cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A006094 Products of 2 successive primes.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Haskell
    a006094 n = a006094_list !! (n-1)
a006094_list = zipWith (*) a000040_list a065091_list
-- Reinhard Zumkeller, Mar 13 2011
  • Haskell
    a006094_list = pr a000040_list
  where pr (n:m:tail) = n*m : pr (m:tail)
        pr _ = []
-- Jean-François Antoniotti, Jan 08 2020
  • Magma
    [NthPrime(n)*NthPrime(n+1): n in [1..41]]; // Bruno Berselli, Feb 24 2011
  • Maple
    a:= n-> (p-> p(n)*p(n+1))(ithprime):
seq(a(n), n=1..43);  # Alois P. Heinz, Jan 02 2021
  • Mathematica
    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 *)
  • MuPAD
    ithprime(i)*ithprime(i+1) $ i = 1..41 // Zerinvary Lajos, Feb 26 2007
  • PARI
    g(n) = for(x=1,n,print1(prime(x)*prime(x+1)",")) \\ Cino Hilliard, Jul 28 2006
  • PARI
    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
  • Python
    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
  • Python
    from sympy import prime, nextprime
def A006094(n): return (p:=prime(n))*nextprime(p) # Chai Wah Wu, Oct 18 2024

Formula

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

A176506 Difference between the prime indices of the two factors of the n-th semiprime.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 2, 4, 0, 5, 3, 6, 1, 7, 4, 8, 0, 5, 2, 6, 9, 10, 3, 7, 11, 1, 12, 4, 13, 8, 2, 9, 14, 5, 15, 10, 6, 16, 3, 0, 17, 11, 12, 4, 18, 13, 19, 1, 7, 20, 8, 21, 14, 5, 22, 0, 15, 23, 16, 9, 2, 24, 17, 25, 6, 10, 26, 3, 18, 27, 11, 7, 28, 19, 1, 29, 12, 20, 2, 21, 4, 30, 8, 31, 13, 22
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 19 2010

Keywords

Comments

Are there no adjacent equal terms? I have verified this up to n = 10^6. - Gus Wiseman, Dec 04 2020

Examples

			From _Gus Wiseman_, Dec 04 2020: (Start)
The sequence of semiprimes together with the corresponding differences begins:
   4: 1 - 1 = 0
   6: 2 - 1 = 1
   9: 2 - 2 = 0
  10: 3 - 1 = 2
  14: 4 - 1 = 3
  15: 3 - 2 = 1
  21: 4 - 2 = 2
  22: 5 - 1 = 4
  25: 3 - 3 = 0
  26: 6 - 1 = 5
  33: 5 - 2 = 3
(End)

Links

Crossrefs

Cf. A109313.
Cf. A001358, A049084, A084126, A084127.
A087794 is product of the same indices.
A176504 is the sum of the same indices.
A115392 lists positions of first appearances.
A128301 lists positions of 0's.
A172348 lists positions of 1's.
A338898 has this sequence as row differences.
A338900 is the squarefree case.
A338912/A338913 give the two prime indices of semiprimes.
A006881 lists squarefree semiprimes.
A024697 is the sum of semiprimes of weight n.
A056239 gives sum of prime indices (Heinz weight).
A087112 groups semiprimes by greater factor.
A270650/A270652/A338899 give the prime indices of squarefree semiprimes.
A338904 groups semiprimes by weight.
A338907/A338906 list semiprimes of odd/even weight.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A062198, A112141, A112798, A325699, A320655, A339005.

Programs

  • Maple
    isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A176506 := proc(n) numtheory[pi](A084127(n)) - numtheory[pi](A084126(n)) ; end proc: seq(A176506(n),n=1..120) ; # R. J. Mathar, Apr 22 2010
# Alternative:
N:= 500: # to use the first N semiprimes
Primes:= select(isprime, [2,seq(i,i=3..N/2,2)]):
SP:= NULL:
for i from 1 to nops(Primes) do
  for j from 1 to i do
    sp:= Primes[i]*Primes[j];
    if sp > N then break fi;
    SP:= SP, [sp, i-j]
od od:
SP:= sort([SP],(s,t) -> s[1] t[2], SP); # Robert Israel, Jan 17 2019
  • Mathematica
    M = 500; (* to use the first M semiprimes *)
primes = Select[Join[{2}, Range[3, M/2, 2]], PrimeQ];
SP = {};
For[i = 1, i <= Length[primes], i++,
  For[j = 1, j <= i, j++,
    sp = primes[[i]] primes[[j]];
    If[sp > M, Break []];
    AppendTo[SP, {sp, i - j}]
]];
SortBy[SP, First][[All, 2]] (* Jean-François Alcover, Jul 18 2020, after Robert Israel *)
Table[If[!SquareFreeQ[n],0,-Subtract@@PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)
  • PARI
    lista(nn) = {my(vsp = select(x->(bigomega(x)==2), [1..nn])); vector(#vsp, k, my(f=factor(vsp[k])[,1]); primepi(vecmax(f)) - primepi(vecmin(f)));} \\ Michel Marcus, Jul 18 2020

Formula

a(n) = A049084(A084127(n)) - A049084(A084126(n)). [corrected by R. J. Mathar, Apr 22 2010]
a(n) = A338913(n) - A338912(n). - Gus Wiseman, Dec 04 2020

Extensions

a(51) and a(69) corrected by R. J. Mathar, Apr 22 2010
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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