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 15 results. Next

A003261 Woodall (or Riesel) numbers: n*2^n - 1.

Original entry on oeis.org

1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519, 44040191, 92274687, 192937983, 402653183, 838860799, 1744830463, 3623878655, 7516192767, 15569256447, 32212254719, 66571993087
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

For n>1, 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) = n^2 - 1. - Henry Bottomley, Aug 06 2000
Sequence corresponds also to the maximum chain length of the classic puzzle whereby, under agreed commercial terms, an asset of unringed golden chain, when judiciously fragmented into as few as n pieces and n-1 opened links (through n-1 cuts), might be used to settle debt sequentially, with a golden link covering for unit cost. Here beside the n-1 opened links, the n fragmented pieces have lengths n, 2*n, 4*n, ..., 2^(n-1)*n. For instance, the chain of original length a(5)=159, if segregated by 4 cuts into 5+1+10+1+20+1+40+1+80, may be used to pay sequentially, i.e., a link-cost at a time, for an equivalent cost up to 159 links, to the same creditor. - Lekraj Beedassy, Feb 06 2003

Examples

			G.f. = x + 7*x^2 + 23*x^3 + 63*x^4 + 159*x^5 + 383*x^6 + 895*x^7 + ... - _Michael Somos_, Nov 04 2018

References

  • A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 159.
  • K. R. Bhutani and A. B. Levin, "The Problem of Sawing a Chain", Journal of Recreational Mathematics 2002-3 31(1) 32-35.
  • G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • M. Gardner, Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American, "Gold Links", Problem 4, pp. 50-51; 57-58, University of Chicago Press, 1983.
  • O. O'Shea, Mathematical Brainteasers with Surprising Solutions, Problem 76, pp. 183-185, Prometheus Books, Guilford, Connecticut, 2020.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 241.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002234, A002064, A005849, A050918, A006127.
a(n) = A036289(n) - 1 = A002064(n) - 2.
Cf. A133653.

Programs

Formula

G.f.: x*(-1-2*x+4*x^2) / ( (x-1)*(-1+2*x)^2 ). - Simon Plouffe in his 1992 dissertation
Binomial transform of A133653 and double binomial transform of [1, 5, -1, 1, -1, 1, ...]. - Gary W. Adamson, Sep 19 2007
a(n) = -(2)^n * A006127(-n) for all n in Z. - Michael Somos, Nov 04 2018
E.g.f.: 1 + exp(x)*(2*exp(x)*x - 1). - Stefano Spezia, Nov 24 2024

A002234 Numbers k such that the Woodall number k*2^k - 1 is prime.

Original entry on oeis.org

2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, 17016602
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

a(34) = 17016602 is tentative until the range 16838832..17016601 is fully searched. - Eric W. Weisstein, Mar 22 2018

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 115, p. 40, Ellipses, Paris 2008.
  • R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B20.
  • F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 95, 1983.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 241-242.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 139.

Links

Crossrefs

Cf. A050918 (for the actual primes), A003261, A005849.

Programs

  • PARI
    is(n)=isprime(n<Charles R Greathouse IV, Feb 07 2017

Extensions

a(27) communicated by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 15 2004
a(28) = 1195203 found by M. Rodenkirch; contributed by Eric W. Weisstein, Nov 29 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(30)-a(33) from John Blazek, May 14 2009
a(34) = 17016602 communicated by Eric W. Weisstein, Mar 22 2018

A182352 Primes of the form n*2^n - 3.

Original entry on oeis.org

5, 61, 157, 229373, 1048573, 2228221, 584115552253, 10445360463869, 1448241753615514100500122329229605507956733, 8380834989110329694147210304728253347914979574441574397
Offset: 1

Views

Author

Patrick Devlin, Apr 25 2012

Keywords

Comments

These are similar to the Woodall primes, A050918, which are primes of the form n*2^n-1.

Examples

			5 = 2*2^2 - 3;  61 = 4*2^4 - 3.

Crossrefs

Cf. A050918.

Programs

  • Maple
    #choose N large, then S is the desired set
f:=n->n*2^n - 3:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od

A182353 Primes of the form n*2^n - 5.

Original entry on oeis.org

3, 19, 59, 379, 4603, 1048571, 44040187, 7516192763, 6614661952700411, 13510798882111483, 477381560501272571, 16717361816799281147, 4869940435459321626619, 802726744224113772004900859
Offset: 1

Views

Author

Patrick Devlin, Apr 25 2012

Keywords

Comments

These are similar to the Woodall primes, A050918, and to sequence A182352, which are primes of the form n*2^n - 1 and of the form n*2^n - 3 respectively. However, this sequence seems to grow rather more slowly than those.

Examples

			3 = 2*2^2 - 5;  19 = 3*2^3 - 5;  59 = 4*2^4 - 5.

Links

Crossrefs

Cf. A050918, A182352.

Programs

  • Maple
    #choose N large, then S is the desired set
f:=n->n*2^n - 5:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od
  • Mathematica
    Select[Table[n*2^n-5,{n,2,100}],PrimeQ] (* Harvey P. Dale, Aug 06 2013 *)

A182373 Positive integers k such that k*3^k - 2 is prime.

Original entry on oeis.org

3, 5, 7, 37, 45, 53, 179, 277, 721, 2087, 6197, 6317, 8775, 12781, 38943, 47273, 50507, 66693
Offset: 1

Views

Author

Patrick Devlin, Apr 26 2012

Keywords

Comments

Similar to A060353, and to the Woodall primes, A050918. The next term in the sequence is unknown; if the sequence is infinite, the next term is greater than 5000.
a(15) > 30000. - Tyler NeSmith, Apr 22 2022
a(19) > 10^5. - Michael S. Branicky, Sep 22 2024

Examples

			79 = 3*3^3 - 2; 1213 = 5*3^5 - 2; 15307 = 7*3^7 - 2.

Crossrefs

Cf. A050918, A060353.

Programs

  • Maple
    #choose N large, then S is the desired set
f:=n->n*3^n - 2:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {n}: fi: od
  • Mathematica
    Select[Range[2100], PrimeQ[#*3^# - 2] &] (* Jayanta Basu, Jun 01 2013 *)
  • PARI
    for(n=1, 1e6, if(ispseudoprime(3^n*n - 2), print1(n, ", "))) \\ Altug Alkan, Dec 01 2015

Extensions

a(11)-a(14) from Altug Alkan, Dec 01 2015
a(15)-a(17) from Michael S. Branicky, Apr 23 2023
a(18) from Michael S. Branicky, Sep 22 2024

A210340 Generalized Woodall primes: any primes that can be written in the form n*b^n - 1 with n+2 > b > 2.

Original entry on oeis.org

17, 191, 4373, 5119, 524287, 590489, 3124999, 14680063, 3758096383, 6973568801, 34867844009, 85449218749, 824633720831, 1099999999999, 1618481116086271, 11577835060199423, 14999999999999999, 29311444762388081, 73123168801259519
Offset: 1

Views

Author

Arkadiusz Wesolowski, Mar 20 2012

Keywords

Examples

			167*2^668 - 1 is a prime number and 167*2^668 - 1 = 167*16^167 - 1, so this number is in the sequence.

Links

Crossrefs

Cf. A050918, A002234, A006553, A086661, A059676, A059675.

Programs

  • Mathematica
    lst = {}; Do[p = n*b^n - 1; If[p < 10^200 && PrimeQ[p], AppendTo[lst, p]], {b, 3, 100}, {n, b - 1, 413}]; Sort@lst

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

A288158 a(n) = smallest k such that k*n*2^n-1 is prime.

Original entry on oeis.org

2, 1, 1, 2, 3, 1, 3, 3, 5, 2, 9, 5, 7, 4, 3, 14, 8, 4, 4, 9, 11, 8, 3, 7, 9, 10, 2, 36, 3, 1, 4, 6, 4, 2, 17, 3, 7, 4, 6, 14, 17, 14, 51, 3, 9, 12, 38, 8, 13, 36, 64, 30, 14, 14, 4, 19, 13, 41, 5, 19, 4, 60, 8, 20, 3, 13, 24, 10, 31, 15, 14, 42, 19, 13, 1, 30, 11, 11
Offset: 1

Views

Author

Pierre CAMI, Jun 06 2017

Keywords

Comments

If k = 1 then n*2^n-1 is a Woodall prime (A050918).

Links

Crossrefs

Cf. A050918.

Programs

  • Mathematica
    Table[k = 1; While[! PrimeQ[k n*2^n - 1], k++]; k, {n, 78}] (* Michael De Vlieger, Jun 07 2017 *)
  • PARI
    a(n) = my(k=1); while(1, if(ispseudoprime(k*n*2^n-1), return(k)); k++) \\ Felix Fröhlich, Jun 06 2017

A182354 Primes of the form n*2^n + 3.

Original entry on oeis.org

3, 5, 11, 67, 163, 10243, 22531, 7516192771, 43980465111043, 142788163609707759784588649053552643, 2637188343637273091841153207596203638787
Offset: 1

Views

Author

Patrick Devlin, Apr 25 2012

Keywords

Comments

These are similar to the Woodall primes, A050918, which are primes of the form n*2^n - 1.

Examples

			3 = 0*2^0 + 3; 5 = 1*2^1 + 3; 11 = 2*2^2 + 3; 67 = 4*2^4 + 3

Crossrefs

Cf. A050918, A182352, A182353.

Programs

  • Maple
    #choose N large, then S is the desired set
f:=n->n*2^n + 3:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od

A182375 Positive integers k such that k*2^k - 3 is prime.

Original entry on oeis.org

2, 4, 5, 14, 16, 17, 34, 38, 133, 175, 218, 284, 1036, 1441, 1550, 2893, 2933, 3770
Offset: 1

Views

Author

Patrick Devlin, Apr 26 2012

Keywords

Comments

Similar to the Woodall primes, A050918.
The next term in this sequence is unknown; if it is infinite, the next term is greater than 8000.

Examples

			5 = 2*2^2 - 3;  61 = 4*2^4 - 3;  157 = 5*2^5 - 3

Crossrefs

Cf. A050918. Values for primes are A182352.

Programs

  • Maple
    f:=n->n*2^n - 3:
S:={}:
for n from 0 to 300 do if(isprime(f(n))) then S:=S union {n}: fi: od
Showing 1-10 of 15 results. Next

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

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