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.

Previous Showing 11-19 of 19 results.

A268186 Numbers n such that n^2 + 2, n^2 - 2, n + 2 and n - 2 are all semiprimes.

Original entry on oeis.org

12, 53, 84, 204, 207, 251, 379, 413, 456, 471, 483, 631, 687, 705, 765, 783, 1079, 1135, 1140, 1167, 1269, 1335, 1347, 1395, 1475, 1515, 1587, 1641, 1709, 1767, 1851, 1855, 1943, 1959, 2049, 2157, 2319, 2325, 2575, 2843, 2865, 3099, 3153, 3225, 3267, 3601, 3779
Offset: 1

Views

Author

K. D. Bajpai, Jan 28 2016

Keywords

Examples

			12 appears in the sequence because:
  12^2 + 2 = 146 = 2*73
  12^2 - 2 = 142 = 2*71
  12 + 2   = 14  = 2*7
  12 - 2   = 10  = 2*5 are all semiprimes.

Links

Crossrefs

Subsequence of A105571.
Cf. A001358, A086005, A096173, A096175, A105571, A124936, A237037, A268043.

Programs

  • Magma
    IsSemiprime:=func;[ n : n in [2..10000] | IsSemiprime(n^2 + 2) and  IsSemiprime(n^2 - 2) and  IsSemiprime(n + 2) and  IsSemiprime(n - 2)];
  • Maple
    with(numtheory): select(n -> (bigomega(n^2 + 2)=2 and bigomega(n^2 - 2)=2 and bigomega(n + 2)=2 and bigomega(n - 2)=2), [seq(n, n=1..10000)]);
  • Mathematica
    Select[Range[10000], PrimeOmega[#^2 + 2] == PrimeOmega[#^2 - 2] == PrimeOmega[# + 2] == PrimeOmega[# - 2] == 2 &]
  • PARI
    for(n = 1, 10000,if(bigomega(n^2 + 2) == 2 && bigomega(n^2 - 2) == 2  && bigomega(n + 2) == 2 && bigomega(n - 2) == 2, print1(n, ", ")))

A276564 Perfect powers k (exponent greater than 1) such that k-1 and k+1 are both semiprime.

Original entry on oeis.org

144, 216, 900, 1764, 2048, 3600, 10404, 11664, 39204, 97344, 213444, 248832, 272484, 360000, 656100, 685584, 1040400, 1102500, 1127844, 1633284, 2108304, 2214144, 3504384, 3802500, 4112784, 4536900, 4588164, 5475600, 7784100, 7851204, 8388608, 8820900, 9000000, 9734400
Offset: 1

Views

Author

Antonio Roldán, Nov 16 2016

Keywords

Comments

Intersection of A001597 and A124936. - Michel Marcus, Dec 03 2016

Examples

			2048 = 2^11, and both 2047 = 23*89 and 2049 = 3*683 are semiprimes.

Links

Crossrefs

Cf. A001358, A001597, A108278, A121898, A124936, A167023, A268043, A276565.

Programs

  • Mathematica
    Select[Range[10^7], And[GCD @@ FactorInteger[#][[All, 2]] > 1, Union@ # == {2} &@ Map[PrimeOmega, {# - 1, # + 1}]] &] (* Michael De Vlieger, Dec 07 2016, after Ant King at A001597 *)
  • PARI
    for(i=2,10^7,if(ispower(i)&&bigomega(i-1)==2&&bigomega(i+1)==2,print1(i,", ")))

A276565 Oblong numbers n such that n - 1 and n + 1 are both semiprime.

Original entry on oeis.org

56, 552, 870, 1056, 1190, 1640, 1892, 2652, 4032, 5256, 5402, 6806, 8372, 9120, 9506, 9702, 10920, 11772, 12656, 12882, 15006, 15252, 15500, 16256, 16770, 17556, 18632, 23256, 24492, 27722, 29070, 30800, 33306, 33672, 34410, 36290, 40200, 40602, 44310, 45582, 46872, 49506
Offset: 1

Views

Author

Antonio Roldán, Nov 16 2016

Keywords

Comments

Intersection of A002378 and A124936. - Michel Marcus, Nov 26 2016

Examples

			1640 is oblong (1640 = 40*41) and 1639 = 11*149, 1641 = 3*547 are both semiprime.

Links

Crossrefs

Cf. A002378, A124936.
Cf. A001597, A108278, A121898, A167023, A268043, A276564.

Programs

  • Maple
    select(t -> numtheory:-bigomega(t+1)=2 and numtheory:-bigomega(t-1)=2, [seq(i*(i+1),i=1..1000)]); # Robert Israel, Nov 28 2016
  • PARI
    for(i=1,250,n=i*(i+1);if(bigomega(n-1)==2&&bigomega(n+1)==2,print1(n,", ")))

A378627 Products of 6 distinct primes that are sandwiched between semiprime numbers.

Original entry on oeis.org

39270, 66990, 71610, 79170, 82110, 99330, 110670, 122430, 123690, 125970, 129030, 132090, 136290, 144690, 152490, 163590, 166530, 167790, 180642, 182910, 190190, 191730, 215670, 220110, 222222, 226590, 227766, 231990, 235410, 239190, 247170, 248710, 249690, 254562, 258258, 260130
Offset: 1

Views

Author

Massimo Kofler, Dec 02 2024

Keywords

Comments

All terms are even.
Not all terms are divisible by 6: the first that is not is a(21) = 190190. The first term that is deficient is a(1966) = 4739702. - Robert Israel, Feb 03 2025

Examples

			39270 is a term because 39270=2*3*5*7*11*17 is the product of six distinct primes, 39269=107*367 and 39271=173*227 are both semiprimes.
66990 is a term because 66990=2*3*5*7*11*29 is the product of six distinct primes, 66989=13*5153 and 66991=31*2161 are both semiprimes.

Links

Crossrefs

Intersection of A067885 and A124936.
Cf. A001358, A378097.

Programs

  • Maple
    with(priqueue):
children:= proc(t) local R,i,pp;
   R:= NULL:
   pp:= nextprime(t[6]);
   for i from 6 to 2 by -1 do
     R:= R, [t[1]*pp/t[i], op(t[2..i-1]),op(t[i+1..6]),pp];
     if t[i-1] <> prevprime(t[i]) then break fi;
   od;
   {R}
end proc:
Res:= NULL: count:= 0:
initialize(pq):
insert([-2*mul(ithprime(i),i=2..6),3,5,7,11,13],pq);
while count < 100 do
  t:= extract(pq);
  if numtheory:-bigomega(-t[1]-1) = 2 and numtheory:-bigomega(-t[1]+1) = 2 then
    Res:= Res, -t[1]; count:= count+1;
  fi;
  for tt in children(t) do insert(tt,pq) od:
od:
Res; # Robert Israel, Feb 03 2025
  • Mathematica
    SequencePosition[Array[FactorInteger[#][[;; , 2]] &, 270000] /. {2} -> {1, 1}, {{1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1}}][[;; , 1]] + 1 (* Amiram Eldar, Dec 02 2024 *)

Extensions

Edited by Robert Israel, Feb 03 2025

A157487 Numbers k such that k-1 and k+1 are each the product of exactly 7 primes, counted with multiplicity.

Original entry on oeis.org

10529, 15391, 17983, 18751, 22049, 23489, 24751, 26081, 29249, 32561, 35153, 43471, 49951, 52975, 58049, 58481, 67229, 67231, 70687, 71873, 72415, 76049, 77921, 79001, 79649, 82783, 83249, 85751, 88289, 93799, 95551, 97471, 102545
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

Keywords

Links

Crossrefs

Cf. A124936, A014612, A157483, A157484, A157485, A157486.

Programs

  • Maple
    with(numtheory): a := proc (n) if bigomega(n-1) = 7 and bigomega(n+1) = 7 then n else end if end proc: seq(a(n), n = 2 .. 120000); # Emeric Deutsch, Mar 07 2009
  • Mathematica
    q=7;lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==q&&Plus@@Last/@FactorInteger[n+1]==q,AppendTo[lst,n]],{n,9!}];lst
Select[Range[110000],PrimeOmega[#+{1,-1}]=={7,7}&] (* Harvey P. Dale, Apr 04 2015 *)
Mean/@SequencePosition[PrimeOmega[Range[105000]],{7,,7}] (* _Harvey P. Dale, Sep 10 2022 *)

Extensions

More terms from Emeric Deutsch, Mar 07 2009

A157489 Numbers n such that n-+5 are divisible by exactly 5 primes, counted with multiplicity.

Original entry on oeis.org

275, 373, 445, 755, 985, 1165, 1245, 1475, 1535, 1643, 1645, 1705, 1715, 1745, 2219, 2305, 2317, 2389, 2445, 2455, 2543, 2579, 2845, 2855, 2893, 3229, 3299, 3325, 3371, 3565, 3613, 3659, 3695, 3757, 3829, 3875, 4255, 4285, 4295, 4345, 4355, 4477, 4745, 5003, 5065
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

Keywords

Comments

Let a, b and 10 be pairwise coprime, with A001222(a) = A001222(b) = 4. There exists c such that c == 5 (mod a) and c == -5 (mod b). Dickson's conjecture implies that (c+k*a*b-5)/a and (c+k*a*b+5)/b are prime for infinitely many k; for such k, c+k*a*b is in the sequence. - Robert Israel, Mar 22 2020

Links

Crossrefs

Cf. A001222, A124936, A014612, A157483, A157484, A157485, A157486

Programs

  • Maple
    N:= 10^4: # for terms <= N
T5:= select(t -> numtheory:-bigomega(t)=5, {$1..N+5}):
S:= T5 intersect map(`+`,T5,10):
sort(convert(map(`-`,S,5),list)); # Robert Israel, Mar 22 2020
  • Mathematica
    q=5;lst={};Do[If[Plus@@Last/@FactorInteger[n-q]==q&&Plus@@Last/@FactorInteger[n+q]==q,AppendTo[lst,n]],{n,8!}];lst
SequencePosition[PrimeOmega[Range[5100]],{5,,,_,,,_,,,_,5}][[All,1]]+5 (* Harvey P. Dale, Sep 23 2021 *)

A166988 Products n of a square of a prime and a cube of a prime such that n-1 and n+1 are semiprimes.

Original entry on oeis.org

392, 14792, 19652, 48668, 55112, 197192, 291848, 783752, 908552, 963272, 1203052, 1541768, 1670792, 5081672, 5903048, 8193532, 9732872, 10089032, 10285412, 12241352, 13333448, 13960328, 14087432, 14818568, 15882248, 16290632
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Oct 26 2009

Keywords

Comments

Intersection of A143610 and A124936.

Examples

			392 = 7^2*2^3; 391 = 17*23 and 393 = 3*131 are semiprimes, hence 392 is in the sequence.
14792 = 2^3*43^2 is in the sequence because 14791=7*2113 and 14793=3*4931 are semiprimes.

Links

Crossrefs

Cf. A001248 (squares of primes), A030078 (cubes of primes), A001358 (semiprimes).

Programs

  • Mathematica
    f2[n_]:=Last/@FactorInteger[n]=={2,3}||Last/@FactorInteger[n]=={3,2}; f1[n_]:=Plus@@Last/@FactorInteger[n]==2; lst={};Do[If[f2[n],If[f1[n-1]&&f1[n+1],AppendTo[lst,n]]],{n,10!}];lst
With[{prs=Prime[Range[300]]},Union[Select[Times@@@Tuples[{prs^2, prs^3}], PrimeOmega[#-1] == PrimeOmega[#+1]==2&]]] (* Harvey P. Dale, Aug 13 2013 *)
  • PARI
    {m=17000000; v=[]; forprime(j=2, sqrtint(m\8), a=j^2; g=sqrtn(m\a, 3); forprime(k=2, g, n=a*k^3; if(nKlaus Brockhaus, Oct 29 2009

Extensions

Edited by Klaus Brockhaus and R. J. Mathar, Oct 28 2009

A276905 Numbers k such that k^5-1 and k^5+1 are semiprimes.

Original entry on oeis.org

12, 1452, 11352, 79398, 146520, 281622, 352110, 536778, 643302, 680988, 723492, 739200, 878988, 992112, 1115268, 1189650, 1397022, 1698378, 1698510, 1728540, 1806222, 2486220, 2873178, 3031578, 3571458, 3946140, 4467012, 4983858, 5064510, 5135658, 5567562, 5753352
Offset: 1

Views

Author

Gary E. Davis, Sep 21 2016

Keywords

Crossrefs

Cf. A001358, A108278, A124936, A268043.
Intersection of A104238 and A261435.

Programs

  • Mathematica
    upper=600000;
Select[Range[upper],
PrimeOmega[#^5 - 1] == PrimeOmega[#^5 + 1] == 2 &]
  • PARI
    isok(n) = (bigomega(n^5-1)==2) && (bigomega(n^5+1)==2); \\ Michel Marcus, Sep 22 2016

Extensions

More terms from Altug Alkan, Sep 30 2016

A358686 Numbers sandwiched between two semiprimes, one of which is a square.

Original entry on oeis.org

5, 50, 120, 122, 288, 290, 528, 842, 960, 1370, 1680, 1850, 2808, 2810, 4488, 5328, 5330, 6240, 6242, 6888, 6890, 9408, 9410, 11880, 12768, 18770, 22200, 22800, 26568, 27888, 36482, 38808, 39600, 52440, 54290, 58080, 63000, 63002, 69170, 72360, 72362, 73442, 76730, 78960
Offset: 1

Views

Author

Tanya Khovanova, Nov 26 2022

Keywords

Comments

Numbers in A124936 but not in A358665.
All numbers except 5 (the first term) are even.
Subsequence of A124936.

Examples

			5 is sandwiched between two semiprimes 4 = 2*2 and 6 = 3*2, one of which is a square. Thus, 5 is in this sequence.
34 is sandwiched between squarefree semiprimes 33 = 3*11 and 35 = 5*7. Thus, 34 is not in this sequence.

Links

Crossrefs

Cf. A001358, A006881, A124936, A358665.

Programs

  • Mathematica
    Select[Range[100000], Total[Transpose[FactorInteger[# - 1]][[2]]] == 2 && Total[Transpose[FactorInteger[# + 1]][[2]]] == 2 && ! (Transpose[FactorInteger[# - 1]][[2]] == {1, 1} && Transpose[FactorInteger[# + 1]][[2]] == {1, 1}) &]
Mean/@Select[SequencePosition[PrimeOmega[Range[80000]],{2,,2}],AnyTrue[Sqrt[#],IntegerQ]&] (* _Harvey P. Dale, Jun 14 2023 *)
  • Python
    from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    nxt = []
    yield 5
    for k in count(6, 2):
        prv, nxt = nxt, list(factorint(k+1).values())
        if (prv==[1, 1] and nxt==[2]) or (prv==[2] and nxt==[1, 1]): yield k
print(list(islice(agen(), 44))) # Michael S. Branicky, Nov 26 2022
Previous Showing 11-19 of 19 results.

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

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