A003261 -id:A003261 - cp's OEIS frontend

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A098713 a(n) = (2n+1)*2^(2n+1) - 1.

1, 23, 159, 895, 4607, 22527, 106495, 491519, 2228223, 9961471, 44040191, 192937983, 838860799, 3623878655, 15569256447, 66571993087, 283467841535, 1202590842879, 5085241278463, 21440476741631, 90159953477631, 378231999954943, 1583296743997439, 6614661952700415, 27584547717644287, 114841790497947647

Offset: 0

Parthasarathy Nambi, Sep 28 2004

nonn

Cf. A003261.

[ (2*n+1)*2^(2*n+1) - 1: n in [0..30] ]; // Vincenzo Librandi, Feb 04 2011

A098713 := proc(n) (2*n+1)*2^(2*n+1)-1 ; end proc: # R. J. Mathar, Feb 04 2011

More terms from Sam Alexander, Jan 06 2005

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

K. D. Bajpai, May 04 2014

nonn

The n-th Woodall number is Wn = n*2^n - 1.

If Wn is semiprime, it is in the sequence.

			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.

Amiram Eldar, Table of n, a(n) for n = 1..47 (terms 1..26 from K. D. Bajpai)

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

with(numtheory): A242115:= proc(); if bigomega(x*2^x-1)=2 then RETURN (x*2^x-1); fi; end: seq(A242115 (),x=1..200);

Select[Table[n*2^n-1,{n,100}],PrimeOmega[#]==2&] (* Harvey P. Dale, Jan 03 2019 *)

for(n=1, 1000, if(bigomega(n*2^n-1)==2, print1(n*2^n-1, ", "))) \\ Colin Barker, May 07 2014

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

A242116 Cullen semiprimes: Semiprimes of the form k*2^k + 1.

Original entry on oeis.org

9, 25, 65, 161, 2049, 4609, 22529, 1048577, 44040193, 283467841537, 1202590842881, 256065421246102339102334047485953, 4259306016766850789028922770063361, 356615920533143509709616588588493085605889, 57729314674570665269045550892293179276409335447553

Offset: 1

K. D. Bajpai, May 04 2014

nonn

The k-th Cullen number Cullen(k) = k*2^k + 1.
If Cullen(k) is semiprime, it is in the sequence.
The next term, a(16), has 52 digits.

			a(4) = 161 = (5*2^5+1) is 5th Cullen number and 161 = 7 * 23 is semiprime.
a(5) = 2049 = (8*2^8+1) is 8th Cullen number and 2049 = 3 * 683 is semiprime.

Tyler Busby, Table of n, a(n) for n = 1..35 (terms 1..16 from K. D. Bajpai, terms 17..34 from Amiram Eldar)

Cf. A001358, A002064, A003261, A005849, A242175.

IsSemiprime:=func; [s: n in [1..200] | IsSemiprime(s) where s is n*2^n+1]; // // Vincenzo Librandi, May 07 2014

Maple

with(numtheory): A242116:= proc(); if bigomega(x*2^x+1) = 2 then RETURN (x*2^x+1); fi; end: seq(A242116 (), x=1..200);

Mathematica

cullen[n_] := n * 2^n + 1; Select[cullen[Range[35]], PrimeOmega[#] == 2 &] (* Amiram Eldar, Nov 27 2019 *)

PARI

select(n->bigomega(n)==2, vector(90,n,n<Charles R Greathouse IV, May 06 2014

Formula

a(n) = A002064(A242175(n)). - Amiram Eldar, Nov 27 2019

A373100 Last digit of n*2^n - 1.

Original entry on oeis.org

1, 7, 3, 3, 9, 3, 5, 7, 7, 9, 7, 1, 5, 5, 9, 5, 3, 1, 1, 9, 1, 7, 3, 3, 9, 3, 5, 7, 7, 9, 7, 1, 5, 5, 9, 5, 3, 1, 1, 9, 1, 7, 3, 3, 9, 3, 5, 7, 7, 9, 7, 1, 5, 5, 9, 5, 3, 1, 1, 9, 1, 7

Offset: 1

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Author

Javier Rodríguez Ríos, May 23 2024

Keywords

nonn
base
easy

Comments

This is a cyclic sequence of 20 numbers, using only 1,3,5,7 and 9 (4 times each).

References

Richard K. Guy (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7.

Links

Wikipedia, Woodall number.

Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1,1,1,-1,-1,1,0,-1,0,1).

Crossrefs

Cf. A010879, A003261.

Cf. A373098, A373099.

Programs

Maple

lastDigit := proc(n) return (n * 2^n - 1) mod 10; end proc: # Example usage minN := 1; maxN := 10; lastDigits := [seq(lastDigit(n), n = minN .. maxN)]; print(lastDigits);

Mathematica

lastDigit[n_] := Mod[n * 2^n - 1, 10] (* Example usage *) minN = 1; maxN = 10; lastDigits = Table[lastDigit[n], {n, minN, maxN}] Print[lastDigits]

PARI

a(n) = lift(Mod(n*2^n - 1, 10))

Python

def last_digit(n): return (n * 2**n - 1) % 10 # Example usage min_n, max_n = 1, 10 last_digits = [last_digit(n) for n in range(min_n, max_n + 1)] print(last_digits)

Formula

a(n) = A010879(A003261(n)).

From Chai Wah Wu, Jul 06 2024: (Start)

a(n) = a(n-2) - a(n-4) + a(n-5) + a(n-6) - a(n-7) - a(n-8) + a(n-9) - a(n-11) + a(n-13) for n > 13.

G.f.: x*(-9*x^12 - x^11 + 8*x^10 - 2*x^9 - 13*x^8 + 2*x^7 + 9*x^6 - 6*x^5 - 7*x^4 + 4*x^3 - 2*x^2 - 7*x - 1)/((x - 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^6 + x^4 - x^2 + 1)). (End)

A131838 Multiplicative persistence of Woodall numbers.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 5, 2, 2, 1, 1, 8, 3, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

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Author

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

Keywords

easy
nonn
base

Comments

After the 111th term, all the numbers have some digits equal to zero, thus the persistence is equal to 1.

Examples

Woodall number 159 --> 1*5*9=45 --> 4*5=20 --> 2*0=0 thus persistence is 3.

Crossrefs

Cf. A003261, A031346, A131841.

Programs

Maple

P:=proc(n) local i,k,w,ok,cont; for i from 1 by 1 to n do w:=1; k:=i*2^i-1; ok:=1; if k<10 then print(0); else cont:=1; while ok=1 do while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w<10 then ok:=0; print(cont); else cont:=cont+1; k:=w; w:=1; fi; od; fi; od; end: P(120);

Mathematica

Table[wn=n*2^n-1;Length[NestWhileList[Times@@IntegerDigits[#]&, wn, #>=10&]], {n, 105}]-1 (* James C. McMahon, Mar 01 2025 *)

Formula

a(n) = A031346(A003261(n)). - Michel Marcus, Mar 01 2025

A131841 Additive persistence of Woodall numbers.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 3

Offset: 1

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Author

Paolo P. Lava and Giorgio Balzarotti, Jul 20 2007

Keywords

easy
nonn
base

Examples

Woodall number 159 --> 1+5+9=15 --> 1+5=6 thus persistence is 2

Crossrefs

Cf. A003261, A031286, A131838.

Programs

Maple

with(numtheory): with(combinat): P:=proc(n) local a,t;t:=0; a:=n*2^n-1; while a>9 do t:=t+1; a:=convert(convert(a,base,10),`+`); od; t; end: seq(P(i),i=1..10^2);

Mathematica

f[n_] := Length@ NestWhileList[Plus @@ IntegerDigits@# &, n*2^n - 1, UnsameQ@## &, All] - 2; Array[f, 88] (* James C. McMahon, Mar 01 2025 *)

Formula

a(n) = A031286(A003261(n)). - James C. McMahon, Mar 01 2025

Extensions

Corrected entries and Maple code by Paolo P. Lava, Dec 19 2017

A229018 Primes of the form (3*x + 2)*2^x - 1.

Original entry on oeis.org

31, 223, 1279, 3276799, 14680063, 420906795007, 2357352929951743, 32326824857489154029020587706017980088319, 173918694842377447266238495093237679339055972614143

Offset: 1

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Author

K. D. Bajpai, Sep 11 2013

Keywords

nonn

Comments

Also primes of the form W(n) + W(n+1) + 1 where W(n) and W(n+1) are consecutive Woodall numbers. The n-th Woodall number = n*2^n-1.

Examples

a(2) = 223: for x=4: R= x*2^x-1 = 4*2^4-1 = 63 and S= (x+1)*2^(x+1)-1 = 5*2^5-1 = 159. R+S+1 = 63+159+1 = 223 which is prime.

Links

K. D. Bajpai, Table of n, a(n) for n = 1..19

Wikipedia, Woodall number

Crossrefs

Cf. A003261, A072669.

Programs

Maple

KD:= proc() local a,b,d; a:= x*2^x-1; b:=(x+1)*2^(x+1)-1; d:=a+b+1; if isprime(d) then RETURN(d): fi; end: seq(KD(),x=1..1000);

Mathematica

Select[Table[(3*x + 2)*2^x - 1, {x, 200}], PrimeQ] (* T. D. Noe, Sep 20 2013 *)

A382646 Numbers k such that (k*2^d - 1)*(d*2^k - 1) is semiprime for some divisor d of k.

Original entry on oeis.org

2, 3, 6, 7, 12, 18, 19, 21, 30, 31, 42, 60, 75, 81, 115, 123, 126, 132, 133, 225, 249, 306, 324, 362, 384, 462, 468, 512, 606, 607, 612, 751, 822, 1279, 2170, 2202, 2281, 5312, 7755, 9531, 12379, 14898, 15822, 18123, 18819, 18885, 22971, 23005, 23208, 41628, 44497, 51384, 52540, 98726

Offset: 1

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Author

Juri-Stepan Gerasimov, Apr 01 2025

Keywords

nonn

Comments

No further terms <= 10^5. - Michael S. Branicky, Apr 07 2025

Examples

7 is in this sequence because (7*2^1-1)*(1*2^7-1) = 13*127 is semiprime for divisor 1 of 7.

Crossrefs

Supersequence of A002234.

Cf. A001358, A003261, A382887.

Programs

Magma

[n: n in [1..1000] | not #[d: d in Divisors(n) | IsPrime(d*2^n-1) and IsPrime(n*2^d-1)] eq 0];

PARI

isok(k) = fordiv(k, d, if (ispseudoprime(k*2^d - 1) && ispseudoprime(d*2^k - 1), return(1))); \\ Michel Marcus, Apr 02 2025

Python

from itertools import count, islice from sympy import isprime, divisors def A382646_gen(): # generator of terms yield from filter(lambda k:any(isprime((k<A382646_list = list(islice(A382646_gen(), 30)) # Chai Wah Wu, Apr 15 2025

Extensions

a(40) from Michel Marcus, Apr 02 2025

a(41)-a(54) from Michael S. Branicky, Apr 07 2025

A056235 n*2^(n*2^n+n)-1.

Original entry on oeis.org

7, 2047, 402653183, 1180591620717411303423, 233840261972944466912589573234605283144949206876159

Offset: 1

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Author

Henry Bottomley, Aug 06 2000

Keywords

nonn

Comments

For n>2, a(n) is base at which zero is reached for the function "write f(j) in base j, read as base j+1 and then subtract 1 to give f(j+1)" starting from f(n)=2n^2-1.

Examples

a(3)=3*2^(3*2^3+3)-1=3*2^27-1=402653184-1=402653183

Formula

a(n) =A003261(A036289(n)) =A036289(n)*A001146(n)^n-1

A099051 p*2^p - 1 where p is prime.

Original entry on oeis.org

7, 23, 159, 895, 22527, 106495, 2228223, 9961471, 192937983, 15569256447, 66571993087, 5085241278463, 90159953477631, 378231999954943, 6614661952700415, 477381560501272575, 34011184385901985791

Offset: 1

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Author

Parthasarathy Nambi, Nov 13 2004

Keywords

nonn
easy

Comments

This is the subset of Woodall numbers of prime index. The 9th largest known Woodall prime is in this sequence: 12379*2^12379-1, where 12379 is prime, as found by Wilfrid Keller in 1984. Smaller primes are when p = 2, 3, 751. These numbers can also be semiprime, as when p = 159, 163, or 211 and hard to factor as when n = 349 (108 digits). - Jonathan Vos Post, Nov 19 2004

Examples

If p=3, 3*2^3 - 1 = 23. If p=11, 11*2^11 - 1 = 22527.

References

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 360-361, 1996

Links

Eric Weisstein's World of Mathematics, Woodall Numbers.

Crossrefs

Similar to Woodall numbers (A003261). Cf. A002234.

Programs

Mathematica

Table[ Prime[n]*2^Prime[n] - 1, {n, 17}] (* Robert G. Wilson v, Nov 16 2004 *)

Extensions

More terms from Robert G. Wilson v, Nov 15 2004

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A098713 a(n) = (2n+1)*2^(2n+1) - 1.

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Crossrefs

Programs

Magma

Maple

Extensions

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

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A242116 Cullen semiprimes: Semiprimes of the form k*2^k + 1.

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Programs

Magma

Maple

Mathematica

PARI

Formula

A373100 Last digit of n*2^n - 1.

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References

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Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A131838 Multiplicative persistence of Woodall numbers.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A131841 Additive persistence of Woodall numbers.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A229018 Primes of the form (3*x + 2)*2^x - 1.

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Examples

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A229018 Primes of the form (3x + 2)2^x - 1.

A382646 Numbers k such that (k2^d - 1)(d*2^k - 1) is semiprime for some divisor d of k.

A056235 n2^(n2^n+n)-1.