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.

A109287 4-almost primes equal to p*q + 1, where p and q are (not necessarily distinct) primes.

Original entry on oeis.org

16, 36, 40, 56, 88, 135, 156, 184, 204, 210, 220, 248, 250, 260, 296, 306, 315, 328, 330, 340, 342, 372, 414, 459, 472, 490, 516, 536, 546, 580, 584, 636, 650, 686, 690, 708, 714, 732, 735, 738, 804, 808, 819, 836, 850, 852, 870, 872, 940, 950, 966, 975, 996
Offset: 1

Views

Author

Giovanni Teofilatto and Jonathan Vos Post, Aug 20 2005

Keywords

Comments

4-almost primes of the form semiprime + 1.

Examples

			a(1) = 16 because (3*5+1)=(2^4) = 16.
a(2) = 36 because (5*7+1)=((2^2)*(3^2)) = 36.
a(3) = 40 because (3*13+1)=((2^3)*5) = 40.
a(4) = 56 because (5*11+1)=((2^3)*7) = 56.
a(5) = 88 because (3*29+1)=((2^3)*11) = 88.
a(6) = 135 because (2*67+1)=((3^3)*5) = 135.
a(7) = 156 because (5*31+1)=((2^2)*3*13) = 156.
a(8) = 184 because (3*61+1)=((2^3)*23) = 184.

Links

Crossrefs

Primes are in A000040. Semiprimes are in A001358. 4-almost primes are in A014613.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in this sequence.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Similar to A076153; after A076153(0)=3 next difference is A076153(100)=1792 whereas A109287(100)=1794.
Cf. A077065, A079148, A092307, A109288, A109289, A109290.

Programs

  • Mathematica
    bo[n_] := Plus @@ Last /@ FactorInteger[n]; Select[Range[1000], bo[ # ] == 4 && bo[ # - 1] == 2 &] (* Ray Chandler, Aug 27 2005 *)
  • PARI
    is(n)=bigomega(n)==4 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Sep 16 2015

Formula

a(n) is in this sequence iff a(n) is in A014613 and (a(n)-1) is in A001358.

Extensions

Extended by Ray Chandler, Aug 27 2005
Edited by Ray Chandler, Mar 20 2007

A109373 Semiprimes of the form semiprime + 1.

Original entry on oeis.org

10, 15, 22, 26, 34, 35, 39, 58, 86, 87, 94, 95, 119, 122, 123, 134, 142, 143, 146, 159, 178, 202, 203, 206, 214, 215, 218, 219, 254, 299, 302, 303, 327, 335, 362, 382, 394, 395, 446, 447, 454, 482, 502, 515, 527, 538, 543, 554, 566, 623, 634, 635, 695, 698
Offset: 1

Views

Author

Jonathan Vos Post, Aug 24 2005

Keywords

Examples

			a(1) = 10 because (3*3+1)=(2*5) = 10.
a(2) = 15 because (2*7+1)=(3*5) = 15.
a(3) = 22 because (3*7+1)=(2*11) = 22.
a(4) = 26 because (5*5+1)=(2*13) = 26.
a(5) = 34 because (3*11+1)=(2*17) = 34.

Links

Crossrefs

Primes are in A000040. Semiprimes are in A001358.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in this sequence.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A056809, A077065, A079148, A086005, A086006, A092307.
Subsequence of A088707; A064911.

Programs

  • Haskell
    a109373 n = a109373_list !! (n-1)
a109373_list = filter ((== 1) . a064911) a088707_list
-- Reinhard Zumkeller, Feb 20 2012
  • Mathematica
    fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[ 700], fQ[ # - 1] && fQ[ # ] &] (* Robert G. Wilson v *)
With[{sps=Select[Range[700],PrimeOmega[#]==2&]},Transpose[Select[ Partition[ sps,2,1],#[[2]]-#[[1]]==1&]][[2]]] (* Harvey P. Dale, Sep 05 2012 *)
  • PARI
    is(n)=bigomega(n)==2 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Jan 31 2017

Formula

a(n) is in this sequence iff a(n) is in A001358 and (a(n)-1) is in A001358.
a(n) = A070552(n) + 1.

Extensions

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005
Edited by Ray Chandler, Mar 20 2007

A109067 3-almost primes of the form semiprime + 1.

Original entry on oeis.org

27, 50, 52, 63, 66, 70, 75, 78, 92, 116, 124, 130, 147, 170, 186, 188, 195, 207, 222, 236, 238, 255, 266, 268, 275, 279, 290, 292, 310, 322, 356, 363, 366, 387, 399, 404, 412, 418, 423, 428, 438, 452, 455, 470, 474, 483, 494, 498, 506, 518, 530, 534, 539, 555
Offset: 1

Views

Author

Jonathan Vos Post, Aug 24 2005

Keywords

Examples

			a(1) = 27 because (2*13+1)=(3^3) = 27.
a(2) = 50 because (7*7+1)=(2*5^2) = 50.
a(3) = 52 because (3*17+1)=(2^2*13) = 52.
a(4) = 63 because (2*31+1)=(3^2*7) = 63.
a(5) = 66 because (5*13+1)=(2*3*11) = 66.
a(6) = 70 because (3*23+1)=(2*5*7) = 70.
a(7) = 75 because (2*37+1)=(3*5^2) = 75.
a(8) = 78 because (7*11+1)=(2*3*13) = 78.

Links

Crossrefs

Primes are in A000040. Semiprimes are in A001358. 3-almost primes are in A014612.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in this sequence.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A077065, A079148, A092307.

Programs

  • Mathematica
    f[n_] := Plus @@ Last /@ FactorInteger[n];Select[Range[600], f[ # ] == 3 && f[ # - 1] == 2 &] (* Ray Chandler, Mar 20 2007 *)
Select[Select[Range[600],PrimeOmega[#]==2&]+1,PrimeOmega[#]==3&] (* Harvey P. Dale, Nov 24 2013 *)
  • PARI
    list(lim)=my(v=List(),t); forprime(p=2,lim, forprime(q=2,min(p,lim\p), if(bigomega(t=p*q+1)==3, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Formula

a(n) is in this sequence iff a(n) is in A014612 and a(n)-1 is in A001358.

Extensions

Edited and extended by Ray Chandler, Mar 20 2007

A109383 5-almost primes of the form semiprime + 1.

Original entry on oeis.org

112, 120, 162, 300, 304, 378, 392, 396, 408, 520, 552, 567, 592, 612, 630, 656, 675, 680, 688, 696, 700, 750, 780, 918, 924, 944, 952, 980, 990, 1044, 1100, 1116, 1136, 1140, 1160, 1168, 1170, 1242, 1264, 1272, 1300, 1323, 1352, 1372, 1380, 1386, 1416, 1470
Offset: 1

Views

Author

Jonathan Vos Post, Aug 25 2005

Keywords

Examples

			a(1) = 112 because (3*37)+1 = (2^4) * 7 = 112.
a(2) = 120 because (7*17)+1 = (2^3) * 3 * 5 = 120.
a(3) = 162 because (7*23)+1 = 2 * (3^4) = 162.

Links

Crossrefs

Primes are in A000040. Semiprimes are in A001358. 5-almost primes are in A014614.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in A109373.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in this sequence.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A077065, A079148, A092307.

Programs

  • Mathematica
    f[n_] := Plus @@ Last /@ FactorInteger[n];Select[Range[1500], f[ # ] == 5 && f[ # - 1] == 2 &] (* Ray Chandler, Mar 20 2007 *)
  • PARI
    v=vector(10000);i=0; for(n=1,9e99, if(issemi(n)&bigomega(n+1)==5, v[i++]=n+1;if(i==#v, return))); v \\ Charles R Greathouse IV, Feb 14 2011

Formula

a(n) is in this sequence iff a(n) is in A014614 and (a(n)-1) is in A001358.

Extensions

Extended by Ray Chandler, Mar 20 2007

A128686 Least number of the form semiprime - 1 which is the product of exactly n primes.

Original entry on oeis.org

3, 9, 8, 24, 32, 64, 128, 864, 1792, 1536, 2048, 4096, 8192, 24576, 73728, 98304, 131072, 393216, 524288, 1048576, 3145728, 6291456, 8388608, 37748736, 50331648, 150994944, 301989888, 268435456, 1207959552, 1610612736, 2147483648, 4294967296, 19327352832, 25769803776, 115964116992, 171798691840, 343597383680, 927712935936, 2886218022912, 1099511627776, 20890720927744, 24189255811072
Offset: 1

Views

Author

Ray Chandler, Mar 20 2007

Keywords

Crossrefs

Cf. A005383, A070552, A128665.

Extensions

a(27)-a(42) from Donovan Johnson, Nov 24 2010
Showing 1-5 of 5 results.

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