A006254 -id:A006254 - cp's OEIS frontend

A098090 Numbers k such that 2k-3 is prime.

3, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 23, 25, 28, 31, 32, 35, 37, 38, 41, 43, 46, 50, 52, 53, 55, 56, 58, 65, 67, 70, 71, 76, 77, 80, 83, 85, 88, 91, 92, 97, 98, 100, 101, 107, 113, 115, 116, 118, 121, 122, 127, 130, 133, 136, 137, 140, 142, 143, 148, 155, 157, 158

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 14 2004

Supersequence of A063908.

Left edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A085090, A086801.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), this sequence (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([1..200], k-> IsPrime(2*k-3) ); # G. C. Greubel, May 21 2019

[ n: n in [1..200] | IsPrime(2*n-3) ]; // Vincenzo Librandi, Dec 26 2010

(Prime[Range[2,100]]+3)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[200],PrimeQ[2#-3]&] (* Harvey P. Dale, Mar 05 2022 *)

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

[n for n in (1..200) if is_prime(2*n-3) ] # G. C. Greubel, May 21 2019

Half of p + 3, where p is a prime greater than 2.

A122845(a(n), 3) = 3; a(n) = A113935(n+1)/2. - Reinhard Zumkeller, Sep 14 2006

A024702 a(n) = (prime(n)^2 - 1)/24.

Original entry on oeis.org

1, 2, 5, 7, 12, 15, 22, 35, 40, 57, 70, 77, 92, 117, 145, 155, 187, 210, 222, 260, 287, 330, 392, 425, 442, 477, 495, 532, 672, 715, 782, 805, 925, 950, 1027, 1107, 1162, 1247, 1335, 1365, 1520, 1552, 1617, 1650, 1855, 2072, 2147, 2185, 2262, 2380, 2420, 2625, 2752, 2882, 3015

Offset: 3

Clark Kimberling, Dec 11 1999

Note that p^2 - 1 is always divisible by 24 since p == 1 or 2 (mod 3), so p^2 == 1 (mod 3) and p == 1, 3, 5, or 7 (mod 8) so p^2 == 1 (mod 8). - Michael B. Porter, Sep 02 2016
For n > 3 and m > 1, a(n) = A000330(m)/(2*m + 1), where 2*m + 1 = prime(n). For example, for m = 8, 2*m + 1 = 17 = prime(7), A000330(8) = 204, 204/17 = 12 = a(7). - Richard R. Forberg, Aug 20 2013
For primes => 5, a(n) == 0 or 2 (mod 5). - Richard R. Forberg, Aug 28 2013
The only primes in this sequence are 2, 5 and 7 (checked up to n = 10^7). The set of prime factors, however, appears to include all primes. - Richard R. Forberg, Feb 28 2015
Subsequence of generalized pentagonal numbers (cf. A001318): a(n) = k_n*(3*k_n - 1)/2, for k_n in {1, -1, 2, -2, 3, -3, 4, 5, -5, -6, 7, -7, 8, 9, 10, -10, ...} = A024699(n-2)*((A000040(n) mod 6) - 3)/2, n >= 3. - Daniel Forgues, Aug 02 2016
The only primes in this sequence are indeed 2, 5 and 7. For a prime p >= 5, if both p + 1 and p - 1 contains a prime factor > 3, then (p^2 - 1)/24 = (p + 1)*(p - 1)/24 contains at least 2 prime factors, so at least one of p +  1 and p - 1 is 3-smooth. Let's call it s. Also, If (p^2 - 1)/24 is a prime, then A001222(p^2-1) = 5. Since A001222(p+1) and A001222(p-1) are both at least 2, A001222(s) <= 5 - 2 = 3. From these we can see the only possible cases are p = 7, 11 and 13. - Jianing Song, Dec 28 2018

			For n = 6, the 6th prime is 13, so a(6) = (13^2 - 1)/24 = 168/24 = 7.

Charles R Greathouse IV, Table of n, a(n) for n = 3..10000
Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).
Carlos Rivera, Puzzle 1060. Can you find more solutions?, The Prime Puzzles and Problems Connection. [Asks for squares in this sequence]

Subsequence of generalized pentagonal numbers A001318.

Cf. A075888.

A024702:=n->(ithprime(n)^2-1)/24: seq(A024702(n), n=3..70); # Wesley Ivan Hurt, Mar 01 2015

(Prime[Range[3,100]]^2-1)/24 (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *)

a(n)=prime(n)^2\24 \\ Charles R Greathouse IV, May 30 2013

is(n)=my(k);issquare(24*n+1,&k)&&isprime(k) \\ Charles R Greathouse IV, May 31 2013

a(n) = (A000040(n)^2 - 1)/24 = (A001248(n) - 1)/24. - Omar E. Pol, Dec 07 2011
a(n) = A005097(n-1)*A006254(n-1)/6. - Bruno Berselli, Dec 08 2011
a(n) = A084920(n)/24. - R. J. Mathar, Aug 23 2013
a(n) = A127922(n)/A000040(n) for n >= 3. - César Aguilera, Nov 01 2019

A105760 Nonnegative numbers k such that 2k+7 is prime.

Original entry on oeis.org

0, 2, 3, 5, 6, 8, 11, 12, 15, 17, 18, 20, 23, 26, 27, 30, 32, 33, 36, 38, 41, 45, 47, 48, 50, 51, 53, 60, 62, 65, 66, 71, 72, 75, 78, 80, 83, 86, 87, 92, 93, 95, 96, 102, 108, 110, 111, 113, 116, 117, 122, 125, 128, 131, 132, 135, 137, 138, 143, 150, 152, 153, 155, 162

Offset: 1

Parthasarathy Nambi, May 04 2005

			If n=0, then 2*0 + 7 = 7 (prime).
If n=15, then 2*15 + 7 = 37 (prime).
If n=27, then 2*27 + 7 = 61 (prime).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

Cf. A153053 (Numbers n such that 2n+7 is not a prime)
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), this seq(k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+7) ); # G. C. Greubel, May 21 2019

[n: n in [0..200]| IsPrime(2*n+7)]; // Vincenzo Librandi, Dec 21 2010

(Prime[Range[4,100]]-7)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 200], PrimeQ[2 # + 7] &] (* Vincenzo Librandi, May 20 2014 *)

is(n)=isprime(2*n+7) \\ Charles R Greathouse IV, Feb 16 2017

[n for n in (0..200) if is_prime(2*n+7) ] # G. C. Greubel, May 21 2019

More terms from Rick L. Shepherd, May 18 2005

A089253 Numbers n such that 2n - 5 is a prime.

Original entry on oeis.org

4, 5, 6, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 26, 29, 32, 33, 36, 38, 39, 42, 44, 47, 51, 53, 54, 56, 57, 59, 66, 68, 71, 72, 77, 78, 81, 84, 86, 89, 92, 93, 98, 99, 101, 102, 108, 114, 116, 117, 119, 122, 123, 128, 131, 134, 137, 138, 141, 143, 144, 149, 156, 158, 159, 161

Offset: 1

Giovanni Teofilatto, Dec 12 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

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).

Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), this sequence (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+5)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[2,200],PrimeQ[2#-5]&] (* Harvey P. Dale, Dec 29 2024 *)

a(n)=prime(n+1)\2+3 \\ Charles R Greathouse IV, Oct 30 2012

a(n) = A067076(n) + 4. - Giovanni Teofilatto, Dec 14 2003

Corrected by Ralf Stephan, Mar 03 2004

Further correction from Jeremy Gardiner, Sep 11 2004

A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 86, 89, 92, 94, 97, 101, 102, 103, 106, 107, 109, 113, 114, 116, 118, 122, 124, 127, 131, 134, 136, 137, 138

Offset: 1

Omar E. Pol, Apr 06 2014

nonn

All odd primes are in the sequence because the parts of the symmetric representation of sigma(prime(i)) are [m, m], where m = (1 + prime(i))/2, for i >= 2.
There are no odd composite numbers in this sequence.
First differs from A173708 at a(13).
Since sigma(p*q) >= 1 + p + q + p*q for odd p and q, the symmetric representation of sigma(p*q) has more parts than the two extremal ones of size (p*q + 1)/2; therefore, the above comments are true. - Hartmut F. W. Hoft, Jul 16 2014
From Hartmut F. W. Hoft, Sep 16 2015: (Start)
The following two statements are equivalent:
  (1) The symmetric representation of sigma(n) has two parts, and
  (2) n = q * p where q is in A174973, p is prime, and 2 * q < p.
For a proof see the link and also the link in A071561.
This characterization allows for much faster computation of numbers in the sequence - function a239929F[] in the Mathematica section - than computations based on Dyck paths. The function a239929Stalk[] gives rise to the associated irregular triangle whose columns are indexed by A174973 and whose rows are indexed by A065091, the odd primes. (End)
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
For the respective columns of the irregular triangle with fixed m: k = 2^m * p, m >= 1, 2^(m+1) < p and p prime:
(a) each number k is representable as the sum of 2^(m+1) but no fewer consecutive positive integers [since 2^(m+1) < p].
(b) each number k has 2^m as largest divisor <= sqrt(k) [since 2^m < sqrt(k) < p].
(c) each number k is of the form 2^m * p with p prime [by definition].
m = 1: (a) A100484 even semiprimes (except 4 and 6)
       (b) A161344 (except 4, 6 and 8)
       (c) A001747 (except 2, 4 and 6)
m = 2: (a) A270298
       (b) A161424 (except 16, 20, 24, 28 and 32)
       (c) A001749 (except 8, 12, 20 and 28)
m = 3: (a) A270301
       (b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)
       (c) sequence not in OEIS
b(i,j) = A174973(j) * {1,5) mod 6 * A174973(j), for all i,j >= 1; see A091999 for j=2. (End)

			From _Hartmut F. W. Hoft_, Sep 16 2015: (Start)
a(23) = 52 = 2^2 * 13 = q * p with q = 4 in A174973 and 8 < 13 = p.
a(59) = 136 = 2^3 * 17 = q * p with q = 8 in A174973 and 16 < 17 = p.
The first six columns of the irregular triangle through prime 37:
   1    2    4    6    8   12 ...
  -------------------------------
   3
   5   10
   7   14
  11   22   44
  13   26   52   78
  17   34   68  102  136
  19   38   76  114  152
  23   46   92  138  184
  29   58  116  174  232  348
  31   62  124  186  248  372
  37   74  148  222  296  444
  ...
(End)

Hartmut F. W. Hoft, Proof of Characterization Theorem

Column 2 of A240062.
Cf. A000203, A006254, A065091, A071561, A174973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A238443, A239660, A239663, A239665, A239931-A239934, A244050, A245062, A262626.
Cf. A000040, A001747, A001749, A091999, A100484, A161344, A161424, A162528, A270298, A270301. - Hartmut F. W. Hoft, Dec 06 2016

isA174973 := proc(n)
    option remember;
    local k,dvs;
    dvs := sort(convert(numtheory[divisors](n),list)) ;
    for k from 2 to nops(dvs) do
        if op(k,dvs) > 2*op(k-1,dvs) then
            return false;
        end if;
    end do:
    true ;
end proc:
A174973 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA174973(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
isA239929 := proc(n)
    local i,p,j,a73;
    for i from 1 do
        p := ithprime(i+1) ;
        if p > n then
            return false;
        end if;
        for j from 1 do
            a73 := A174973(j) ;
            if a73 > n then
                break;
            end if;
            if p > 2*a73 and n = p*a73 then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 200 do
    if isA239929(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 04 2018

(* sequence of numbers k for m <= k <= n having exactly two parts *)
(* Function a237270[] is defined in A237270 *)
a239929[m_, n_]:=Select[Range[m, n], Length[a237270[#]]==2&]
a239929[1, 260] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
(* test for membership in A174973 *)
a174973Q[n_]:=Module[{d=Divisors[n]}, Select[Rest[d] - 2 Most[d], #>0&]=={}]
a174973[n_]:=Select[Range[n], a174973Q]
(* compute numbers satisfying the condition *)
a239929Stalk[start_, bound_]:=Module[{p=NextPrime[2 start], list={}}, While[start p<=bound, AppendTo[list, start p]; p=NextPrime[p]]; list]
a239929F[n_]:=Sort[Flatten[Map[a239929Stalk[#, n]&, a174973[n]]]]
a239929F[138] (* data *)(* Hartmut F. W. Hoft, Sep 16 2015 *)

Entries b(i, j) in the irregular triangle with rows indexed by i>=1 and columns indexed by j>=1 (alternate indexing of the example):

b(i,j) = A000040(i+1) * A174973(j) where A000040(i+1) > 2 * A174973(j). - Hartmut F. W. Hoft, Dec 06 2016

Extended beyond a(56) by Michel Marcus, Apr 07 2014

A089192 Numbers n such that 2n - 7 is a prime.

Original entry on oeis.org

5, 6, 7, 9, 10, 12, 13, 15, 18, 19, 22, 24, 25, 27, 30, 33, 34, 37, 39, 40, 43, 45, 48, 52, 54, 55, 57, 58, 60, 67, 69, 72, 73, 78, 79, 82, 85, 87, 90, 93, 94, 99, 100, 102, 103, 109, 115, 117, 118, 120, 123, 124, 129, 132, 135, 138, 139, 142, 144, 145, 150, 157, 159, 160

Offset: 1

Giovanni Teofilatto, Dec 08 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

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).

Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), this sequence (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Select[Range[3,200],PrimeQ[2#-7]&] (* Harvey P. Dale, Aug 24 2014 *)

is(n)=isprime(2*n - 7) \\ Charles R Greathouse IV, Apr 28 2015

Corrected by Ralf Stephan, Mar 03 2004

Further correction from Jeremy Gardiner, Sep 11 2004

A153143 Nonnegative numbers k such that 2k + 19 is prime.

Original entry on oeis.org

0, 2, 5, 6, 9, 11, 12, 14, 17, 20, 21, 24, 26, 27, 30, 32, 35, 39, 41, 42, 44, 45, 47, 54, 56, 59, 60, 65, 66, 69, 72, 74, 77, 80, 81, 86, 87, 89, 90, 96, 102, 104, 105, 107, 110, 111, 116, 119, 122, 125, 126, 129, 131, 132, 137, 144, 146, 147, 149, 156, 159, 164, 165

Offset: 1

Vincenzo Librandi, Dec 19 2008

Or, (p-19)/2 for primes p >= 19.
a(n) = (A000040(n+7) - 19)/2.
a(n) = A005097(n+6) - 9.
a(n) = A067076(n+6) - 8.
a(n) = A089038(n+5) - 7.
a(n) = A105760(n+4) - 6.
a(n) = A101448(n+3) - 4.
a(n) = A089559(n+1) - 2.

			For k = 4, 2*k+19 = 27 is not prime, so 4 is not in the sequence;
for k = 17, 2*k+19 = 53 is prime, so 17 is in the sequence.

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

Cf. A000040 (prime numbers), A153144 (2n+19 is not prime).
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), this seq (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+19) ); # G. C. Greubel, May 22 2019

[ n: n in [0..165] | IsPrime(2*n+19) ];

(Prime[Range[8,100]]-19)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 170], PrimeQ[(2*#)+19]&] (* Vincenzo Librandi, Sep 24 2012 *)

is(n)=isprime(2*n+19) \\ Charles R Greathouse IV, Feb 17 2017

list(lim)=apply(p->p\2-9, primes([19,lim\1*2+19])) \\ Charles R Greathouse IV, Jan 03 2025

[n for n in (0..200) if is_prime(2*n+19) ] # G. C. Greubel, May 22 2019

a(n) ~ (n/2) log n. - Charles R Greathouse IV, Jan 03 2025

Edited, corrected and extended by Klaus Brockhaus, Dec 22 2008

Definition clarified by Zak Seidov, Jul 11 2014

A089559 Nonnegative numbers n such that 2*n + 15 is prime.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 23, 26, 28, 29, 32, 34, 37, 41, 43, 44, 46, 47, 49, 56, 58, 61, 62, 67, 68, 71, 74, 76, 79, 82, 83, 88, 89, 91, 92, 98, 104, 106, 107, 109, 112, 113, 118, 121, 124, 127, 128, 131, 133, 134, 139, 146, 148, 149, 151, 158, 161, 166

Offset: 1

Ray Chandler, Nov 29 2003

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

Cf. A086303.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), this seq (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+15) ); # G. C. Greubel, May 22 2019

[n: n in [0..200] |  IsPrime(2*n+15)]; // Vincenzo Librandi, Apr 28 2014

(Prime[Range[7,100]]-15)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 200], PrimeQ[2 # + 15] &] (* Vincenzo Librandi, Apr 28 2014 *)

is(n)=isprime(2*n+15) \\ Charles R Greathouse IV, Apr 28 2015

[n for n in (0..200) if is_prime(2*n+15) ] # G. C. Greubel, May 22 2019

a(n) = A086303(n)/2.

Definition clarified by Zak Seidov, Jul 11 2014

A101448 Nonnegative numbers k such that 2k + 11 is prime.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 10, 13, 15, 16, 18, 21, 24, 25, 28, 30, 31, 34, 36, 39, 43, 45, 46, 48, 49, 51, 58, 60, 63, 64, 69, 70, 73, 76, 78, 81, 84, 85, 90, 91, 93, 94, 100, 106, 108, 109, 111, 114, 115, 120, 123, 126, 129, 130, 133, 135, 136, 141, 148, 150, 151, 153, 160, 163

Offset: 1

Parthasarathy Nambi, Jan 24 2005

2 is the smallest single-digit prime and 11 is the smallest two-digit prime.

			If n=1, then 2*1 + 11 = 13 (prime).
If n=49, then 2*49 + 11 = 109 (prime).
If n=69, then 2*69 + 11 = 149 (prime).

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A101123, A101086.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), this seq (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+11) ); # G. C. Greubel, May 21 2019

[n: n in [0..200] | IsPrime(2*n+11)]; // Vincenzo Librandi, Nov 17 2010

select(k-> isprime(11+2*k), [$0..200])[];  # Alois P. Heinz, Jun 02 2022

Select[Range[0, 200], PrimeQ[2# + 11] &] (* Stefan Steinerberger, Feb 28 2006 *)

is(n)=isprime(2*n+11) \\ Charles R Greathouse IV, Apr 29 2015

[n for n in (0..200) if is_prime(2*n+11) ] # G. C. Greubel, May 21 2019

More terms from Stefan Steinerberger, Feb 28 2006

Definition clarified by Zak Seidov, Jul 11 2014

A240062 Square array read by antidiagonals in which T(n,k) is the n-th number j with the property that the symmetric representation of sigma(j) has k parts, with j >= 1, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 6, 7, 15, 21, 8, 10, 25, 27, 63, 12, 11, 35, 33, 81, 147, 16, 13, 45, 39, 99, 171, 357, 18, 14, 49, 51, 117, 189, 399, 903, 20, 17, 50, 55, 153, 207, 441, 987, 2499, 24, 19, 70, 57, 165, 243, 483, 1029, 2709, 6069, 28, 22, 77, 65, 195, 261, 513, 1113

Offset: 1

Omar E. Pol, Apr 06 2014

This is a permutation of the positive integers.
All odd primes are in column 2 (together with some even composite numbers) because the symmetric representation of sigma(prime(i)) is [m, m], where m = (1 + prime(i))/2, for i >= 2.
The union of all odd-indexed columns gives A071562, the positive integers that have middle divisors. The union of all even-indexed columns gives A071561, the positive integers without middle divisors. - Omar E. Pol, Oct 01 2018
Each column in the table of A357581 is a subsequence of the respective column in the table of this sequence; however, the first row in the table of A357581 is not a subsequence of the first row in the table of this sequence. - Hartmut F. W. Hoft, Oct 04 2022
Conjecture: T(n,k) is the n-th positive integer with k 2-dense sublists of divisors. - Omar E. Pol, Aug 25 2025

			Array begins:
   1,  3,  9, 21,  63, 147, 357,  903, 2499, 6069, ...
   2,  5, 15, 27,  81, 171, 399,  987, 2709, 6321, ...
   4,  7, 25, 33,  99, 189, 441, 1029, 2793, 6325, ...
   6, 10, 35, 39, 117, 207, 483, 1113, 2961, 6783, ...
   8, 11, 45, 51, 153, 243, 513, 1197, 3025, 6875, ...
  12, 13, 49, 55, 165, 261, 567, 1239, 3087, 6909, ...
  16, 14, 50, 57, 195, 275, 609, 1265, 3249, 7011, ...
  18, 17, 70, 65, 231, 279, 621, 1281, 3339, 7203, ...
  20, 19, 77, 69, 255, 297, 651, 1375, 3381, 7353, ...
  24, 22, 91, 75, 273, 333, 729, 1407, 3591, 7581, ...
  ...
[Lower right hand triangle of array completed by _Hartmut F. W. Hoft_, Oct 04 2022]

Index entries for sequences that are permutations of the natural numbers

For programs see A237271 and A237593.
Row 1 is A239663.
Column 1 is A174973.
Columns 2..7: A239929, A279102, A280107, A320066, A320511, A357775.
For more information see A239663 and A239665.
Cf. A000203, A006254, A065091, A067742, A071561, A071562, A196020, A236104, A235791, A237048, A237270, A239660, A239929, A239931-A239934, A245092, A262626, A319529, A319796, A319801, A319802.
Cf. A341969, A341970, A341971, A357581.

(* function a341969 and support functions are defined in A341969, A341970 and A341971 *)
partsSRS[n_] := Length[Select[SplitBy[a341969[n], #!=0&], #[[1]]!=0&]]
widthTable[n_, {r_, c_}] := Module[{k, list=Table[{}, c], parts}, For[k=1, k<=n, k++, parts=partsSRS[k]; If[parts<=c&&Length[list[[parts]]]=1, j--, vec[[PolygonalNumber[i+j-2]+j]]=arr[[i, j]]]]; vec]
a240062T[n_, r_] := TableForm[widthTable[n, {r, r}]]
a240062[6069, 10] (* data *)
a240062T[7581, 10] (* 10 X 10 array - Hartmut F. W. Hoft, Oct 04 2022 *)

a(n) > 128 from Michel Marcus, Apr 08 2014

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