A030664 -id:A030664 - 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

A013638 a(n) = prevprime(n)*nextprime(n).

Original entry on oeis.org

10, 15, 21, 35, 55, 77, 77, 77, 91, 143, 187, 221, 221, 221, 247, 323, 391, 437, 437, 437, 551, 667, 667, 667, 667, 667, 713, 899, 1073, 1147, 1147, 1147, 1147, 1147, 1271, 1517, 1517, 1517, 1591, 1763, 1927

Offset: 3

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 3..10000

Cf. A151799, A151800, A030664.

a013638 n = a151799 n * a151800 n  -- Reinhard Zumkeller, May 22 2015

[ seq(prevprime(i)*nextprime(i),i=3..70) ];

a[n_] := NextPrime[n, -1] NextPrime[n];
Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Aug 02 2018 *)

a(n) = A151799(n)*A151800(n). - Reinhard Zumkeller, May 22 2015

A056139 a(n) = n^2 - primefloor(n)*primeceiling(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, -13, 4, 23, 0, 1, 0, -25, 4, 35, 0, 1, 0, -37, 4, 47, 0, -91, -42, 9, 62, 117, 0, 1, 0, -123, -58, 9, 78, 149, 0, -73, 4, 83, 0, 1, 0, -85, 4, 95, 0, -187, -90, 9, 110, 213, 0, -211, -102, 9, 122, 237, 0, 1, 0, -243, -118, 9, 138, 269, 0, -133, 4, 143, 0, 1, 0, -291, -142, 9, 162, 317, 0, -157, 4, 167, 0, -331, -162, 9

Offset: 2

Henry Bottomley, Jun 15 2000

a(n)= 0 iff n is prime.

			a(3)=3^2-3*3=0, a(4)=4^2-3*5=1

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A007917, A007918, A030664, A056140, A056141.

A030664(n) = if (n < 2, 1, precprime(n)*nextprime(n)); \\ From A030664
A056139(n) = (n^2 - A030664(n)); \\ Antti Karttunen, Mar 20 2018

a(n) = n^2 - A007917(n)*A007918(n) = A000290(n) - A030664(n).

More terms from Antti Karttunen, Mar 20 2018

A056141 a(n) = primefloor(n)primeceiling(n) - previousprime(n)nextprime(n).

Original entry on oeis.org

-1, 0, 4, 0, -6, 0, 0, 0, 30, 0, -18, 0, 0, 0, 42, 0, -30, 0, 0, 0, -22, 0, 0, 0, 0, 0, 128, 0, -112, 0, 0, 0, 0, 0, 98, 0, 0, 0, 90, 0, -78, 0, 0, 0, -70, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 248, 0, -232, 0, 0, 0, 0, 0, 158, 0, 0, 0, 150, 0, -280, 0, 0, 0, 0, 0, 182

Offset: 3

Henry Bottomley, Jun 15 2000

			a(3)=3*3-2*5=-1, a(4)=3*5-3*5=0

Cf. A056139, A056140.

Cf. A056221 (nonzero terms).

a(n) = if (isprime(n), n^2 - precprime(n-1)*nextprime(n+1), 0); \\ Michel Marcus, Mar 22 2020

a(n) = A007917(n)*A007918(n) - A007917(n-1)*A007918(n+1).
a(n) = A030664(n) - A013638(n).
a(n) = A056140(n) - A056139(n).
a(n) = A056140(n) if n is prime, a(n)=0 otherwise.

More terms from Michel Marcus, Mar 22 2020

A140135 Product of largest semiprime <= n and smallest semiprime >= n.

Original entry on oeis.org

16, 24, 36, 54, 54, 81, 100, 140, 140, 140, 196, 225, 315, 315, 315, 315, 315, 441, 484, 550, 550, 625, 676, 858, 858, 858, 858, 858, 858, 1089, 1156, 1225, 1330, 1330, 1444, 1521, 1794, 1794, 1794, 1794, 1794, 1794, 2116, 2254, 2254, 2401, 2499, 2601, 2805

Offset: 4

Jonathan Vos Post, May 09 2008

This is to A030664 as semiprimes A001358 are to primes A000040. Subset of A014613.

			a(10) = 100 because the largest semiprime <= 10 is 10, the smallest semiprime >= 10 is 10 and 10*10=100.

Cf. A001358, A014613, A030664.

isA001358 := proc(n) RETURN( numtheory[bigomega](n) = 2) ; end: A001358 := proc(n) option remember ; local a; if n = 1 then 4; else for a from A001358(n-1)+1 do if isA001358(a) then RETURN(a) ; fi ; od: fi ; end: prevsemiprime := proc(n) local a; for a from n to 4 by -1 do if isA001358(a) then RETURN(a) ; fi ; od: RETURN(-1) ; end: nextsemiprime := proc(n) local a; for a from n do if isA001358(a) then RETURN(a) ; fi ; od: RETURN(-1) ; end: A140135 := proc(n) prevsemiprime(n)*nextsemiprime(n) ; end: seq(A140135(n),n=4..80) ; # R. J. Mathar, May 11 2008

ls[n_]:=Module[{i=0},While[PrimeOmega[n+i]!=2,i++];n+i]; ss[n_]:=Module[ {i=0}, While[PrimeOmega[n-i]!=2,i++];n-i]; Table[ls[n]*ss[n],{n,4,60}] (* Harvey P. Dale, Oct 18 2013 *)

a(n) = MAX{j in A001358 and j <= n} * MIN{j in A001358 and j >= n}

More terms from R. J. Mathar, May 11 2008

cp's OEIS Frontend

A006094 Products of 2 successive primes.

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A013638 a(n) = prevprime(n)*nextprime(n).

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A056139 a(n) = n^2 - primefloor(n)*primeceiling(n).

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A056141 a(n) = primefloor(n)*primeceiling(n) - previousprime(n)*nextprime(n).

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A140135 Product of largest semiprime <= n and smallest semiprime >= n.

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Maple

Mathematica

Formula

Extensions

A056141 a(n) = primefloor(n)primeceiling(n) - previousprime(n)nextprime(n).