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 1-10 of 12 results. Next

A366898 Number of divisors of n*2^n - 1, the Woodall (or Riesel) numbers.

Original entry on oeis.org

1, 2, 2, 6, 4, 2, 4, 4, 4, 4, 6, 4, 12, 10, 8, 48, 8, 4, 8, 4, 16, 16, 8, 8, 4, 8, 8, 16, 24, 2, 8, 4, 8, 32, 8, 32, 4, 8, 4, 24, 16, 8, 32, 8, 16, 24, 40, 16, 16, 8, 8, 16, 24, 8, 16, 8, 16, 6, 32, 8, 8, 16, 8, 512, 48, 16, 12, 48, 16, 8, 8, 4, 24, 16, 2, 256
Offset: 1

Views

Author

Tyler Busby, Oct 26 2023

Keywords

Links

Crossrefs

Cf. A000005, A003261, A002234, A242273, A063515, A056821, A367006, A366899.

Programs

  • Mathematica
    Table[DivisorSigma[0, n*2^n - 1], {n, 1, 100}] (* Amiram Eldar, Dec 11 2023 *)
  • PARI
    a(n) = numdiv(n*2^n - 1); \\ Amiram Eldar, Dec 11 2023

Formula

a(n) = sigma0(n*2^n - 1) = A000005(A003261(n)).

A242175 Numbers k such that k*2^k + 1 is a semiprime.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 11, 16, 21, 33, 35, 101, 105, 131, 158, 165, 191, 234, 251, 435, 453, 459, 561, 579, 604, 671, 744, 753, 933, 963, 1041, 1146, 1168, 1254, 1794
Offset: 1

Views

Author

Vincenzo Librandi, May 07 2014

Keywords

Comments

The semiprimes of this form are 9, 25, 65, 161, 2049, 4609, 22529, ... (A242116).
a(35) >= 1528. Below 2000, 1794 and 1961 are in the sequence. Unknown factorization for 1528, 1576, 1908. - Hugo Pfoertner, Jul 29 2019
The k*2^k + 1 corresponding to 1528 and 1576 each have at least three prime factors. - Tyler Busby, Mar 16 2025

Links

Crossrefs

Cf. A001358, A002064, A005849, A242116, A242273.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..230] | IsSemiprime(s) where s is n*2^n+1]; // Bruno Berselli, May 08 2014
  • Mathematica
    Select[Range[165], Plus@@Last/@FactorInteger[# * 2^# + 1]==2&]

Formula

A002064(a(n)) = A242116(n). - Amiram Eldar, Nov 27 2019

Extensions

a(17) from Bruno Berselli, May 08 2014
a(18)-a(30) from Luke March, Aug 13 2015
a(31)-a(34) from Hugo Pfoertner, Jul 29 2019
Wrong term 941 removed by Amiram Eldar, Nov 27 2019
a(35) from Tyler Busby, Mar 16 2025

A242115 Woodall semiprimes: Semiprimes of the form n*2^n - 1.

Original entry on oeis.org

159, 895, 2047, 4607, 10239, 49151, 4718591, 20971519, 838860799, 137438953471, 5085241278463, 21440476741631, 340010386766614455386111, 96714065569170333976494079, 3288278229351791355200798719, 111414603535684224740921180159, 15370263527767281493147526365183
Offset: 1

Views

Author

K. D. Bajpai, May 04 2014

Keywords

Comments

The n-th Woodall number is Wn = n*2^n - 1.
If Wn is semiprime, it is in the sequence.

Examples

			a(1) = 159 = (5*2^5 - 1) is 5th Woodall number and 159 = 3*53 which is semiprime.
a(2) = 895 = (7*2^7 - 1) is 7th Woodall number and 895 = 5*179 which is semiprime.

Links

Crossrefs

Cf. A001358, A002234, A003261, A050918, A242273.

Programs

  • Maple
    with(numtheory): A242115:= proc(); if bigomega(x*2^x-1)=2 then RETURN (x*2^x-1); fi; end: seq(A242115 (),x=1..200);
  • Mathematica
    Select[Table[n*2^n-1,{n,100}],PrimeOmega[#]==2&] (* Harvey P. Dale, Jan 03 2019 *)
  • PARI
    for(n=1, 1000, if(bigomega(n*2^n-1)==2, print1(n*2^n-1, ", "))) \\ Colin Barker, May 07 2014

Formula

a(n) = A003261(A242273(n)). - Amiram Eldar, Nov 27 2019

A242274 Numbers k such that k*3^k - 1 is semiprime.

Original entry on oeis.org

4, 5, 8, 12, 20, 24, 25, 28, 32, 38, 42, 44, 60, 62, 66, 70, 72, 80, 122, 125, 148, 228, 244, 270, 389, 390, 432, 464, 470, 488, 549, 560, 804, 862
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are 323, 1214, 52487, 6377291, 69735688019, 6778308875543, 21182215236074, 640550188738907, 59296646043258911, ...
804 is a term of this sequence. - Luke March, Aug 22 2015
The smallest unresolved value of k is now 862. - Sean A. Irvine, Jun 20 2022
The smallest unresolved value of k is now 866. - Tyler Busby, Oct 06 2023
From Jon E. Schoenfield, Oct 06 2023: (Start)
After the possible term 866, the only remaining 3-digit terms are 912 and 984, unless 920 is a term.
If k is an odd term, then k*3^k - 1 is even, so (k*3^k - 1)/2 is a prime. The next odd terms after 549 are 1125 and 12889. Odd terms are in A366323. (End)
26925 is a term. - Michael S. Branicky, Oct 08 2024

Crossrefs

Cf. similar sequence listed in A242273.
Cf. A006553, A060352, A366323.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..241] | IsSemiprime(s) where s is n*3^n-1];
  • Mathematica
    Select[Range[241], PrimeOmega[# 3^# - 1]==2&]
  • PARI
    isok(n)=bigomega(n*3^n-1)==2 /* Anders Hellström, Aug 18 2015 */

Extensions

a(21)-a(23) from Carl Schildkraut, Aug 18 2015
a(24)-a(32) from Luke March, Aug 22 2015
a(32) = 804 removed by Sean A. Irvine, Apr 25 2022
a(32)-a(33) from Sean A. Irvine, Jun 20 2022
a(34) from Tyler Busby, Oct 06 2023

A242335 Numbers k such that k*4^k-1 is semiprime.

Original entry on oeis.org

10, 12, 18, 24, 27, 44, 47, 65, 71, 82, 84, 131, 134, 138, 143, 155, 164, 168, 197, 212, 227, 243, 248, 293, 302, 384, 401
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 10485759, 201326591, 1236950581247, 6755399441055743, 486388759756013567, 13617340432139183023890366463, ...

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A060416, A086661.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..100] | IsSemiprime(s) where s is n*4^n-1];
  • Mathematica
    Select[Range[100], PrimeOmega[# 4^# - 1]==2&]
  • PARI
    isok(n)=bigomega(n*4^n-1)==2 \\ Anders Hellström, Aug 18 2015

Extensions

a(12)-a(16) from Carl Schildkraut, Aug 18 2015
a(17)-a(27) from Charles R Greathouse IV, Aug 18 2015

A242336 Numbers k such that k*5^k-1 is semiprime.

Original entry on oeis.org

1, 2, 6, 12, 15, 19, 20, 26, 50, 55, 66, 68, 96, 99, 150, 166, 228, 459
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 4, 49, 93749, 2929687499, 457763671874, 362396240234374, 1907348632812499, 38743019104003906249, 4440892098500626161694526672363281249, 15265566588595902430824935436248779296874, ...
a(19) >= 534. - Daniel Suteu, Aug 05 2019

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A059676, A064751.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [1..400] | IsSemiprime(s) where s is n*5^n-1];
  • Maple
    select(t -> (numtheory:-bigomega(t*5^t-1)=2), [$1..400]); # Robert Israel, Aug 18 2015
  • Mathematica
    Select[Range[400], PrimeOmega[# 5^# - 1]==2&]

Extensions

1 prepended by Carl Schildkraut, Aug 18 2015
a(13)-a(17) from Carl Schildkraut, Aug 18 2015
a(18) from Daniel Suteu, Aug 05 2019

A242337 Numbers k such that k*6^k - 1 is semiprime.

Original entry on oeis.org

4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 18, 22, 23, 25, 32, 35, 38, 45, 51, 54, 57, 68, 72, 82, 97, 110, 138, 155, 234, 254
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are 5183, 38879, 279935, 1959551, 90699263, 604661759, 3990767615, 26121388031, 169789022207, 1097098297343, ...
a(32) >= 423. - Tyler Busby, Mar 19 2023

Links

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A059675, A064752.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..352] | IsSemiprime(s) where s is n*6^n-1];
  • Mathematica
    Select[Range[352], PrimeOmega[# 6^# - 1]==2&]

Extensions

a(26)-a(31) from Carl Schildkraut, Aug 17 2015

A242338 Numbers k such that k*7^k-1 is semiprime.

Original entry on oeis.org

1, 5, 12, 21, 42, 50, 60, 242, 272
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 6, 84034, 166095446411, 11729463145748964146, 13102886255950779594655873516522994057, ...
From Robert Israel, Aug 19 2014: (Start)
If k is odd, k is in the sequence iff (k*7^k-1)/2 is prime.
If k == 1 (mod 3), k is in the sequence iff (k*7^k-1)/3 is prime.
a(10) >= 506. 506*7^506 - 1 is a 431-digit composite which apparently has not been factored.
(End)

Links

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A001358, A064753, A242200.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..80] | IsSemiprime(s) where s is n*7^n-1];
  • Maple
    issemiprime:= proc(n) local F,t;
    F:= ifactors(n,easy)[2];
    t:= add(f[2],f=F);
    if t = 1 then
       if type(F[1][1],integer) then return false fi
    elif t = 2 then
       return not hastype(F,name)
    else # t > 2
       return false
    fi;
    F:= ifactors(n)[2];
    return evalb(add(f[2],f=F)=2);
end proc:
select(n -> `if`(n::odd, isprime((n*7^n-1)/2),
              issemiprime(n*7^n-1)), [$1..100]); # Robert Israel, Aug 19 2014
  • Mathematica
    Select[Range[80], PrimeOmega[# 7^# - 1]==2&]
  • PARI
    for(n=1,100,if(bigomega(n*7^n-1)==2,print1(n,", "))) \\ Derek Orr, Aug 20 2014

Extensions

a(1) = 1 prepended and comment amended by Harvey P. Dale, Aug 12 2014
a(8) and a(9) from Robert Israel, Aug 20 2014

A242339 Numbers k such that k*8^k-1 is semiprime.

Original entry on oeis.org

3, 6, 9, 13, 21, 24, 32, 45, 61, 62, 73, 132, 150, 174, 187, 217, 266, 331, 534
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 1535, 1572863, 1207959551, 7146825580543, 193690812773950291967, 113336795588871485128703, 2535301200456458802993406410751, 1960026433464605549549037738806984897986559, ...

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A064754, A242201.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..80] | IsSemiprime(s) where s is n*8^n-1];
  • Mathematica
    Select[Range[80], PrimeOmega[# 8^# - 1]==2&]
  • PARI
    isok(n)=bigomega(n*8^n-1)==2 /* Anders Hellström, Aug 18 2015 */

Extensions

a(12)-a(19) from Luke March, Aug 14 2015

A242340 Numbers k such that k*9^k-1 is semiprime.

Original entry on oeis.org

2, 3, 6, 8, 11, 18, 28, 38, 48, 168, 254, 274, 310, 430
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 161, 2186, 3188645, 344373767, 345191655698, 2701703435345984177, 14653733724766095041978322587, ...
a(15) >= 512. - Daniel Suteu, Aug 05 2019

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A064755, A242202.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..70] | IsSemiprime(s) where s is n*9^n-1];
  • Mathematica
    Select[Range[70], PrimeOmega[# 9^# - 1]==2&]

Extensions

a(10)-a(14) from Daniel Suteu, Aug 05 2019
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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