A152241 -id:A152241 - 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

A166329 Products of squares of 2 successive primes.

Original entry on oeis.org

36, 225, 1225, 5929, 20449, 48841, 104329, 190969, 444889, 808201, 1315609, 2301289, 3108169, 4084441, 6205081, 9778129, 12952801, 16703569, 22629049, 26863489, 33258289, 42994249, 54567769, 74528689, 95981209, 108222409

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 11 2009

			2^2*3^2 = 36, 3^2*5^2 = 225, 5^2*7^2 = 1225, ..

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A006094, A152241.

[NthPrime(n)^2*NthPrime(n+1)^2: n in [1..30]]; // Vincenzo Librandi, May 10 2016

Array[Prime[ # ]*Prime[ # ]*Prime[ #+1]*Prime[ #+1]&,5! ]
Times@@@(Partition[Prime[Range[30]],2,1]^2) (* Harvey P. Dale, Apr 12 2018 *)

a(n) = prime(n)^2*prime(n+1)^2; \\ Michel Marcus, May 10 2016

a(n) = A006094(n)^2. - Michel Marcus, May 10 2016

Edited by N. J. A. Sloane, Oct 13 2009

A291339 Primes p such that p^3*q^3 + p^3 + q^3 is prime, where q is the next prime after p.

Original entry on oeis.org

2, 3, 7, 19, 37, 47, 83, 89, 107, 137, 181, 251, 257, 349, 379, 569, 631, 653, 677, 691, 797, 823, 839, 863, 883, 919, 1009, 1021, 1223, 1229, 1361, 1423, 1571, 1609, 1831, 1873, 1907, 1993, 2053, 2113, 2207, 2239, 2293, 2309, 2579, 2833, 3137, 3319, 3593, 3673

Offset: 1

K. D. Bajpai, Aug 22 2017

nonn

			a(2) = 3 is prime; 5 is the next prime: 3^3*5^3 + 3^3 + 5^3 = 27*125 + 27 + 125 = 3527 that is a prime.
a(3) = 7 is prime; 11 is the next prime: 7^3*11^3 + 7^3 + 11^3 = 343*1331 + 343 + 1331 = 458207 that is a prime.

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

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241.

[p: p in PrimesUpTo (5000) | IsPrime(p^3*q^3+p^3+q^3)];

select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^3+nextprime(p)^3)]), [seq(p, p=1..10^4)]);

Prime@Select[Range[1000], PrimeQ[Prime[#]^3*Prime[# + 1]^3 + Prime[#]^3 + Prime[# + 1]^3] &]
Select[Partition[Prime[Range[600]],2,1],PrimeQ[Times@@(#^3)+Total[#^3]]&][[;;,1]] (* Harvey P. Dale, Apr 28 2025 *)

is(n) = my(q=nextprime(n+1)); ispseudoprime(n^3*q^3+n^3+q^3)
forprime(p=1, 3700, if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Aug 22 2017

list(lim)=my(v=List(),p=2,p3=8,q3); forprime(q=3,nextprime(lim\1+1), q3=q^3; if(isprime(p3*q3+p3+q3), listput(v,p)); p=q; p3=q3); Vec(v) \\ Charles R Greathouse IV, Aug 23 2017

A291374 Primes p such that p^3*q^3 + p + q is prime, where q is next prime after p.

Original entry on oeis.org

11, 17, 41, 43, 47, 137, 313, 359, 389, 401, 491, 557, 577, 709, 757, 829, 863, 929, 937, 953, 1129, 1163, 1249, 1301, 1307, 1439, 1597, 1627, 1693, 1847, 2087, 2311, 2351, 2437, 2663, 2731, 2741, 3109, 3119, 3217, 3253, 4027, 4219, 4271, 4547, 4637, 5189, 5237

Offset: 1

K. D. Bajpai, Aug 23 2017

nonn

			a(1) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11 + 13 = 1331*2197 + 11 + 13 = 2924231 that is a prime.
a(2) = 17 is prime; 19 is the next prime: 17^3*19^3 + 17 + 19 = 4913*6859 + 17 + 19 = 33698303 that is a prime.

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

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339.

[p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p + q) where q is NextPrime(p)];

select(p -> andmap(isprime, [p,(p^3*nextprime(p)^3+p+nextprime(p))]), [seq(p,p=1..10^4)]);

Prime@Select[Range[1000], PrimeQ[Prime[#]^3* Prime[# + 1]^3 + Prime[#] + Prime[# + 1]] &]

forprime(p=1,5000, q=nextprime(p+1); if(ispseudoprime(p^3*q^3 + p + q), print1(p, ", ")));

list(lim)=my(v=List(),p=2,p3=8,q3); forprime(q=3,nextprime(lim\1+1), q3=q^3; if(isprime(p3*q3+p+q), listput(v,p)); p=q; p3=q3); Vec(v) \\ Charles R Greathouse IV, Aug 23 2017

A166502 The n-th power of the product prime(n)*prime(n+1) of 2 successive primes.

Original entry on oeis.org

6, 225, 42875, 35153041, 59797108943, 116507435287321, 366790143213462347, 1329999555322686599521, 26129584584668699724236347, 344823548950275944213556441001, 4520615782446712879799718786455203, 148534373731547764810930925932451123761

Offset: 1

Jonathan Vos Post, Oct 15 2009

			A[k,n] = n-th product of k-th power of 2 successive primes begins:
===============================================================================
...|.n=1|..n=2|....n=3|.....n=4|......n=5|......n=6|......n=7|......n=8|.in.OEIS
================================================================================
k=1|...6|...15|.....35|......77|......143|......221|......323|......437|A006094
k=2|..36|..225|...1225|....5929|....20449|....48841|...104329|...104329|A166329
k=3|.216|.3375|..42875|..456533|..2924207|.10793861|.33698267|.83453453|A152241
k=4|1296|50625|1500625|35153041|418161601|.........|.........|.........|.......
================================================================================

Harvey P. Dale, Table of n, a(n) for n = 1..166

Cf. A000040, A006094, A166329, A152241.

A166502 := proc(n) ithprime(n)*ithprime(n+1) ; %^n ; end: seq(A166502(n),n=1..15) ; # R. J. Mathar, Oct 16 2009

With[{nn=20},(Times@@#[[2]])^#[[1]]&/@Thread[{Range[nn-1],Partition[ Prime[ Range[ nn]],2,1]}]] (* Harvey P. Dale, Jan 12 2015 *)

a(n) = (prime(n)*prime(n+1))^n; \\ Michel Marcus, May 05 2019

a(n) = Product_{i=n..n+1} prime(i)^n = (A000040(n)*A000040(n+1))^n. [corrected by R. J. Mathar, Oct 16 2009]

a(n) = A006094(n)^n. - Michel Marcus, May 05 2019

A258155 Products of squares of three successive primes.

Original entry on oeis.org

900, 11025, 148225, 1002001, 5909761, 17631601, 55190041, 160604929, 427538329, 1106427169, 2211538729, 4255083361, 6865945321, 11473194769, 21599886961, 36384418009, 58145123689, 84202691329, 120590202121, 167655034849, 229116352921, 340557446329, 513428138521

Offset: 1

K. D. Bajpai, May 22 2015

			a(1) = 900 = 2^2 * 3^2 * 5^2;
a(2) = 11025 = 3^2 * 5^2 * 7^2.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A006094, A046301, A152241, A166329.

Table[Prime[n]^2 Prime[n + 1]^2 Prime[n + 2]^2, {n, 50}]

for( n= 1,100, k= prime(n)^2 * prime(n+1)^2 * prime(n+2)^2; print1(k, ", "))

a(n) = A046301(n)^2. - Michel Marcus, May 22 2015

a(n) ~ n^3 log^3 n. - Charles R Greathouse IV, May 22 2015

A291464 Primes p such that p^3*q^3 + p^2 + q^2 is prime, where q is next prime after p.

Original entry on oeis.org

2, 11, 13, 41, 97, 277, 389, 1093, 1229, 1409, 1429, 1627, 1823, 1931, 1979, 2437, 2521, 2549, 2657, 2689, 2719, 2729, 2731, 2969, 3019, 3413, 3539, 3593, 3613, 3623, 3697, 4003, 4027, 4289, 4327, 4583, 4751, 5051, 5323, 5503, 5657, 5783, 6143, 6221, 6299, 6329

Offset: 1

K. D. Bajpai, Aug 24 2017

nonn

			a(1) = 2 is prime; 3 is the next prime: 2^3*3^3 + 2^2 + 3^2 = 8*27 + 4 + 9 = 229 that is a prime.
a(2) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11^2 + 13^2 = 1331*2197 + 121 + 169 = 2924497 that is a prime.

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

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339, A291374.

[p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p^2 + q^2) where q is NextPrime(p)];

select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^2+nextprime(p)^2)]), [seq(p, p=1..10^4)]);

Select[Prime[Range[5000]], PrimeQ[#^3*NextPrime[#]^3 + #^2 + NextPrime[#]^2] &]
Select[Partition[Prime[Range[1000]],2,1],PrimeQ[#[[1]]^3 #[[2]]^3+#[[1]]^2+#[[2]]^2]&][[;;,1]] (* Harvey P. Dale, Sep 11 2023 *)

forprime(p=1, 5000, q=nextprime(p+1); p3=p^3; p2=p^2; q3=q^3; q2=q^2; if(ispseudoprime(p3*q3 + p2 + q2), print1(p, ", ")));

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A006094 Products of 2 successive primes.

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A166329 Products of squares of 2 successive primes.

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A291339 Primes p such that p^3*q^3 + p^3 + q^3 is prime, where q is the next prime after p.

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A291374 Primes p such that p^3*q^3 + p + q is prime, where q is next prime after p.

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A166502 The n-th power of the product prime(n)*prime(n+1) of 2 successive primes.

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A258155 Products of squares of three successive primes.

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