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 1-8 of 8 results.

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

A109288 Semiprimes equal to p*q + 1, where p and q are distinct primes.

Original entry on oeis.org

15, 22, 34, 35, 39, 58, 86, 87, 94, 95, 119, 123, 134, 142, 143, 146, 159, 178, 202, 203, 206, 214, 215, 218, 219, 254, 299, 302, 303, 327, 335, 382, 394, 395, 446, 447, 454, 482, 502, 515, 527, 538, 543, 554, 566, 623, 634, 635, 695, 698, 699, 707, 718, 746
Offset: 1

Views

Author

Giovanni Teofilatto, Aug 20 2005

Keywords

Links

Crossrefs

Cf. A109287, A109289, A109290, A263990.

Programs

  • Mathematica
    With[{nn=60},Take[Select[Union[Times@@#+1&/@Subsets[Prime[Range[2nn]],{2}]],PrimeOmega[#]==2&],nn]] (* Harvey P. Dale, Apr 28 2018 *)
Select[Range[1000], !IntegerQ[Sqrt[#-1]] && PrimeOmega[#] == PrimeOmega[#-1] == 2 &] (* Amiram Eldar, Aug 08 2025 *)
  • PARI
    for(i=1,1000,if(bigomega(i)==2&&bigomega(i+1)==2&&issquare(i)==0,print1(i+1,","))) \\ Lambert Klasen (lambert.klasen(AT)gmx.net), Aug 21 2005

Formula

a(n) = A263990(n) + 1. - Amiram Eldar, Aug 08 2025

Extensions

More terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Aug 21 2005

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

A128665 Least n-almost prime of the form semiprime + 1.

Original entry on oeis.org

5, 10, 27, 16, 112, 96, 288, 896, 512, 1536, 2048, 10240, 12288, 36864, 49152, 98304, 196608, 393216, 1769472, 1572864, 3145728, 6291456, 8388608, 41943040, 50331648, 234881024, 201326592, 905969664, 805306368, 6039797760, 3221225472, 6442450944, 19327352832, 64424509440, 173946175488, 257698037760, 137438953472, 412316860416, 1236950581248, 3848290697216, 2199023255552, 10995116277760, 29686813949952
Offset: 1

Views

Author

Ray Chandler, Mar 19 2007

Keywords

Crossrefs

Cf. A005385, A109373, A109067, A109287, A109383.

Extensions

a(28)-a(43) from Donovan Johnson, Nov 24 2010

A109290 Composite numbers which are not of the forms p*q -+ 1, where p and q are (not necessarily distinct) primes.

Original entry on oeis.org

4, 6, 12, 18, 28, 30, 42, 44, 46, 49, 51, 55, 60, 62, 65, 69, 72, 74, 77, 80, 82, 91, 98, 99, 100, 102, 104, 106, 108, 111, 115, 125, 126, 129, 136, 138, 148, 150, 152, 153, 155, 161, 164, 166, 169, 171, 172, 174, 175, 180, 183, 185, 187, 189, 190, 192, 194, 196
Offset: 1

Views

Author

Giovanni Teofilatto, Aug 20 2005

Keywords

Links

Crossrefs

Cf. A109287, A109288, A109289.

Programs

  • Maple
    filter:= proc(n)
   if n::even then numtheory:-bigomega(n+1) <> 2 and numtheory:-bigomega(n-1) <> 2
   elif n mod 4 = 1 then not isprime(n) and not isprime((n+1)/2)
   else not isprime(n) and not isprime((n-1)/2)
   fi
end proc:
select(filter, [$4..200]); # Robert Israel, Apr 20 2021
  • Mathematica
    bo[n_] := Plus @@ Last /@ FactorInteger[n]; Select[Range[2, 200], ! (PrimeQ[ # ] || bo[ # - 1] == 2 || bo[ # + 1] == 2) &] (* Ray Chandler, Aug 27 2005 *)

Extensions

Corrected and extended by Ray Chandler, Aug 27 2005

A109289 Composite numbers which are not of the form p*q + 1, where p and q are distinct primes.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 18, 20, 21, 24, 25, 26, 28, 30, 32, 33, 38, 42, 44, 45, 46, 48, 49, 50, 51, 54, 55, 57, 60, 62, 64, 65, 68, 69, 72, 74, 76, 77, 80, 81, 82, 84, 85, 90, 91, 93, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 115, 117, 118, 121, 122, 125, 126, 128
Offset: 1

Views

Author

Giovanni Teofilatto, Aug 20 2005

Keywords

Crossrefs

Cf. A109287, A109288, A109290.

Programs

  • Mathematica
    Select[Range[2, 130], ! (PrimeQ[ # ] || (Plus @@ Last /@ FactorInteger[ # - 1] == 2 && Length[FactorInteger[ # - 1]] == 2)) &] (* Ray Chandler, Aug 25 2005 *)
fQ[n_] := Last /@ FactorInteger[n] != {1, 1}; Select[ Range[2, 128], !PrimeQ[ # ] && fQ[ # - 1] &] (* Robert G. Wilson v *)

Extensions

Extended by Robert G. Wilson v and Ray Chandler, Aug 25 2005

A076153 Numbers n such that Omega(n) = Omega(n-1)^2, where Omega(m) (A001222) denotes the number of prime factors of m, counted with multiplicity.

Original entry on oeis.org

3, 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, 999
Offset: 1

Views

Author

Joseph L. Pe, Oct 31 2002

Keywords

Examples

			Omega(996) = 2^2 = Omega(995)^2, so 996 is a term of the sequence.

Links

Crossrefs

Similar to A109287; after a(1)=3 next difference is a(101)=1792 whereas A109287(100)=1794. - Ray Chandler, Mar 20 2007
Cf. A001222.

Programs

Extensions

Edited by Ray Chandler, Mar 20 2007
Showing 1-8 of 8 results.

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