A087370 -id:A087370 - cp's OEIS frontend

A016789 a(n) = 3*n + 2.

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179

Offset: 0

N. J. A. Sloane

Except for 1, n such that Sum_{k=1..n} (k mod 3)*binomial(n,k) is a power of 2. - Benoit Cloitre, Oct 17 2002
The sequence 0,0,2,0,0,5,0,0,8,... has a(n) = n*(1 + cos(2*Pi*n/3 + Pi/3) - sqrt(3)*sin(2*Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - Paul Barry, Jan 28 2004 [Artur Jasinski, Dec 11 2007, remarks that this should read (3*n + 2)*(1 + cos(2*Pi*(3*n + 2)/3 + Pi/3) - sqrt(3)*sin(2*Pi*(3*n + 2)/3 + Pi/3))/3.]
Except for 2, exponents e such that x^e + x + 1 is reducible. - N. J. A. Sloane, Jul 19 2005
The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the first of the second kind). - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
Union of A165334 and A165335. - Reinhard Zumkeller, Sep 17 2009
a(n) is the set of numbers congruent to {2,5,8} mod 9. - Gary Detlefs, Mar 07 2010
It appears that a(n) is the set of all values of y such that y^3 = k*n + 2 for integer k. - Gary Detlefs, Mar 08 2010
These numbers do not occur in A000217 (triangular numbers). - Arkadiusz Wesolowski, Jan 08 2012
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
Also indices of even Bell numbers (A000110). - Enrique Pérez Herrero, Sep 10 2013
Central terms of the triangle A108872. - Reinhard Zumkeller, Oct 01 2014
A092942(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n-1), n >= 1, is also the complex dimension of the manifold E(S), the set of all second-order irreducible Fuchsian differential equations defined on P^1 = C U {oo}, having singular points at most in S = {a_1, ..., a_n, a_{n+1} = oo}, a subset of P^1. See the Iwasaki et al. reference, Proposition 2.1.3., p. 149. - Wolfdieter Lang, Apr 22 2016
Except for 2, exponents for which 1 + x^(n-1) + x^n is reducible. - Ron Knott, Sep 16 2016
The reciprocal sum of 8 distinct items from this sequence can be made equal to 1, with these terms: 2, 5, 8, 14, 20, 35, 41, 1640. - Jinyuan Wang, Nov 16 2018
There are no positive integers x, y, z such that 1/a(x) = 1/a(y) + 1/a(z). - Jinyuan Wang, Dec 31 2018
As a set of positive integers, it is the set sum S + S where S is the set of numbers in A016777. - Michael Somos, May 27 2019
Interleaving of A016933 and A016969. - Leo Tavares, Nov 16 2021
Prepended with {1}, these are the denominators of the elements of the 3x+1 semigroup, the numerators being A005408 prepended with {2}. See Applegate and Lagarias link for more information. - Paolo Xausa, Nov 20 2021
This is also the maximum number of moves starting with n + 1 dots in the game of Sprouts. - Douglas Boffey, Aug 01 2022 [See the Wikipedia link. - Wolfdieter Lang, Sep 29 2022]
a(n-2) is the maximum sum of the span (or L(2,1)-labeling number) of a graph of order n and its complement. The extremal graphs are stars and their complements. For example, K_{1,2} has span 3, and K_2 has span 2. Thus a(3-1) = 5. - Allan Bickle, Apr 20 2023

			G.f. = 2 + 5*x + 8*x^2 + 11*x^3 + 14*x^4 + 17*x^5 + 20*x^6 + ... - _Michael Somos_, May 27 2019

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
H. Balakrishnan and N. Deo, Parallel algorithm for radiocoloring a graph, Congr. Numer. 160 (2003), 193-204.
Allan Bickle, Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392.
L. Euler, Observatio de summis divisorum p. 9.
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 937
L. B. W. Jolley, Summation of Series, Dover, 1961, p. 16
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Capped Triangular Frames
Wikipedia, Sprouts (game)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

First differences of A005449.
Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.
Cf. A087370.
Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.
Cf. A000096, A000217.
Cf. A016933, A016969.

List([0..70],n->3*n+2); # Muniru A Asiru, Nov 02 2018

a016789 = (+ 2) . (* 3)  -- Reinhard Zumkeller, Jul 05 2013

[3*n+2: n in [0..80]]; // Vincenzo Librandi, Apr 14 2015

seq(3*n+2, n = 0 .. 50); # Matt C. Anderson, May 18 2017

Range[2, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
LinearRecurrence[{2,-1},{2,5},70] (* Harvey P. Dale, Aug 11 2021 *)

vector(100,n,3*n-1) \\ Derek Orr, Apr 13 2015

for n in range(0,100): print(3*n+2, end=', ') # Stefano Spezia, Nov 21 2018

G.f.: (2+x)/(1-x)^2.
a(n) = 3 + a(n-1).
a(n) = 1 + A016777(n).
a(n) = A124388(n)/9.
a(n) = A125199(n+1,1). - Reinhard Zumkeller, Nov 24 2006
Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) - log(2)). - Benoit Cloitre, Apr 05 2002
1/2 - 1/5 + 1/8 - 1/11 + ... = (1/3)*(Pi/sqrt(3) - log 2). [Jolley] - Gary W. Adamson, Dec 16 2006
Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3) - log 2)/9 = 0.12451569... (see A196548). [Jolley p. 48 eq (263)]
a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - Philippe Deléham, Nov 03 2008
a(n) = 6*n - a(n-1) + 1 with a(0)=2. - Vincenzo Librandi, Aug 25 2010
Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1). - Gilian Breysens, Jul 21 2017
E.g.f.: (2 + 3*x)*exp(x). - G. C. Greubel, Nov 02 2018
a(n) = A005449(n+1) - A005449(n). - Jinyuan Wang, Feb 03 2019
a(n) = -A016777(-1-n) for all n in Z. - Michael Somos, May 27 2019
a(n) = A007310(n+1) + (1 - n mod 2). - Walt Rorie-Baety, Sep 13 2021
a(n) = A000096(n+1) - A000217(n-1). See Capped Triangular Frames illustration. - Leo Tavares, Oct 05 2021

A003627 Primes of the form 3n-1.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 431, 443, 449, 461, 467, 479, 491, 503, 509, 521, 557, 563, 569, 587

Offset: 1

N. J. A. Sloane and Mira Bernstein

Inert rational primes in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p such that 1+x+x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1. - Benoit Cloitre, Feb 08 2003
A039701(A049084(a(n))) = 2; A134323(A049084(a(n))) = -1. - Reinhard Zumkeller, Oct 21 2007
The set of primes of the form 3n - 1 is a superset of the set of lesser of twin primes larger than three (A001359). - Paul Muljadi, Jun 05 2008
Primes of this form do not occur in or as divisors of {n^2+n+1}. See A002383 (n^2+n+1 = prime), A162471 (prime divisors of n^2+n+1 not in A002383), and A002061 (numbers of the form n^2-n+1). - Daniel Tisdale, Jul 04 2009
Or, primes not in A007645. A003627 UNION A007645 = A000040. Also, primes of the form 6*k-5/2-+3/2. - Juri-Stepan Gerasimov, Jan 28 2010
Except for first term "2", all these prime numbers are of the form: 6*n-1. - Vladimir Joseph Stephan Orlovsky, Jul 13 2011
A088534(a(n)) = 0. - Reinhard Zumkeller, Oct 30 2011
For n>1: Numbers k such that (k-4)! mod k =(-1)^(floor(k/3)+1)*floor((k+1)/6), k>4. - Gary Detlefs, Jan 02 2012
Binomial(a(n),3)/a(n)= (3*A024893(n)^2+A024893(n))/2, n>1. - Gary Detlefs, May 06 2012
For every prime p in this sequence, 3 is a 9th power mod p. See Williams link. - Michel Marcus, Nov 12 2017
2 adjoined to A007528. - David A. Corneth, Nov 12 2017
For n >= 2 there exists a polygonal number P_s(3) = 3s - 3 = a(n) + 1. These are the only primes p with P_s(k) = p + 1, s >= 3, k >= 3, since P_s(k) - 1 is composite for k > 3. - Ralf Steiner, May 17 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Eisenstein Prime
Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand., Vol 35 (1974), 309-317.
Index to sequences related to decomposition of primes in quadratic fields

Primes of form 3n+1 give A002476.
These are the primes arising in A024893, A087370, A088879. A091177 gives prime index.
Cf. A001359, A007528, A007645, A221717, A057145.
Subsequence of A034020.

a003627 n = a003627_list !! (n-1)
a003627_list = filter ((== 2) . (`mod` 3)) a000040_list
-- Reinhard Zumkeller, Oct 30 2011

[n: n in PrimesUpTo(720) | n mod 3 eq 2]; // Bruno Berselli, Apr 05 2011

t1 := {}; for n from 0 to 500 do if isprime(3*n+2) then t1 := {op(t1),3*n+2}; fi; od: A003627 := convert(t1,list);

Select[Range[-1, 600, 3], PrimeQ[#] &] (* Vincenzo Librandi, Jun 17 2015 *)
Select[Prime[Range[200]],Mod[#,3]==2&] (* Harvey P. Dale, Jan 31 2023 *)

is(n)=n%3==2 && isprime(n) \\ Charles R Greathouse IV, Mar 20 2013

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n>=1} 1/a(n)^2 = 0.30792... = A085548 - 1/9 - A175644.
Sum_{n>=1} 1/a(n)^3 = 0.134125... = A085541 - 1/27 - A175645. (End)

A024893 Numbers k such that 3*k+2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 13, 15, 17, 19, 23, 27, 29, 33, 35, 37, 43, 45, 49, 55, 57, 59, 63, 65, 75, 77, 79, 83, 85, 87, 89, 93, 97, 103, 105, 115, 117, 119, 127, 129, 133, 139, 143, 147, 149, 153, 155, 159, 163, 167, 169, 173, 185, 187, 189, 195, 197, 199, 205, 213, 215, 217, 219, 225, 227

Offset: 1

Clark Kimberling

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

Cf. A003627 (associated primes), A091177 (gives prime index).

Cf. A087370, A088879.

[n: n in [0..1000] | IsPrime(3*n+2)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[0, 250], PrimeQ[3# + 2] &] (* Vincenzo Librandi, Sep 25 2012 *)

is(n)=isprime(3*n+2) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A087370(n)-1 = A088879(n)+1.

A091177 Numbers m such that the m-th prime is of the form 3*k-1.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 13, 15, 16, 17, 20, 23, 24, 26, 28, 30, 32, 33, 35, 39, 40, 41, 43, 45, 49, 51, 52, 54, 55, 56, 57, 60, 62, 64, 66, 69, 71, 72, 76, 77, 79, 81, 83, 86, 87, 89, 91, 92, 94, 96, 97, 98, 102, 103, 104, 107, 108, 109, 113, 116, 118, 119, 120, 123

Offset: 1

Ray Chandler, Dec 26 2003

A003627 indexed by A000040.

The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Feb 28 2021

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

Cf. A003627 (primes of the form 3*k-1), A024893, A087370, A088879.

A133677 is another version.

PrimePi/@Select[3Range[0,250]-1,PrimeQ]  (* Harvey P. Dale, Apr 26 2011 *)
Select[Range[150],IntegerQ[(Prime[#]+1)/3]&] (* Harvey P. Dale, Dec 14 2021 *)

a091177(limit)={my(m=0);forprime(p=2,prime(limit),m++;if(p%3==2,print1(m,", ")))};
a091177(123) \\ Hugo Pfoertner, Aug 03 2021

a(n) = k such that A000040(k) = A003627(n).

A088879 Numbers n such that 3n + 5 is a prime.

Original entry on oeis.org

-1, 0, 2, 4, 6, 8, 12, 14, 16, 18, 22, 26, 28, 32, 34, 36, 42, 44, 48, 54, 56, 58, 62, 64, 74, 76, 78, 82, 84, 86, 88, 92, 96, 102, 104, 114, 116, 118, 126, 128, 132, 138, 142, 146, 148, 152, 154, 158, 162, 166, 168, 172, 184, 186, 188, 194, 196, 198, 204, 212, 214, 216, 218

Offset: 1

Giovanni Teofilatto, Nov 27 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A003627 gives primes, A091177 gives prime index.

Cf. A016789.

lst={};Do[p=n+(n+2)+(n+3);If[PrimeQ[p],AppendTo[lst,n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 22 2009 *)
Select[Range[-1,300],PrimeQ[3#+5]&] (* Harvey P. Dale, Jun 08 2020 *)

a(n) = A024893(n) - 1 = A087370(n) - 2.

Edited and extended by Ray Chandler, Dec 26 2003

A259645 Numbers m such that m^2 + 1, 3*m - 1 and m^2 + m + 41 are all prime.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 16, 20, 24, 36, 66, 90, 94, 116, 120, 134, 150, 156, 160, 206, 240, 280, 340, 350, 384, 396, 430, 436, 470, 536, 634, 690, 700, 714, 780, 826, 864, 930, 946, 960, 1004, 1124, 1150, 1176, 1294, 1316, 1376, 1410, 1430, 1494, 1644, 1674

Offset: 1

Reinhard Zumkeller, Jul 03 2015

nonn

This sequence is infinite if the generalized Dickson's conjecture holds.

			.            | (i, j, k) such that |        corresponding
.            | a(n) = A005574(i)   |        prime triples
.     |      |      = A087370(j)   |        let m = a(n):
.   n | a(n) |      = A056561(k)   |  (m^2+1, 3*m-1, m^2+m+41)
.  ---+------+---------------------+--------------------------
.   1 |    1 |     (1,  1,  2)     |        (2,   2,   43)
.   2 |    2 |     (2,  2,  3)     |        (5,   5,   47)
.   3 |    4 |     (3,  3,  5)     |       (17,  11,   61)
.   4 |    6 |     (4,  4,  7)     |       (37,  17,   83)
.   5 |   10 |     (5,  6, 11)     |      (101,  29,  151)
.   6 |   14 |     (6,  7, 13)     |      (197,  41,  251)
.   7 |   16 |     (7,  8, 15)     |      (257,  47,  313)
.   8 |   20 |     (8, 10, 21)     |      (401,  59,  461)
.   9 |   24 |     (9, 11, 25)     |      (597,  71,  641)
.  10 |   36 |    (11, 15, 37)     |     (1297, 107, 1373)
.  11 |   66 |    (15, 24, 61)     |     (4357, 197, 4463)
.  12 |   90 |    (18, 31, 79)     |     (8101, 269, 8231)  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Bunyakovsky conjecture
Wikipedia, Dickson's conjecture

Intersection of A005574, A087370 and A056561.

import Data.List.Ordered (isect)
a259645 n = a259645_list !! (n-1)
a259645_list = a005574_list `isect` a087370_list `isect` a056561_list

Select[Range[100], AllTrue[{#^2 + 1, 3 # - 1, #^2 + # + 41}, PrimeQ] &] (* Robert Price, Apr 19 2025 *)

A255844 a(n) = 2*n^2 + 6.

Original entry on oeis.org

6, 8, 14, 24, 38, 56, 78, 104, 134, 168, 206, 248, 294, 344, 398, 456, 518, 584, 654, 728, 806, 888, 974, 1064, 1158, 1256, 1358, 1464, 1574, 1688, 1806, 1928, 2054, 2184, 2318, 2456, 2598, 2744, 2894, 3048, 3206, 3368, 3534, 3704, 3878, 4056, 4238, 4424, 4614

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.
Equivalently, numbers m such that 2*m-12 is a square.
For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016825 (first differences), A087370, A107303, A114949, A117950.
Cf. A152811: nonnegative numbers of the form 2*m^2-6.
Subsequence of A000378.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+6: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 6, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+6)
```
Sage
```
[2*n^2+6 for n in (0..50)]
```

G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117950(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)
E.g.f.: 2*exp(x)*(3 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Corrected and extended by Bruno Berselli, Mar 11 2015

A281702 Numbers k such that A001221(k) = A001221(A000326(k)).

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 30, 34, 38, 42, 46, 50, 58, 66, 78, 86, 90, 94, 98, 106, 118, 130, 134, 150, 154, 170, 174, 186, 190, 198, 206, 214, 218, 226, 234, 254, 258, 266, 270, 274, 286, 294, 310, 314, 318, 326, 338, 350, 354, 366, 370, 394, 398, 406, 410, 426

Offset: 1

Colin Barker, Jan 28 2017

nonn

Numbers k such that the number of distinct prime divisors of k is equal to the number of distinct prime divisors of the pentagonal number k*(3*k-1)/2.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000326, A001221, A087370, A281703.

Select[Range[100], PrimeNu[#] == PrimeNu[#*(3*# - 1)/2] &] (* G. C. Greubel, Apr 23 2017 *)

select(k->omega(k)==omega(k*(3*k-1)/2), vector(500, k, k))

A281703 Numbers k such that A000005(k) = A000005(A000326(k)).

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 30, 34, 38, 46, 50, 58, 66, 78, 86, 90, 94, 98, 106, 118, 130, 134, 150, 154, 170, 174, 186, 190, 198, 206, 214, 218, 226, 234, 254, 258, 266, 270, 274, 286, 294, 310, 314, 318, 326, 338, 350, 354, 366, 370, 394, 398, 406, 410, 426, 430

Offset: 1

Colin Barker, Jan 28 2017

nonn

Numbers k such that the number of divisors of k is equal to the number of divisors of the pentagonal number k*(3*k-1)/2.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000005, A000326, A087370, A281702.

Select[Range[500],DivisorSigma[0,#]==DivisorSigma[0,(#(3#-1))/2]&] (* Harvey P. Dale, Nov 12 2017 *)

select(k->numdiv(k)==numdiv(k*(3*k-1)/2), vector(500, k, k))

A246907 Numbers n such that sigma(n + sigma(n)) = 3n.

Original entry on oeis.org

1, 2, 4, 8, 16, 64, 128, 2048, 262144, 17179869184, 274877906944, 8796093022208, 36028797018963968

Offset: 1

Jaroslav Krizek, Sep 07 2014

Conjecture: for n >= 2; numbers n of the form 2^k such that 3*(2^k) - 1 is prime. The next terms: 18446744073709551616, 75557863725914323419136, 19807040628566084398385987584, … Sequence of numbers k: 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, … Subsequence of A087370 (numbers n such that 3n - 1 is a prime).

a(14) > 5*10^17. - Hiroaki Yamanouchi, Sep 11 2015

			Number 16 is in sequence because sigma(16 + sigma(16)) = sigma(16 + 31) = sigma(47) = 48 = 3 * 16.

Cf. A000203, A087370, A246906.

[n:n in[1..10000000] | SumOfDivisors(n+SumOfDivisors(n))eq 3*n]

Select[Range[300000], DivisorSigma[1, # + DivisorSigma[1, #]] == 3 # &] (* Harvey P. Dale, Jul 19 2015 *)

for(n=1,10^7,if(sigma(n+sigma(n))==3*n,print1(n,", "))) \\ Derek Orr, Sep 07 2014

a(10)-a(13) from Hiroaki Yamanouchi, Sep 11 2015

cp's OEIS Frontend

A016789 a(n) = 3*n + 2.

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A003627 Primes of the form 3n-1.

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A024893 Numbers k such that 3*k+2 is prime.

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A091177 Numbers m such that the m-th prime is of the form 3*k-1.

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A088879 Numbers n such that 3n + 5 is a prime.

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A259645 Numbers m such that m^2 + 1, 3*m - 1 and m^2 + m + 41 are all prime.

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A255844 a(n) = 2*n^2 + 6.

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