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 1-6 of 6 results.

A103533 Even semiprimes of the form prime(n)*prime(n+1) - 1.

Original entry on oeis.org

14, 34, 142, 898, 1762, 5182, 19042, 79522, 213442, 359998, 412162, 627238, 685582, 777922, 1192462, 1299478, 1695202, 2005006, 2585662, 2663398, 3849322, 4536898, 5143822, 5588446, 5673922, 6594502, 7225342, 8363638, 8538058, 12110278
Offset: 1

Views

Author

Giovanni Teofilatto, Mar 22 2005

Keywords

Comments

5 is the only odd number of the form prime(n)*prime(n+1) - 1. - Klaus Brockhaus, Mar 29 2005
2*A086870(n) is a subsequence of this sequence. They first differ when 313619 is not in A086870, but 2*313619 = 627238 = a(12). This is because 787 and 797 are the first such pair of consecutive primes that are not twins and (787*797-1)/2 is prime. - Jason Kimberley, Oct 22 2015

Examples

			a(1)=14 because prime(2)*prime(3)- 1=3*5-1=14=2*7;
a(2)=34 because prime(3)*prime(4)- 1=5*7-1=34=2*17;
a(3)=142 because prime(5)*prime(6)-1=11*13-1=142=2*71.

Links

Crossrefs

Cf. A001358, A006881, A086870, A103746, A269663.

Programs

  • Magma
    [a:n in[2..1000]|IsPrime(a div 2)where a is NthPrime(n)*NthPrime(n+1)-1]; // Jason Kimberley, Oct 22 2015
  • Mathematica
    fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Prime[ Range[490]]*Prime[ Range[2, 491]] - 1, fQ[ # ] &] (* Robert G. Wilson v, Mar 24 2005 *)
Select[Times@@#-1&/@Partition[Prime[Range[500]],2,1],EvenQ[#] && PrimeOmega[ #]==2&] (* Harvey P. Dale, Apr 24 2018 *)
  • PARI
    for(n=1,490,if(bigomega(k=prime(n)*prime(n+1)-1)==2,print1(k,","))) \\ Klaus Brockhaus, Mar 24 2005

Extensions

More terms from Robert G. Wilson v and Klaus Brockhaus, Mar 24 2005

A103614 Semiprimes of the form prime(n)*prime(n+1)*prime(n+2) - 1.

Original entry on oeis.org

4198, 33262, 1564258, 6672202, 7566178, 18181978, 20193022, 178433278, 187466722, 229580146, 293158126, 467821918, 1125878062, 1341880018, 4317369778, 5198554618, 8493529942, 10138087306, 10594343758, 20940647698
Offset: 1

Views

Author

Jonathan Vos Post, Mar 24 2005

Keywords

Comments

This is the three-consecutive-prime minus one equivalent of A103533, which is Giovanni Teofilatto's two-consecutive-prime minus one sequence.

Examples

			n: prime(n) * prime(n+1) * prime(n+2) - 1
6: 13 *17 *19 - 1 = 4198 = 2 * 2099
10: 29 * 31 * 37 - 1 = 33262 = 2 * 16631
29: 109 * 113 * 127 - 1 = 1564258 = 2 * 782129
42: 181 * 191 * 193 -1 = 6672202 = 2 * 3336101
44: 193 * 197 * 199 -1 = 7566178 = 2 * 3783089
55: 257 * 263 * 269 -1 = 18181978 = 2 * 9090989
57: 269 * 271 * 277 -1 = 20193022 = 2 * 10096511
102: 557 * 563 * 569 -1 = 178433278 = 2 * 89216639

Links

Crossrefs

Cf. A000040, A001358, A006881, A103533, A103746, A104874, A104875.

Programs

  • Mathematica
    Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Select[Table[Prime[n]*Prime[n+1]*Prime[n+2]-1, {n, 1000}], SemiprimeQ] (* Ray Chandler, Mar 29 2005 *)
Select[Times@@#-1&/@Partition[Prime[Range[500]],3,1],PrimeOmega[#]==2&] (* Harvey P. Dale, Mar 03 2025 *)
  • PARI
    for(n=1,420,if(bigomega(k=prime(n)*prime(n+1)*prime(n+2)-1)==2,print1(k,","))) (Brockhaus)

Extensions

Extended by Ray Chandler and Klaus Brockhaus, Mar 29 2005

A104874 Semiprimes of the form prime(n)*prime(n+1)*prime(n+2)*prime(n+3) - 1.

Original entry on oeis.org

209, 1154, 645328246, 2445956098, 2337448622686, 19317973275826, 22894376863198, 32220239865718, 51087435019342, 78382834887262, 163068083613646, 176031800345938, 622751201209726, 1292966939911018
Offset: 1

Views

Author

Jonathan Vos Post, Mar 29 2005

Keywords

Comments

This is the four-consecutive-prime minus one equivalent of A103533.

Examples

			n: prime(n) * prime(n+1) * prime(n+2) * prime(n+3) - 1
1: 2 * 3 * 5 * 7 - 1 = 209 = 11 * 19
2: 3 * 5 * 7 * 11 - 1 = 1154 = 2 * 577
36: 151 * 157 * 163 * 167 - 1 = 645328246 = 2 * 322664123
47: 211 * 223 * 227 * 229 - 1 = 2445956098 = 2 * 1222978049
201: 1229 * 1231 * 1237 * 1249 - 1 = 2337448622686 = 2 * 1168724311343.

Links

Crossrefs

Cf. A000040, A001358, A006881, A103533, A103614, A103746, A104875.

Programs

  • Mathematica
    Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Select[Table[Prime[n]*Prime[n+1]*Prime[n+2]*Prime[n+3]-1, {n, 1000}], SemiprimeQ] (* Ray Chandler, Mar 29 2005 *)

Extensions

Extended by Ray Chandler, Mar 29 2005

A104875 Semiprimes of the form prime(n)*prime(n+1)*prime(n+2)*prime(n+3)*prime(n+4) - 1.

Original entry on oeis.org

15014, 1062346, 600662302, 2224636919002, 118335570521086, 168652154886862, 3790374062238502, 6290838589498366, 127018534712243098, 131125107904515418, 190740905520325018, 2057351971883521282, 3151949824862998762
Offset: 1

Views

Author

Jonathan Vos Post, Mar 29 2005

Keywords

Comments

This is the five-consecutive-prime minus one equivalent of A103533.

Examples

			n prime(n) * prime(n+1) * prime(n+2) * prime(n+3) * prime(n+4) - 1
1: 2 * 3 * 5 * 7 * 11 - 1 = 2309 is prime; examples hereafter are semiprime
2: 3 * 5 * 7 * 11 * 13 - 1 = 15014 = 2 * 7507
5: 11 * 13 * 17 * 19 * 23 - 1 = 1062346 = 2 * 531173
15: 47 * 53 * 59 * 61 * 67 - 1 = 600662302 = 2 * 300331151
60: 281 * 283 * 293 * 307 * 311 - 1 = 2224636919002 = 2 * 1112318459501
117: 643 * 647 * 653 * 659 * 661 - 1 = 118335570521086 = 2 * 59167785260543

Links

Crossrefs

Cf. A000040, A001358, A006881, A103533, A103614, A103746, A104874.

Programs

  • Mathematica
    Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Select[Table[Prime[n]*Prime[n+1]*Prime[n+2]*Prime[n+3]*Prime[n+4]-1, {n, 1000}], SemiprimeQ] (* Ray Chandler, Mar 29 2005 *)

Extensions

Extended by Ray Chandler, Mar 29 2005

A023521 Sum of distinct prime divisors of prime(n)*prime(n-1) - 1.

Original entry on oeis.org

5, 9, 19, 21, 73, 18, 32, 111, 42, 451, 196, 381, 883, 108, 93, 526, 266, 232, 72, 2593, 36, 162, 1236, 98, 112, 752, 55, 26, 3081, 55, 4161, 1002, 9523, 135, 1616, 444, 863, 1368, 117, 415, 266, 3464, 2642, 1908, 1172, 3504, 1312, 2538, 135
Offset: 2

Views

Author

Clark Kimberling

Keywords

Comments

If n-1 is in A103746, then a(n) = (prime(n)*prime(n-1)+3)/2. - Robert Israel, Jun 03 2020

Links

Crossrefs

Cf. A103746.

Programs

  • Maple
    p:= 2: R:= NULL:
for n from 2 to 50 do
  q:= p; p:= nextprime(p);
  R:= R, convert(numtheory:-factorset(p*q-1),`+`)
od:
R; # Robert Israel, Jun 03 2020
  • PARI
    sopf(n) = my(fac=factor(n)); sum(i=1, matsize(fac)[1], fac[i, 1]) ;
a(n) = sopf(prime(n)*prime(n-1) - 1); \\ Michel Marcus, Sep 30 2013

Formula

a(n) = A008472(A023515(n)). - Michel Marcus, Sep 30 2013

Extensions

Offset set to 2 and a(1) removed by Michel Marcus, Sep 30 2013

A103767 Numbers n such that prime(n)*prime(n+1)*prime(n+2) - 1 is semiprime.

Original entry on oeis.org

6, 10, 29, 42, 44, 55, 57, 102, 104, 111, 120, 136, 174, 184, 257, 269, 308, 325, 327, 401, 426, 504, 514, 565, 571, 594, 595, 652, 717, 755, 864, 882, 901, 907, 985, 1014, 1074, 1134, 1141, 1156, 1198, 1301, 1327, 1346, 1362, 1654, 1670, 1674, 1778, 1897
Offset: 1

Views

Author

Klaus Brockhaus, Mar 29 2005

Keywords

Examples

			prime(10)*prime(11)*prime(12) - 1 = 29*31*37 - 1 = 33262= 2*16631, hence 10 is
a term.

Links

Crossrefs

Cf. A001358, A006881, A103614, A103746.

Programs

  • Mathematica
    PrimePi/@Transpose[Select[Partition[Prime[Range[2000]],3,1], PrimeOmega[ Times@@#-1]==2&]][[1]] (* Harvey P. Dale, Mar 15 2015 *)
  • PARI
    for(n=1,1900,if(bigomega(prime(n)*prime(n+1)*prime(n+2)-1)==2,print1(n,",")))
Showing 1-6 of 6 results.

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

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