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 1-5 of 5 results.

A144482 Semiprimes that are a product of Mersenne primes.

Original entry on oeis.org

9, 21, 49, 93, 217, 381, 889, 961, 3937, 16129, 24573, 57337, 253921, 393213, 917497, 1040257, 1572861, 3670009, 4063201, 16252897, 16646017, 66584449, 67092481, 1073602561, 4294434817, 6442450941, 15032385529, 17179607041
Offset: 1

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Author

G. L. Honaker, Jr., Oct 12 2008

Keywords

Comments

As the product of any two primes is semiprime by definition, this is also the list of composite numbers n=x*y where both x and y are Mersenne primes. - Christian N. K. Anderson, Mar 25 2013

Links

Crossrefs

Cf. A000668, A001358, A144856, A333788.
Subsequence of A335882.

Programs

  • Mathematica
    Take[Times@@@Tuples[2^# -1&/@MersennePrimeExponent[Range[10]],2]//Union,30] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 30 2020 *)
  • PARI
    isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA144482(n) = ((2==bigomega(n))&&isA000668(vecmin(factor(n)[,1]))&&isA000668(vecmax(factor(n)[,1]))); \\ Antti Karttunen, Jun 28 2020

A279389 3 times Mersenne primes A000668.

Original entry on oeis.org

9, 21, 93, 381, 24573, 393213, 1572861, 6442450941, 6917529027641081853, 1856910058928070412348686333, 486777830487640090174734030864381, 510423550381407695195061911147652317181
Offset: 1

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Author

Omar E. Pol, Dec 20 2016

Keywords

Comments

Also sum of n-th Mersenne prime and the radical of n-th even perfect number.
The binary representation of a(n) has only two zeros, starting with "10" and ending with "01". The sequence begins: 1001, 10101, 1011101, 101111101, 101111111111101,...

Crossrefs

Cf. A000010, A000203, A000396, A000668, A051027, A147757.
Subsequence of A001748, and of A147758, and of A174055, and possibly of other sequences, see below:
Cf. A060482, A068156, A072087, A136252, A144482, A144856, A249780, A269633.

Formula

a(n) = 3*A000668(n) = A000668(n) + A139257(n).
a(n) = phi(M(n)) + sigma(sigma(M(n))) = A000010(A000668(n)) + A000203(A000203(A000668(n))) = A000010(A000668(n)) + A051027(A000668(n)).

A165223 Products of 2 successive Mersenne primes.

Original entry on oeis.org

21, 217, 3937, 1040257, 1073602561, 68718821377, 1125897758834689, 4951760154835678088235319297, 1427247692705959880439315947500961989719490561
Offset: 1

Views

Author

Boris Hostnik (megpplus(AT)siol.net), Sep 08 2009

Keywords

Comments

Subsequence of A144856 and of A046528. [From R. J. Mathar, Sep 17 2009]

Formula

a(n)=A000668(n)*A000668(n+1). [From R. J. Mathar, Sep 17 2009]

Extensions

Extended by R. J. Mathar, Sep 17 2009
Offset corrected by N. J. A. Sloane, Sep 20 2009

A344780 Semiprimes that are product of two distinct Honaker primes.

Original entry on oeis.org

34453, 59867, 120191, 136109, 137419, 142921, 170431, 178291, 187723, 205801, 250603, 253223, 273257, 275887, 280471, 286933, 290951, 297763, 319771, 339421, 342163, 348853, 354617, 356189, 357499, 357943, 367193, 376879, 401777, 410947, 413173, 422999, 449723
Offset: 1

Views

Author

K. D. Bajpai, May 28 2021

Keywords

Comments

Subsequence of A006881.
a(1) = 34453 is the only number <= 5*10^6 that is a triangular number.

Examples

			34453 = 131*263 which are distinct Honaker primes.
120191 = 263*457 which are distinct Honaker primes.

Crossrefs

Cf. A006881, A033548, A144482, A144856, A333788.

Programs

  • Maple
    isA006881 := proc(n)
    if numtheory[bigomega](n) =2 and A001221(n) = 2 then
        true ;
    else
        false ;
    end if;
end proc:
isA344780 := proc(n)
    if isA006881(n) then
        for p in ifactors(n)[2] do
            if not isA033548(op(1,p)) then
                return false;
            end if;
        end do:
        true ;
    else
        false;
    end if;
end proc:
for n from 1  do
    if isA344780(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jul 07 2021
  • Mathematica
    fHQ[n_] := Plus @@ IntegerDigits@n == Plus @@ IntegerDigits@PrimePi@n;
lst = {}; Do[If[Plus @@ Last /@ FactorInteger[n] == 2, a = Length[First /@ FactorInteger[n]]; If[a == 2, b = First /@ FactorInteger[n]; c = b[[1]]; d = b[[2]]; If[fHQ[c] && fHQ[d], AppendTo[lst, {n,c,d}]]]], {n, 2000000}]; lst

A346138 Sphenic numbers that are the product of Mersenne primes.

Original entry on oeis.org

651, 2667, 11811, 27559, 172011, 761763, 1777447, 2752491, 3120771, 7281799, 11010027, 12189603, 28442407, 32247967, 48758691, 49938051, 113770279, 116522119, 199753347, 466091143, 516026527, 2064117919, 3220807683, 7515217927, 12883304451, 30061043719, 33281679391, 45097156587, 133127479327
Offset: 1

Views

Author

Timothy L. Tiffin, Jul 06 2021

Keywords

Comments

The largest known sphenic number is a term of this sequence.
a(n) == 3 (mod 4) for n >= 1.

Examples

			a(1) = 651, since 651 = 3*7*31.
a(2) = 2667, since 2667 = 3*7*127.
a(3) = 11811, since 11811 = 3*31*127.

Links

Crossrefs

Cf. A007304, A046528, A144856.

Programs

  • Mathematica
    M = 2^MersennePrimeExponent[Range[12]] - 1; Take[ Union[ Times @@@ Subsets[ M, {3}]], 50]
Showing 1-5 of 5 results.

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