A219603 -id:A219603 - cp's OEIS frontend

A006094 Products of 2 successive primes.

6, 15, 35, 77, 143, 221, 323, 437, 667, 899, 1147, 1517, 1763, 2021, 2491, 3127, 3599, 4087, 4757, 5183, 5767, 6557, 7387, 8633, 9797, 10403, 11021, 11663, 12317, 14351, 16637, 17947, 19043, 20711, 22499, 23707, 25591, 27221, 28891, 30967, 32399, 34571, 36863

Offset: 1

N. J. A. Sloane

The Huntley reference would suggest prefixing the sequence with an initial 4 - Enoch Haga. [But that would conflict with the definition! - N. J. A. Sloane, Oct 13 2009]
Sequence appears to coincide with the sequence of numbers n such that the largest prime < sqrt(n) and the smallest prime > sqrt(n) divide n. - Benoit Cloitre, Apr 04 2002
This is true: p(n) < [ sqrt(a(n)) = sqrt(p(n)*p(n+1)) ] < p(n+1) by definition. - Jon Perry, Oct 02 2013
a(n+1) = smallest number such that gcd(a(n), a(n+1)) = prime(n+1). - Alexandre Wajnberg and Ray Chandler, Oct 14 2005
Also the area of rectangles whose side lengths are consecutive primes. E.g., the consecutive primes 7,11 produce a 7 X 11 unit rectangle which has area 77 square units. - Cino Hilliard, Jul 28 2006
a(n) = A001358(A172348(n)); A046301(n) = lcm(a(n), a(n+1)); A065091(n) = gcd(a(n), a(n+1)); A066116(n+2) = a(n+1)*a(n); A109805(n) = a(n+1) - a(n). - Reinhard Zumkeller, Mar 13 2011
See A209329 for the sum of the reciprocals. - M. F. Hasler, Jan 22 2013
A078898(a(n)) = 3. - Reinhard Zumkeller, Apr 06 2015

H. E. Huntley, The Divine Proportion, A Study in Mathematical Beauty. New York: Dover, 1970. See Chapter 13, Spira Mirabilis, especially Fig. 13-5, page 173.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
A. Bernoff and R. Pennington, Problems Drive 1984, Archimedeans Problems Drive, Eureka, 45 (1985), 22-25, 50. (Annotated scanned copy)
C. Cobeli and A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11 (2014) pp. 1489-1501, DOI: 10.1080/10236198.2014.940337. Also available as arXiv:1411.1334 [math.NT], 2014.

Subset of the squarefree semiprimes, A006881.
Cf. A090076, A090090.
Cf. A166329, A152241, A030664, A219603.
Cf. A046301, A046302, A046303, A046324, A046325, A046326, A046327.
Subsequence of A256617 and A097889.
Cf. A000040, A078898.

a006094 n = a006094_list !! (n-1)
a006094_list = zipWith (*) a000040_list a065091_list
-- Reinhard Zumkeller, Mar 13 2011

a006094_list = pr a000040_list
  where pr (n:m:tail) = n*m : pr (m:tail)
        pr _ = []
-- Jean-François Antoniotti, Jan 08 2020

[NthPrime(n)*NthPrime(n+1): n in [1..41]]; // Bruno Berselli, Feb 24 2011

a:= n-> (p-> p(n)*p(n+1))(ithprime):
seq(a(n), n=1..43);  # Alois P. Heinz, Jan 02 2021

Table[ Prime[n] Prime[n + 1], {n, 40}] (* Robert G. Wilson v, Jan 22 2004 *)
Times@@@Partition[Prime[Range[60]], 2, 1] (* Harvey P. Dale, Oct 15 2011 *)

ithprime(i)*ithprime(i+1) $ i = 1..41 // Zerinvary Lajos, Feb 26 2007

g(n) = for(x=1,n,print1(prime(x)*prime(x+1)",")) \\ Cino Hilliard, Jul 28 2006

is(n)=my(p=precprime(sqrtint(n))); p>1 && n%p==0 && isprime(n/p) && nextprime(p+1)==n/p \\ Charles R Greathouse IV, Jun 04 2014

from sympy import prime, primerange
def aupton(nn):
    alst, prevp = [], 2
    for p in primerange(3, prime(nn+1)+1): alst.append(prevp*p); prevp = p
    return alst
print(aupton(43)) # Michael S. Branicky, Jun 15 2021

from sympy import prime, nextprime
def A006094(n): return (p:=prime(n))*nextprime(p) # Chai Wah Wu, Oct 18 2024

A209329 = Sum_{n>=2} 1/a(n). - M. F. Hasler, Jan 22 2013

a(n) = A000040(n) * A000040(n+1). - Alois P. Heinz, Jan 02 2021

A031368 Odd-indexed primes: a(n) = prime(2n-1).

Original entry on oeis.org

2, 5, 11, 17, 23, 31, 41, 47, 59, 67, 73, 83, 97, 103, 109, 127, 137, 149, 157, 167, 179, 191, 197, 211, 227, 233, 241, 257, 269, 277, 283, 307, 313, 331, 347, 353, 367, 379, 389, 401, 419, 431, 439, 449, 461, 467, 487, 499, 509, 523, 547, 563

Offset: 1

Jeff Burch

Appeared as a puzzle in "Stickelers", a nationally distributed feature, by Terry Stickels, Sep 28 2006. - Franklin T. Adams-Watters, Sep 28 2006
Also every second prime starting with 2. - Cino Hilliard, Dec 02 2007
Central terms of the triangle in A005145. - Reinhard Zumkeller, Aug 05 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A031215 (even-indexed primes), A005408.

First differences are A155067.

a031368 = a000040 . ((subtract 1) . (* 2))
a031368_list = map a000040 [1, 3 ..]  -- Reinhard Zumkeller, Nov 25 2012

[ NthPrime(2*n-1): n in [1..1000] ]; // Vincenzo Librandi, Apr 11 2011

A031368 := n->ithprime(2*n-1): seq(A031368(n), n=1..100);

Table[ Prime[ 2n -1], {n, 52}] (* Robert G. Wilson v, Dec 01 2013 *)

a(n) = prime(2*n-1) \\ Jianing Song, Jun 03 2021

a(n) = A219603(n) / A000040(n). - Reinhard Zumkeller, Nov 25 2012

A087112 Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.

Original entry on oeis.org

4, 6, 9, 10, 15, 25, 14, 21, 35, 49, 22, 33, 55, 77, 121, 26, 39, 65, 91, 143, 169, 34, 51, 85, 119, 187, 221, 289, 38, 57, 95, 133, 209, 247, 323, 361, 46, 69, 115, 161, 253, 299, 391, 437, 529, 58, 87, 145, 203, 319, 377, 493, 551, 667, 841, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899, 961

Offset: 1

Ray Chandler, Aug 21 2003

Terms through row n, sorted, will provide terms for A077553 through row n*(n+1)/2.

			Triangle begins:
   4;
   6,   9;
  10,  15,  25;
  14,  21,  35,  49;
  22,  33,  55,  77, 121;
  26,  39,  65,  91, 143, 169;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A100484 (left edge), A001248 (right edge), A143215 (row sums), A219603 (central terms of odd-indexed rows); A000040, A065342.

Cf. A276086, A370121.

a087112 n k = a087112_tabl !! (n-1) !! (k-1)
a087112_row n = map (* last ps) ps where ps = take n a000040_list
a087112_tabl = map a087112_row [1..]
-- Reinhard Zumkeller, Nov 25 2012

T := (n, k) -> ithprime(n) * ithprime(k):
seq(print(seq(T(n, k), k = 1..n)), n = 1..11);  # Peter Luschny, Jun 25 2024

Table[ Prime[j]*Prime[k], {j, 11}, {k, j}] // Flatten (* Robert G. Wilson v, Feb 06 2017 *)

A087112(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (prime(1+c) * prime(1+(n-binomial(1+c, 2)))); }; \\ Antti Karttunen, Feb 29 2024

The n-th row consists of n terms, prime(n)*prime(i), i=1..n.
T(n, k) = A000040(n) * A000040(k).
For n >= 2, a(n) = A276086(A370121(n-1)). - Antti Karttunen, Feb 29 2024

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A006094 Products of 2 successive primes.

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A031368 Odd-indexed primes: a(n) = prime(2n-1).

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Haskell

Magma

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A087112 Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula