A156592 -id:A156592 - cp's OEIS frontend

A005384 Sophie Germain primes p: 2p+1 is also prime.

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559

Offset: 1

N. J. A. Sloane

Then 2p+1 is called a safe prime: see A005385.
Primes p such that the equation phi(x) = 2p has solutions, where phi is the totient function. See A087634 for another such collection of primes. - T. D. Noe, Oct 24 2003
Subsequence of A117360. - Reinhard Zumkeller, Mar 10 2006
Let q = 2n+1. For these n (and q), the difference of two cyclotomic polynomials can be written as a cyclotomic polynomial in x^2: Phi(q,x) - Phi(2q,x) = 2x Phi(n,x^2). - T. D. Noe, Jan 04 2008
A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e., p = 5 (mod 6). A prime p of the form 6k+1, k >= 1, i.e., p = 1 (mod 6), cannot be a Sophie Germain prime since 2p+1 is divisible by 3. - Daniel Forgues, Jul 31 2009
Also solutions to the equation: floor(4/A000005(2*n^2+n)) = 1. - Enrique Pérez Herrero, May 03 2012
In the spirit of the conjecture related to A217788, we conjecture that for any integers n >= m > 0 there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) is prime. - Zhi-Wei Sun, Mar 26 2013
If k is the product of a Sophie Germain prime p and its corresponding safe prime 2p+1, then a(n) = (k-phi(k))/3, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013
Giovanni Resta found the first Sophie Germain prime which is also a Brazilian number (A125134), 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. - _Bernard Schott, Mar 07 2019
For all Sophie Germain primes p >= 5, 2*p + 1 = min(A, B) where A is the smallest prime factor of 2^p - 1 and B the smallest prime factor of (2^p + 1) / 3. - Alain Rocchelli, Feb 01 2023
Consider a pair of numbers (p, 2*p+1), with p >= 3. Then p is a Sophie Germain prime iff (p-1)!^2 + 6*p == 1 (mod p*(2*p+1)). - Davide Rotondo, May 02 2024

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.
A. Peretti, The quantity of Sophie Germain primes less than x, Bull. Number Theory Related Topics, Vol. 11, No. 1-3 (1987), pp. 81-92.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 76, 227-230.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

J. S. Cheema, Table of n, a(n) for n = 1..100000. [This replaces an earlier b-file computed by T. D. Noe]
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].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
P. Bruillard, S.-H. Ng, E. Rowell and Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050 [math.QA], 2013.
Chris K. Caldwell, The Prime Glossary, Sophie Germain Prime
Andrea Del Centina, Letters of Sophie Germain preserved in Florence Historia Mathematica, Vol. 32 (2005), pp. 60-75.
Harvey Dubner, Large Sophie Germain Primes, Math. Comp., Vol. 65, No. 213 (1996), pp. 393-396.
Luis H. Gallardo and Olivier Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Reikku Kulon, Sublinear arbitrary precision generator of Sophie Germain and safe primes in C (public domain).
Henri Lifchitz, A new and simpler primality test for Sophie-Germain numbers(q=2*p+1).
Victor Meally, Letter to N. J. A. Sloane, no date.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
Frans Oort, Prime numbers, 2013.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 31, 114.
Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept., Jul, 1999.
Carlos Rivera, Puzzle 1122. OEIS A005385, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 1140. Test for Sophie Germain primes, The Prime Puzzles & Problems Connection.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Anthony G. Shannon, Peter J.-S. Shiue, Tian-Xiao He, and Christopher Saito, On the special cases of Carmichael's totient conjecture, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 504-534.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Agoh Takashi, On Sophie Germain primes, Number theory (Liptovský Ján, 1999), Tatra Mt. Math. Publ., Vol. 20 (2000), pp. 65-73.
Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Vmoraru, PlanetMath.org, Sophie Germain prime.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.
Eric Weisstein's World of Mathematics, Sophie Germain Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Sophie Germain prime.
Samuel Yates, Sophie Germain primes, in "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991, pp. 882-886.

Cf. A005385, A057331, A005602, A087634.
Cf. also A000355, A156541, A156542, A156592, A161896, A156660, A156874, A092816, A023212, A007528 (primes of the form 6k-1).
For primes p that remains prime through k iterations of the function f(x) = 2x + 1: this sequence (k=1), A007700 (k=2), A023272 (k=3), A023302 (k=4), A023330 (k=5), A278932 (k=6), A138025 (k=7), A138030 (k=8).

Filtered([1..1600],p->IsPrime(p) and IsPrime(2*p+1)); # Muniru A Asiru, Mar 06 2019

[ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; // Klaus Brockhaus, Jan 01 2009

A:={}: for n from 1 to 246 do if isprime(2*ithprime(n)+1) then A:=A union {ithprime(n)} fi od: A:=A; # Emeric Deutsch, Dec 09 2004

Select[Prime[Range[1000]],PrimeQ[2#+1]&]
lst = {}; Do[If[PrimeQ[n + 1] && PrimeOmega[n] == 2, AppendTo[lst, n/2]], {n, 2, 10^4}]; lst (* Hilko Koning, Aug 17 2021 *)

select(p->isprime(2*p+1), primes(1000)) \\ In old PARI versions <= 2.4.2, use select(primes(1000), p->isprime(2*p+1)).

forprime(n=2, 10^3, if(ispseudoprime(2*n+1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

is_A005384=(p->isprime(2*p+1)&&isprime(p));
  {A005384_vec(N=100,p=1)=vector(N,i,until(isprime(2*p+1),p=nextprime(p+1));p)} \\ M. F. Hasler, Mar 03 2020

from sympy import isprime, nextprime
def ok(p): return isprime(2*p+1)
def aupto(limit): # only test primes
  alst, p = [], 2
  while p <= limit:
    if ok(p): alst.append(p)
    p = nextprime(p)
  return alst
print(aupto(1559)) # Michael S. Branicky, Feb 03 2021

a(n) mod 10 <> 7. - Reinhard Zumkeller, Feb 12 2009
A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. - Reinhard Zumkeller, Feb 18 2009
tau(4*a(n) + 2) = tau(4*a(n)) - 2, for n > 1. - Arkadiusz Wesolowski, Aug 25 2012
eulerphi(4*a(n) + 2) = eulerphi(4*a(n)) + 2, for n > 1. - Arkadiusz Wesolowski, Aug 26 2012
A005097 INTERSECT A000040. - R. J. Mathar, Mar 23 2017
Sum_{n>=1} 1/a(n) is in the interval (1.533944198, 1.8026367) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021
a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

A068443 Triangular numbers which are the product of two primes.

Original entry on oeis.org

6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, 3403, 5671, 12403, 13861, 15931, 18721, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801, 258121, 269011

Offset: 1

Stephan Wagler (stephanwagler(AT)aol.com), Mar 09 2002

These triangular numbers are equal to p * (2p +- 1).
All terms belong to A006987. For n>2 all terms are odd and belong to A095147. - Alexander Adamchuk, Oct 31 2006
A156592 is a subsequence. - Reinhard Zumkeller, Feb 10 2009
Triangular numbers with exactly 4 divisors. - Jon E. Schoenfield, Sep 05 2018

			Triangular numbers begin 0, 1, 3, 6, 10, ...; 6=2*3, and 2 and 3 are two distinct primes; 10=2*5, and 2 and 5 are two distinct primes, etc. - _Vladimir Joseph Stephan Orlovsky_, Feb 27 2009
a(11) = 1891 and 1891 = 31 * 61.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A005382, A005384, A006987, A095147, A001358, A005385, A006881, A007304, A066179, A111206, A157342, A157344-A157347, A157352-A157357, A164977.

Cf. A075875.

q:= n-> is(numtheory[bigomega](n)=2):
select(q, [i*(i+1)/2$i=0..1000])[];  # Alois P. Heinz, Mar 27 2024

Select[ Table[ n(n + 1)/2, {n, 1000}], Apply[Plus, Transpose[ FactorInteger[ # ]] [[2]]] == 2 &]
Select[Accumulate[Range[1000]],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 03 2016 *)

list(lim)=my(v=List());forprime(p=2,(sqrtint(lim\1*8+1)+1)\4, if(isprime(2*p-1),listput(v,2*p^2-p)); if(isprime(2*p+1), listput(v,2*p^2+p))); Vec(v) \\ Charles R Greathouse IV, Jun 13 2013

A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 2009

a(n) = A000217(A164977(n)). - Zak Seidov, Feb 16 2015

Edited by Robert G. Wilson v, Jul 08 2002

Definition corrected by Zak Seidov, Mar 09 2008

A298856 Triangular numbers n for which A240542(n) = A240542(n-1).

Original entry on oeis.org

3, 10, 21, 55, 78, 105, 136, 171, 253, 351, 406, 465, 595, 666, 741, 820, 903, 1081, 1275, 1378, 1711, 1830, 1953, 2211, 2485, 2628, 2775, 2926, 3081, 3403, 3741, 3916, 4465, 4656, 5050, 5253, 5671, 5886, 6105, 6328, 7021, 7503, 7750, 8001, 8515, 9045, 9316, 9591

Offset: 1

Hartmut F. W. Hoft, Jan 27 2018

nonn

Number n is in this sequence exactly when two parts of the symmetric representation of sigma(n) meet at the diagonal.
Proof: If n = k*(2*k+1) is in this sequence then the length of row n in A240542 is 2*k and that of row n-1 is 2*k-1, i.e., the last leg of the Dyck path for n down to the diagonal is vertical and that for n-1 is horizontal to the same point on the diagonal. Therefore, one part of the symmetric representation of sigma(n) ends at the diagonal and so does its symmetric copy. Conversely, if two parts meet at the diagonal then the number of legs in the Dyck path to the diagonal for n, i.e., the length of row n in A240542, is one larger than that for n-1 and must be even, i.e., n has the form n = k*(2*k+1).
A156592 is a subsequence since for every number of the form n = p*(2*p+1) where both p and 2*p+1 are primes A240542(n) = A240542(n-1). For a proof let T(n,k) = ceiling((n+1)/k - (k+1)/2) for 1 <= k <= floor((sqrt(8*n+1) - 1)/2) = 2*p, see A235791; then T(n,k) = T(n-1,k) + 1 for k = 1, 2, p, 2*p, and T(n,k) = T(n-1,k) for all other k. Therefore, the two alternating sums defining A240542(n) and A240542(n-1) are equal, i.e., their Dyck paths meet at the diagonal.
Except for missing 10 the intersection of this sequence and A298855 equals A156592. Sequence A262259 is a subsequence of this sequence.
The five known members of A191363 belong to this sequence, and since their symmetric representation consists of two parts of width one (the respective rows of triangle A237048 have the form 1 0 ... 0 1) they also belong to A262259.
Subsequence of A014105. - Omar E. Pol, Jan 31 2018
Second hexagonal numbers without middle divisors. - Omar E. Pol, Mar 10 2023

			3, 10 and 21 are in the sequence as the illustration of Dyck paths in sequence A237593 shows.
The sequence contains triangular numbers n*(2n+1) where neither n nor 2n+1 are prime. Numbers 1275=25*51 and 2926=38*77 are examples, however, 36 = 4*9 does not belong to the sequence.
78 is the first number in the sequence whose two parts of its symmetric representation contain pieces of width two.

Cf. A000217, A156592, A191363, A237048, A237270, A237593, A240542, A262259, A298855.

Cf. A067742.

(* Function path[] is defined in A237270 *)
meetAtDiagonalQ[n_] := Module[{diags=Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]}, Length[Union[diags[[n]]]]==1 && First[diags[[n-1]]]!=Last[diags[[n-1]]]]
a298856[m_, n_] := Select[Map[#(2#+1)&, Range[m, n]], meetAtDiagonalQ]
a298856[1, 70] (* data *)

A231966 Squarefree numbers (from A005117) with prime divisors in a 2p+1 progression.

Original entry on oeis.org

10, 21, 55, 110, 253, 1081, 1265, 1711, 2530, 3403, 5671, 11891, 13861, 15931, 25651, 34453, 59455, 60031, 64261, 73153, 108811, 114481, 118910, 126253, 158203, 171991, 258121, 351541, 371953, 392941, 482653, 518671, 568301, 703891, 822403, 853471, 869221, 933661

Offset: 1

Jaroslav Krizek, Nov 16 2013

nonn

Squarefree numbers with k>=2 prime divisors of the form p_1 * p_2 * … * p_k, where p_1 < p_2 < … < p_k = primes with p_k = 2 * p_(k-1) + 1.

Supersequence of A156592 (numbers of the form p*q, p and q prime with q=2*p+1; see A005384 and A005385).

			118910 = 2*5*11*23*47, where 5 = 2*2 + 1, 11 = 2*5 + 1, 23 = 2*11 + 1, 47 = 2*23 + 1.

Cf. A005117, A156592, A005384, A005385, A000040, A231967, A231968, A231969.

A226755 Numbers of the form p*q, p and q prime with q=2p-3.

Original entry on oeis.org

9, 35, 77, 209, 299, 527, 989, 1829, 2627, 3239, 3569, 5459, 8777, 9869, 13529, 18527, 20099, 22577, 25199, 31877, 37127, 48827, 55277, 64979, 72389, 73919, 88409, 98789, 107879, 115439, 125249, 137549, 159329, 192509, 200027, 218129, 239777, 277139, 353219

Offset: 1

José María Grau Ribas, Jun 16 2013

nonn

The smaller prime factor of a(n) = p = sopf(a(n))/3 + 1. The larger prime factor of a(n) = q = 2*sopf(a(n))/3 - 1. Furthermore, 2(sopf(a(n))/3 + 1) is representable as the sum of two primes in at least two ways since 2p = p + p = 3 + q. - Wesley Ivan Hurt, Jun 30 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A129521, A156592, A226754.

fa = FactorInteger; t[n_]:=Length[fa[n]] == 2 && fa[n][[1,2]]== fa[n][[2, 2]] == 1 && 2 fa[n][[1, 1]]-3 == fa[n][[2, 1]]; Select[1+Range[200000], t]

list(lim)=my(v=List(), q); forprime(p=2, (sqrt(8*lim+9)+3)\4, if(isprime(q=2*p-3), listput(v, p*q))); Vec(v) \\ Charles R Greathouse IV, Nov 19 2013

a(1) added by Charles R Greathouse IV, Nov 19 2013

A229965 Numbers n such that A229964(n) = 1.

Original entry on oeis.org

6, 8, 10, 16, 21, 55, 253, 1081, 1711, 3403, 5671, 13861, 15931, 25651, 34453, 60031, 64261, 73153, 108811, 114481, 126253, 158203, 171991, 258121, 351541, 371953, 392941, 482653, 518671, 703891, 822403, 853471, 869221, 933661, 1034641, 1104841, 1159003

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {6, 8, 16} UNION A156592.

Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229966, A229967.

A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.

Original entry on oeis.org

21, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 235, 237, 249, 253, 259, 265, 267, 287, 291, 295, 301, 303, 305, 309, 319, 321, 327, 329, 335, 339, 341, 355, 365, 371, 377, 381, 393, 395

Offset: 1

Hartmut F. W. Hoft, Jan 27 2018

All numbers in this sequence are odd since the symmetric representation of 2*p, p prime > 3, has two parts each of size 3*(p+1)/2, and that for 6 has one part of size 12.
A number in this sequence has the form p*q, p and q prime, 3 <= p and 2*p < q, since in this case 2*p <= floor((sqrt(8*p*q + 1) - 1)/2) < q so that 1's in row p*q of A237048 occur only in positions 1, 2, p and 2*p.
This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107.
The two central parts of the symmetric representation of sigma(p*q), each of size (p+q)/2, meet on the diagonal when q = 2*p + 1 since in this case 2*p = floor((sqrt(8*p*q + 1) - 1)/2). These triangular numbers p*(2p+1) form sequence A156592, except for its first element 10, and form a subsequence of the diagonal in the associated irregular triangle of this sequence given in the Example section. They also are a subsequence of A264104. A function to compute the coordinates on the diagonal where the two central parts meet is defined in sequence A240542.
Except for missing 10 the intersection of this sequence and A298856 equals A156592.

			21=3*7 is the smallest number in the sequence since 2*3<7.
1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal.
The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes:
  q\p|   3    5    7   11   13   17   19   23
  ---+---------------------------------------
   7 |  21
  11 |  33   55
  13 |  39   65
  17 |  51   85  119
  19 |  57   95  133
  23 |  69  115  161  253
  29 |  87  145  203  319  377
  31 |  93  155  217  341  403
  37 | 111  185  259  407  481  629
  41 | 123  205  287  451  533  697  779
  43 | 129  215  301  473  559  731  817
  47 | 141  235  329  517  611  799  893 1081

Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856.

(* Function a237270[] is defined in A237270 *)
a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]]
a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &]
a298855[1, 400] (* data *)
(* column for prime p through number n *)
stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&]

A176045 Numbers n such that n-1 and 2*n-1 are both prime.

Original entry on oeis.org

3, 4, 6, 12, 24, 30, 42, 54, 84, 90, 114, 132, 174, 180, 192, 234, 240, 252, 282, 294, 360, 420, 432, 444, 492, 510, 594, 642, 654, 660, 684, 720, 744, 762, 810, 912, 954, 1014, 1020, 1032, 1050, 1104, 1224, 1230, 1290, 1410, 1440, 1452, 1482, 1500, 1512, 1560

Offset: 1

Michel Lagneau, Apr 07 2010

nonn

Also numbers n such that all eigenvalues of the n X n matrix M_n defined in A176043 are prime. The eigenvalues are 2*n-1, and n-1 with multiplicity n-1.

a(n)^2 = p^2 + q, where both p and q are primes. These are the only squares of this form, and which always yields q > p with a(n) - 1 = p = A005384(n) and 2*a(n) - 1 = q = A005385(n), for the same n. Also: a(n) = q - p; p + q + a(n) = 2q = A194593(n+1); and p*q = A156592 - Richard R. Forberg, Mar 04 2015

			6-1 = 5 and 2*6-1 = 11 are both prime, so 6 is in the sequence. 7-1 = 6 and 2*7-1 = 13 are not both prime, so 7 is not in the sequence.
p = 3, q = 7; p^2 + q = 16, a(n) = sqrt(16) = 4. - _Richard R. Forberg_, Mar 04 2015

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

Cf. A176043, A005384 (Sophie Germain primes), A005385 (Safe Primes), A124485 (2*n-1 and 4*n-1 are prime).

[ n: n in [2..1600] | IsPrime(n-1) and IsPrime(2*n-1) ]; // Klaus Brockhaus, Apr 19 2010

with(numtheory):for n from 2 to 2000 do:if type((2*n-1),prime)=true and type((n-1),prime)=true then print(n):else fi:od:

Select[Prime[Range[250]],PrimeQ[2#+1]&]+1 (* Harvey P. Dale, Jul 31 2013 *)

isok(n) = isprime(n-1) && isprime(2*n-1); \\ Michel Marcus, Apr 06 2016

a(n) = A005384(n)+1.

a(n) = 2*A124485(n-1) for n > 1.

Edited and 1482 inserted by Klaus Brockhaus, Apr 19 2010

A226754 Numbers of the form pq, p and q prime with q=2p+3.

Original entry on oeis.org

14, 65, 119, 377, 629, 779, 1769, 3827, 4559, 5777, 9179, 10877, 16109, 19109, 25877, 32639, 37949, 39059, 49769, 56279, 60377, 75077, 78209, 79799, 100127, 103739, 105569, 145529, 154289, 161027, 189419, 228149, 244649, 250277, 288419, 294527, 316409, 335789

Offset: 1

José María Grau Ribas, Jun 16 2013

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A129521, A156592, A226755.

fa = FactorInteger; t[n_]:=Length[fa[n]] == 2 && fa[n][[1,2]]== fa[n][[2, 2]] == 1 && 2 fa[n][[1, 1]]+3 == fa[n][[2, 1]];Select[1+Range[200000], t]
Times@@#&/@Select[Table[{p,2p+3},{p,Prime[Range[200]]}],PrimeQ[#[[2]]]&] (* Harvey P. Dale, Jul 03 2021 *)

list(lim)=my(v=List(),q); forprime(p=2,(sqrt(8*lim+9)-3)\4, if(isprime(q=2*p+3),listput(v,p*q))); Vec(v) \\ Charles R Greathouse IV, Nov 19 2013

A264104 Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one and two regions meet at the center of the Dyck path.

Original entry on oeis.org

21, 55, 253, 406, 1081, 1378, 1711, 3403, 3916, 5671, 9316, 11026, 13861, 14878, 15931, 25651, 27028, 34453, 36046, 42778, 50086, 60031, 64261, 73153, 75466, 108811, 114481, 126253, 129286, 154846, 158203, 161596, 171991, 175528, 212878, 258121, 298378, 317206, 326836, 351541, 366796, 371953, 392941

Offset: 1

Hartmut F. W. Hoft, Nov 03 2015

This sequence is a subsequence of A264102 and also of A014105, the second hexagonal numbers. Every number in this sequence is a triangular number.
The sequence A156592 of products of a Sophie Germain prime (A005384) and its associated safe prime (A005385) except for the first pair (2, 5) forms a subsequence of this sequence, the first column in the irregular triangular grid in the example.
The areas of the first two regions are (2^(m+1) - 1) * (2^(m+1) * p^2 * p + 1) / 2 and (2^(m+1) - 1) * (2^(m+1) * p + p + 1) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (2^(m+1) * p + 2).
For a proof of the formula for this sequence see the link.

			406 = 2*7*29 is in the sequence since m = 1 and 4 < 7 < 28 < 29. The first two regions in the symmetric representation of sigma(406) = 720 start with legs 1 and 7 and have areas 306 and 54, respectively. Note also that 406 is a triangular number and the middle two regions meet at the center of the Dyck path.
10 does not belong to this sequence since the symmetric representation of sigma(10) has two regions of width 1 that meet at the diagonal.
There is a natural arrangement of the numbers n = 2^m * p * (2^(m+1) * p + 1) as a sparse irregular triangular (p,m)-grid.
p\m| 0      1       2        3        4        5   ...
-------------------------------------------------------
3  | 21
5  | 55
7  |        406
11 | 253            3916
13 |        1378
17 |                9316
19 |
23 | 1081
29 | 1711           27028
31 |
37 |        11026           175528
41 | 3403
43 |        14878
47 |
53 | 5671                           1439056
59 |                                1783216
61 |                        476776
67 |        36046                            9195616
71 |                161596          2582128
73 |        42778                            10916128
...
The first number in the m = 6 column is 181880128 = 2^6*149*19073 in row p = 149 and the second is 228477376 = 2^6*167*21377 in row p = 167.

Hartmut F. W. Hoft, Diagram of symmetric representations of sigma(n), for n = 21, 55, 253, 406
Hartmut F. W. Hoft, Proof of 4 regions width 1 and 2 meet at center

Cf. A005384, A005385, A014105, A156592, A264102.

For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.

mStalk[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*(2^(m+1)*p+1)<=bound, If[PrimeQ[2^(m+1)*p+1], AppendTo[list, 2^m *p*(2^(m+1)*p+1)]]; p=NextPrime[p]]; list]
a264104[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*(2^(m+1)*NextPrime[2^(m+1)]+1)<=bound, list=Union[list, mStalk[m, bound]]; m++]; list]
a264104[400000] (* data *)

n = 2^m * p * (2^(m+1) * p + 1) where m >= 0, 2^(m+1) < p and p as well as 2^(m+1) * p + 1 are prime.

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