A014688 -id:A014688 - cp's OEIS frontend

A262206 Product of prime(n) consecutive numbers starting from n.

2, 24, 2520, 604800, 54486432000, 53353114214400, 35905578804006912000, 80018147048929689600000, 203939450748460387344384000000, 1441310123089178548721360295690240000000, 9218619547278385997621820451234775040000000

Offset: 1

Altug Alkan, Sep 15 2015

a(n) is always divisible by A039716(n).

			For n=1, a(1) = 1*2 = 2.
For n=2, a(2) = 2*3*4 = 24.
For n=3, a(3) = 3*4*5*6*7 = 2520.
For n=4, a(4) = 4*5*6*7*8*9*10 = 604800.

Cf. A075069: product of prime(n) consecutive numbers starting from prime(n).

Cf. A014688, A039716, A075068, A262204.

[Factorial(NthPrime(n)+n-1)/Factorial(n-1): n in [1..15]]; // Vincenzo Librandi, Sep 16 2015

A262206:=n->(ithprime(n)+n-1)! / (n-1)!: seq(A262206(n), n=1..15); # Wesley Ivan Hurt, Sep 15 2015

Table[(Prime[n] + n - 1)!/(n - 1)!, {n, 15}] (* Michael De Vlieger, Sep 15 2015 *)

a(n) = (prime(n)+n-1)! / (n-1)!;
vector(15,n,a(n))

a(n)=factorback([n..n+prime(n)-1]) \\ Charles R Greathouse IV, Sep 21 2015

a(n) = (prime(n) + n - 1)! / (n-1)!.

A267421 Primes of the form prime(n) + n + n^2.

Original entry on oeis.org

17, 41, 73, 113, 139, 163, 193, 223, 491, 859, 919, 1187, 1259, 1409, 1483, 1901, 1987, 2083, 2267, 2467, 2677, 3221, 4339, 4603, 5923, 6079, 7573, 8839, 9421, 9619, 10223, 11489, 11701, 12143, 12589, 13499, 13729, 14449, 15679, 16183, 16703, 17231, 17497, 19121

Offset: 1

Emre APARI, Jan 14 2016

			The ninth prime is 23, and 23 + 9 + 9^2 = 113, which is prime, so 113 is in the sequence.

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

Cf. A014688, A128938.

p:= 0: Res:= NULL:
for n from 1 to 1000 do
  p:= nextprime(p);
  if isprime(p+n+n^2) then Res:= Res, p+n+n^2 fi
od:
Res; # Robert Israel, Jan 08 2017

Select[Table[Prime[n] + n + n^2, {n, 100}], PrimeQ] (* Alonso del Arte, Feb 22 2016 *)

lista(nn) = {for (n=1, nn, if (isprime(p=prime(n)+n+n^2), print1(p, ", ")););} \\ Michel Marcus, Mar 13 2016

More terms from Michel Marcus, Mar 13 2016

A066197 Squarefree kernel of (nprime(n))(n+prime(n)).

Original entry on oeis.org

6, 30, 30, 154, 110, 1482, 714, 114, 138, 11310, 14322, 1554, 3198, 34314, 43710, 7314, 38114, 28914, 109478, 64610, 144102, 175538, 202354, 60342, 59170, 333502, 40170, 22470, 436218, 484770, 622046, 42706, 768570, 817598, 239890, 169422

Offset: 1

Reinhard Zumkeller, Dec 15 2001

nonn

			For n=20 we have: A = n = 20, B = A000040(20) = 71, C = A + B = 20 + 71 = 91 and A*B*C = 129220 with squarefree kernel a(20) = 64610 = 2*5*7*13*71.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ivars Peterson, The Amazing ABC Conjecture
M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Eric Weisstein's World of Mathematics, ABC Conjecture
Eric Weisstein's World of Mathematics, Squarefree
Wikipedia, abc conjecture

a066197 n = a007947 $ a033286 n * a014688 n
-- Reinhard Zumkeller, Jul 24 2013

sfk[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := sfk[n Prime[n] (n+Prime[n])];
Array[a, 40] (* Jean-François Alcover, Feb 04 2019 *)

a(n)=my(p=prime(n),f=vecsort(concat(concat(p, factor(n)[,1]), factor(n+p)[,1]),,8)~); prod(i=1,#f,f[i]) \\ Charles R Greathouse IV, Jul 23 2013

a(n) = A007947(A033286(n) * A014688(n)).

A104860 Prime next to (n + n-th prime).

Original entry on oeis.org

5, 7, 11, 13, 17, 23, 29, 29, 37, 41, 43, 53, 59, 59, 67, 71, 79, 83, 89, 97, 97, 103, 107, 127, 127, 131, 131, 137, 139, 149, 163, 167, 173, 179, 191, 191, 197, 211, 211, 223, 223, 227, 239, 239, 251, 251, 263, 277, 277, 281

Offset: 1

Zak Seidov, Apr 24 2005

nonn

Cf. A014688.

Table[NextPrime[n+Prime[n]],{n,60}] (* Harvey P. Dale, Aug 16 2018 *)

a(n) = nextprime(n + prime(n) + 1); \\ Michel Marcus, Oct 09 2013

a(n) = nextprime(A014688(n)).

A116023 The n-th prime plus n gives a semiprime (A001358).

Original entry on oeis.org

10, 12, 14, 15, 16, 19, 20, 21, 23, 25, 30, 31, 36, 37, 38, 39, 40, 44, 52, 54, 56, 57, 58, 60, 62, 67, 74, 75, 77, 80, 83, 84, 86, 88, 90, 99, 107, 111, 115, 120, 124, 136, 140, 145, 146, 154, 156, 160, 162, 164, 165, 166, 168, 174, 175, 182, 189, 192, 195, 196

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			p(30)+30=143=11*13.

Cf. A001358, A116012, A116024, A014688

A167136 a(n) = b(n)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where b(n) = noncomposite numbers = A008578(n).

Original entry on oeis.org

1, 3, 5, 8, 11, 16, 19, 24, 27, 32, 39, 42, 49, 54, 57, 62, 69, 76, 79, 86, 91, 94, 101, 106, 113, 122, 127, 130, 135, 138, 143, 158, 163, 170, 173, 184, 187, 194, 201, 206, 213, 220, 223, 234, 237, 242, 245, 258, 271, 276, 279, 284, 291, 294, 305, 312, 319, 326

Offset: 1

Jaroslav Krizek, Oct 28 2009

nonn

a(1) = 1, a(n) = A014688(n-1) = (n-1)-th prime + n - 1 for n >= 2. a(n) = A090178(n) - 1 = n-th noncomposite number + n - 1 for n >= 2.

			A008578(4) = 5, so a(4) = 8 = 5th highest positive integer not equal to 1, 3, or 5 (the values of a(k), 1 <= k <= 3).

a(1) = 1, a(n) = a(n-1) + A008578(n+1) - A008578(n) + 1 for n >= 2. a(1) = 1, a(2) = 3, a(n) = a(n-1) + A001223(n) + 1 for n >= 3. a(1) = 1, a(n) = n - 1 + A000040(n-1) = n - 1 + A008578(n) = n - 1 + A158611(n+1) for n >= 2.

A230846 1 + A075526(n).

Original entry on oeis.org

2, 2, 3, 3, 5, 3, 5, 3, 5, 7, 3, 7, 5, 3, 5, 7, 7, 3, 7, 5, 3, 7, 5, 7, 9, 5, 3, 5, 3, 5, 15, 5, 7, 3, 11, 3, 7, 7, 5, 7, 7, 3, 11, 3, 5, 3, 13, 13, 5, 3, 5, 7, 3, 11, 7, 7, 7, 3, 7, 5, 3, 11, 15, 5, 3, 5, 15, 7, 11, 3, 5, 7, 9, 7, 7, 5, 7, 9, 5, 9, 11, 3, 11, 3, 7, 5, 7, 9, 5, 3, 5, 13, 9, 5, 9, 5, 7

Offset: 1

Omar E. Pol, Nov 01 2013

nonn

Partial sums give A095116.

			On the first quadrant of the square grid consider a diagram in which the n-th horizontal bar contains A006093(n) cells and in which the number of cells in the vertical bars gives A000720 as shown below. a(n) is the sum of the length of the n-th horizontal boundary segment and the length of the n-th vertical boundary segment between the structure formed by the horizontal bars and the structure formed by the vertical bars, hence a(n) = A075526(n) + 1. The total length of the boundary segments from [0, 0] after n-th stage is A095116(n).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
30  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
22  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
18  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
16  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
12  |_ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
10  |_ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
6   |_ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
4   |_ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
1   |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
.

Cf. A000720, A006093, A014688, A054541, A075526, A095116.

Essentially the same as A076368.

A254867 Numbers n such that prime(n) + n and prime(n) + n^2 are prime.

Original entry on oeis.org

1, 2, 4, 22, 66, 96, 106, 144, 180, 222, 324, 378, 466, 492, 604, 742, 760, 778, 784, 960, 984, 990, 994, 1050, 1150, 1162, 1186, 1248, 1302, 1308, 1356, 1360, 1380, 1744, 1830, 1866, 1870, 1956, 2052, 2070, 2112, 2182, 2212, 2380, 2470, 2556, 2586, 2638, 2676, 2760, 2766

Offset: 1

Zak Seidov, Feb 09 2015

nonn

			a(4) = 22 = A064402(6): prime(22) = 79, 79 + {22, 22^2} = {101, 563} both prime.

Subsequence of A064402. Cf. A000040, A014688, A061067, A061068.

A254867:=n->`if`(isprime(ithprime(n)+n) and isprime(ithprime(n)+n^2), n, NULL): seq(A254867(n), n=1..10^4); # Wesley Ivan Hurt, Jan 16 2017

Select[Range[1000], PrimeQ[Prime[#] + #] && PrimeQ[Prime[#] + #^2] &] (* Alonso del Arte, Feb 09 2015 *)
Select[Range[3000],AllTrue[Prime[#]+{#,#^2},PrimeQ]&] (* Harvey P. Dale, Jan 17 2023 *)

A259408 a(1) = 1 thereafter a(n) = Sum_{m=1..n-1} prime(a(m)).

Original entry on oeis.org

1, 2, 5, 16, 69, 416, 3277, 33590, 430131, 6700328, 124069971, 2680915918, 66579269891, 1876496610172, 59387269231505, 2091422223924852, 81321166136299741, 3467614972592015460

Offset: 1

Anders Hellström, Jun 26 2015

Cf. A074271.

a = {1}; Do[AppendTo[a, Sum[Prime[a[[m]]], {m, n - 1}]], {n, 2, 15}];
a (* Michael De Vlieger, Aug 06 2015 *)

a(n) = if (n==1, 1, sum(k=1, n-1, prime(a(k)))); \\ Michel Marcus, Jun 26 2015

first(m)=my(v=vector(m)); v[1]=1; print1(1); for(i=2, m, v[i]=sum(k=1, i-1, prime(v[k])); print1(", ", v[i])); v; \\ Anders Hellström, Aug 01 2015

first(n)=my(v=vector(n,i,i)); for(i=3,n,v[i]=v[i-1]+prime(v[i-1])); v \\ Charles R Greathouse IV, Aug 06 2015

use bignum;
use Math::Prime::Util ':all';
print "1\n2\n";
my $a = 2;
while(1){
  $a += nth_prime($a);
  print "$a\n";
} # Charles R Greathouse IV, Aug 06 2015

from sympy import prime
from functools import lru_cache
@lru_cache()
def a(n): return n if n < 3 else a(n-1) + prime(a(n-1))
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Oct 07 2022

a(n) = A014688(a(n-1)) for n>2, a(1)=1, a(2)=2.

a(15) from Michael De Vlieger, Jul 01 2015

a(16)-a(18) from Charles R Greathouse IV, Aug 06 2015

A272245 Cubes of the form prime(n)+n.

Original entry on oeis.org

8, 27, 1000, 2744, 4096, 46656, 68921, 274625, 941192, 1295029, 1481544, 1906624, 14886936, 34328125, 35937000, 45882712, 50243409, 63521199, 64000000, 67917312, 76225024, 95443993, 112678587, 142236648, 143877824, 174676879, 198155287, 203297472, 216000000

Offset: 1

Emre APARI, Apr 23 2016

nonn

The cube root of the first 10 terms are: 2,3,10,14,16,36,41,65,98,109.

			prime(147) + 147 = 853 + 147 = 1000; which is 10^3.

Giovanni Resta, Table of n, a(n) for n = 1..500

Cf. A000578, A014688, A030078, A076147, A053149.

lista(nn) = {for (n=1, nn, if (ispower(p=n+prime(n), 3), print1(p, ", ")););} \\ Michel Marcus, Apr 23 2016

a(n) = A014688(A076147(n)). - Michel Marcus, Apr 23 2016

a(20)-a(29) from Giovanni Resta, Apr 23 2016

cp's OEIS Frontend

A262206 Product of prime(n) consecutive numbers starting from n.

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Magma

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A267421 Primes of the form prime(n) + n + n^2.

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Maple

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PARI

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A066197 Squarefree kernel of (n*prime(n))*(n+prime(n)).

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Haskell

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A104860 Prime next to (n + n-th prime).

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A116023 The n-th prime plus n gives a semiprime (A001358).

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A167136 a(n) = b(n)-th highest positive integer not equal to any a(k), 1 <= k <= n-1, where b(n) = noncomposite numbers = A008578(n).

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A230846 1 + A075526(n).

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A254867 Numbers n such that prime(n) + n and prime(n) + n^2 are prime.

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Examples

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Maple

Mathematica

A259408 a(1) = 1 thereafter a(n) = Sum_{m=1..n-1} prime(a(m)).

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A066197 Squarefree kernel of (nprime(n))(n+prime(n)).