, 8807, 9521, 9689, 9749, 10589, 10631, 11621, 11777, 11927, 12641, 13487, 13931, 14519, 15527, 15797, 16007

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2008

			3^2 +- 4 -> (5,13) primes,
5^2 +- 6 -> (19,31) primes.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A053182, A091567.

Select[Prime[Range[800]], PrimeQ[ #^2 - (# + 1)] && PrimeQ[ #^2 + (# + 1)] &]

A053182 INTERSECT A091567. - R. J. Mathar, Apr 19 2009

More terms from Karl Hovekamp, Jan 24 2009

cp's OEIS Frontend

A253849 Numbers k such that 2^sigma(k) - 1 is a prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

A261810 n and (2*n^2 + 2*n - 1) are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A341659 Primes p such that p^3 - 1 has 8 divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A237613 Numbers k such that tau(sigma(tau(k))) = sigma(tau(sigma(k))), where tau is A000005 and sigma is A000203.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A243471 Primes p such that p^6 - p^5 + 1 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A243472 Primes p such that p^6 - p^5 - 1 is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A261810 n and (2n^2 + 2n - 1) are primes.