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 10 results.

A256381 Numbers n such that n-3 and n+3 are semiprimes.

Original entry on oeis.org

7, 12, 18, 36, 52, 54, 88, 90, 118, 126, 158, 180, 206, 212, 216, 218, 250, 256, 262, 292, 298, 302, 306, 324, 326, 332, 338, 358, 368, 374, 410, 414, 448, 450, 508, 514, 530, 532, 540, 548, 556, 562, 576, 586, 594, 626, 632, 652, 682, 684, 692, 700, 710, 720
Offset: 1

Views

Author

Michel Marcus, Mar 27 2015

Keywords

Comments

All but the first term are even.

Crossrefs

Cf. A001358 (semiprimes).
Cf. A124936 (n-1 and n+1), A105571 (n-2 and n+2).
Cf. A256382 (n-4 and n+4), A256383 (n-5 and n+5).

Programs

  • Magma
    IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..750] | IsSemiprime(n+3) and IsSemiprime(n-3) ]; // Vincenzo Librandi, Mar 28 2015
  • Mathematica
    Select[Range[750], PrimeOmega[# + 3] == PrimeOmega[# - 3] == 2 &] (* Vincenzo Librandi, Mar 28 2015 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,800}],{1,,,_,,,1}][[All,1]]+3 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 21 2017 *)
  • PARI
    lista(nn,m=3) = {for (n=m+1, nn, if (bigomega(n-m)==2 && bigomega(n+m)==2, print1(n, ", ")););}

A256382 Numbers n such that n-4 and n+4 are semiprimes.

Original entry on oeis.org

10, 18, 29, 30, 42, 53, 61, 73, 78, 81, 89, 90, 91, 115, 119, 125, 137, 138, 162, 165, 173, 181, 198, 205, 209, 210, 213, 217, 222, 258, 263, 291, 295, 299, 305, 323, 325, 330, 331, 390, 399, 407, 411, 441, 449, 450, 462, 477, 485, 489, 493, 497, 501, 515, 523
Offset: 1

Views

Author

Michel Marcus, Mar 27 2015

Keywords

Comments

A117328 is the subsequence of primes.

Links

Crossrefs

Cf. A001358 (semiprimes).
Cf. A117328 (with primes rather than semiprimes).
Cf. A124936 (n-1 and n+1), A105571 (n-2 and n+2).
Cf. A256381 (n-3 and n+3), A256383 (n-5 and n+5).

Programs

  • Magma
    IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [6..750] | IsSemiprime(n+4) and IsSemiprime(n-4) ]; // Vincenzo Librandi, Mar 29 2015
  • Mathematica
    Select[Range[600], PrimeOmega[# + 4] == PrimeOmega[# - 4] == 2 &] (* Vincenzo Librandi, Mar 29 2015 *)
Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,600}],9,1],?(#[[1]]==#[[9]]==1&),{1},Heads->False]]+4 (* _Harvey P. Dale, Mar 29 2015 *)
  • PARI
    lista(nn,m=4) = {for (n=m+1, nn, if (bigomega(n-m)==2 && bigomega(n+m)==2, print1(n, ", ")););}

A256383 Numbers n such that n-5 and n+5 are semiprimes.

Original entry on oeis.org

9, 20, 30, 44, 60, 82, 90, 116, 124, 128, 138, 150, 164, 182, 208, 210, 214, 242, 254, 294, 296, 300, 304, 314, 324, 334, 360, 366, 376, 386, 398, 408, 412, 422, 432, 442, 476, 506, 510, 522, 524, 532, 538, 540, 548, 578, 584, 586, 628, 674, 676, 684
Offset: 1

Views

Author

Michel Marcus, Mar 27 2015

Keywords

Comments

It appears that there are no primes in this sequence.
If n is odd, one of n+5 and n-5 is divisible by 4, so unless n = 9 it can't be a semiprime. Thus all terms except 9 are even. - Robert Israel, Apr 13 2020

Links

Crossrefs

Cf. A001358 (semiprimes).
Cf. A124936 (n-1 and n+1), A105571 (n-2 and n+2).
Cf. A256381 (n-3 and n+3), A256382 (n-4 and n+4).

Programs

  • Magma
    IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [6..700] | IsSemiprime(n+5) and IsSemiprime(n-5) ]; // Vincenzo Librandi, Mar 29 2015
  • Maple
    N:= 1000: # for terms <= N-5
PP:= select(isprime, {seq(i,i=3..N/3,2)}):
P:= select(`<=`,PP,floor(sqrt(N))):
SP:= {}:
for p in P do
  PP:= select(`<=`,PP,N/p);
  SP:= SP union map(`*`,PP,p);
od:
R:= {9} union (map(`+`,SP,5) intersect map(`-`,SP,5)):
sort(convert(R,list)); # Robert Israel, Apr 13 2020
  • Mathematica
    Select[Range[2, 700], PrimeOmega[# + 5] == PrimeOmega[# - 5] == 2 &] (* Vincenzo Librandi, Mar 29 2015 *)
  • PARI
    lista(nn,m=5) = {for (n=m+1, nn, if (bigomega(n-m)==2 && bigomega(n+m)==2, print1(n, ", ")););}
  • PARI
    issemi(n)=bigomega(n)==2
list(lim)=my(v=List([9])); forprime(p=5,(lim-5)\3, if(issemi(3*p+10), listput(v,3*p+5))); forprime(p=29,(lim+5)\3, if(issemi(3*p-10), listput(v,3*p-5))); forstep(n=30,lim\=1,6, if(issemi(n-5) && issemi(n+5), listput(v, n))); Set(v) \\ Charles R Greathouse IV, Apr 13 2020

A105572 Numbers m such that m-3 and m+3 have 3 prime factors.

Original entry on oeis.org

15, 47, 73, 95, 102, 113, 127, 150, 151, 167, 168, 185, 233, 239, 241, 258, 276, 282, 287, 289, 313, 319, 335, 360, 366, 407, 409, 415, 426, 431, 432, 433, 439, 480, 521, 527, 552, 558, 593, 599, 601, 606, 607, 612, 642, 648, 649, 654, 655, 660, 708, 713
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 14 2005

Keywords

Comments

A001222(a(n)-3) = A001222(a(n)+3) = 3.
Prime factors counted with multiplicity. - Harvey P. Dale, May 07 2023

Examples

			From _Jon E. Schoenfield_, Jan 19 2015: (Start)
73 - 3 = 70 = 2 * 5 * 7 and 73 + 3 = 76 = 2 * 2 * 19 so 73 is in the sequence.
81 - 3 = 78 = 2 * 3 * 13 but 81 + 3 = 84 = 2 * 2 * 3 * 7 so 81 is not in the sequence. (End)

Links

Crossrefs

Cf. A014574, A105571, A105573.

Programs

A105573 Numbers m such that m-4 and m+4 have 4 prime factors.

Original entry on oeis.org

20, 136, 200, 224, 344, 346, 368, 376, 480, 488, 554, 568, 576, 640, 736, 744, 808, 816, 854, 856, 864, 872, 880, 944, 992, 1064, 1152, 1160, 1200, 1208, 1214, 1216, 1280, 1288, 1312, 1360, 1426, 1568, 1576, 1606, 1621, 1648, 1656, 1664, 1784, 1808, 1858
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 14 2005

Keywords

Comments

A001222(a(n)-4) = A001222(a(n)+4) = 4.
Prime factors counted with multiplicity. - Harvey P. Dale, Feb 06 2020

Examples

			1214 - 4 = 1210 = 2 * 5 * 11 * 11 and 1214 + 4 = 1218 = 2 * 3 * 7 * 29 so 1214 is in the sequence. - _Jon E. Schoenfield_, Jan 19 2015

Links

Crossrefs

Cf. A014574, A105571, A105572.

Programs

A256389 Numbers n such that one or more primes can be the arithmetic mean of 2 semiprimes whose difference is 2*n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110
Offset: 1

Views

Author

Michel Marcus, Mar 27 2015

Keywords

Comments

That is, there are several primes p, such that p+n and p-n are both semiprime.
Complement of A256387.
The terms of this sequence that do not belong to A256388 are even.

Examples

			A256381 is the list of numbers n such that n-3 and n+3 are semiprimes, and it contains a single prime, hence 3 is in the sequence.
A256382 is the list of numbers n such that n-4 and n+4 are semiprimes, and it contains several primes, hence 4 is in the sequence.

Crossrefs

Cf. A124936, A105571, A256382.
Cf. A256387 (no prime), A256388 (a single prime).

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, ", ")))

A278022 Squares s such that s + 2 and s - 2 are semiprime.

Original entry on oeis.org

36, 121, 144, 289, 1764, 2304, 2601, 2809, 4225, 5184, 5329, 6241, 6889, 7056, 8100, 9409, 13225, 15625, 17689, 19881, 23409, 25281, 27225, 30276, 34969, 35721, 38025, 40401, 41616, 42849, 43681, 45369, 63001, 69696, 72361, 78961, 82369, 84681, 87025, 93636, 106929
Offset: 1

Views

Author

K. D. Bajpai, Nov 08 2016

Keywords

Comments

Intersection of A000290 and A105571.

Examples

			a(1) = 36 appears in the list because 36 = 6^2 (square) and 36 + 2 = 38 = 19*2; 36 - 2 = 34 = 17*2; both are semiprime.
a(2) = 121 appears in the list because 121 = 11^2 (square) and 121 + 2 = 123 = 41*3;121 - 2 = 119 = 17*7; both are semiprime.

Links

Crossrefs

Cf. A000290, A001358, A105571.

Programs

  • Mathematica
    Select[Table[n^2, {n, 1000}], PrimeOmega[# + 2] == 2 &&  PrimeOmega[# - 2] == 2 &]
Select[Range[350]^2,PrimeOmega[#+{2,-2}]=={2,2}&] (* Harvey P. Dale, Jun 14 2021 *)
  • PARI
    for(n = 1, 1000, s = n^2; if( bigomega(s+2)==2 && bigomega(s-2)==2, print1(s, ", ")))

A275023 Cubes c such that c + 2 and c - 2 are semiprime.

Original entry on oeis.org

8, 216, 1331, 2197, 4913, 9261, 15625, 35937, 59319, 68921, 117649, 185193, 421875, 531441, 658503, 704969, 1030301, 1367631, 3723875, 5268024, 5359375, 11390625, 13651919, 16581375, 17779581, 19902511, 23149125, 25672375, 29503629, 36264691, 38958219, 40353607
Offset: 1

Views

Author

K. D. Bajpai, Nov 12 2016

Keywords

Comments

Intersection of A000578 and A105571.

Examples

			a(1) = 8 = 2^3. Also, 8 + 2 = 10 = 2*5; 8 - 2 = 6 = 2*3; both are semiprime.
a(2) = 216 = 6^3. Also, 216 + 2 = 218 = 2*109; 216 - 2 = 214 = 2*107; both are semiprime.

Links

Crossrefs

Cf. A000578, A001358, A105571, A278022.

Programs

  • Mathematica
    Select[Table[n^3, {n, 1000}], PrimeOmega[# + 2] == 2 && PrimeOmega[# - 2] == 2 &]
  • PARI
    for (n = 1, 1000, s = n^3; if(bigomega (s+2) == 2 && bigomega (s-2) == 2, print1 (s, ", ")))

A382049 Numbers k such that k +- 2 and k +- 3 are all semiprimes.

Original entry on oeis.org

12, 36, 216, 540, 1044, 4284, 6336, 11304, 17640, 30276, 31284, 34056, 35496, 35820, 37836, 41796, 46080, 47664, 50940, 57240, 62244, 71064, 75096, 80856, 84924, 98820, 100044, 103536, 106344, 143100, 143424, 144936, 149220, 159264, 159804, 162036, 168120, 172584, 175176, 177624, 194760, 195300
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Mar 13 2025

Keywords

Comments

Except for a(1) = 12, these are the numbers k divisible by 36 with (k +- 2)/2 and (k +- 3)/3 prime.

Examples

			a(3) = 216 is a term because 216 - 3 = 213 = 3 * 71, 216 - 2 = 215 = 2 * 107, 216 + 2 = 218 = 2 * 109, and 216 + 3 = 219 = 3 * 73 are all semiprimes.

Links

Crossrefs

Cf. A001358, A105571.

Programs

  • Maple
    filter:= proc(n) andmap(isprime, [(n-3)/3, (n-2)/2,(n+2)/2,(n+3)/3]) end proc:
filter(12):= true:
S:= select(filter, [12, seq(i,i=36 .. 10^6, 36)]):
  • Mathematica
    Select[Range[12, 200000], 2 ==  PrimeOmega[# - 2] ==
PrimeOmega[# + 2] ==  PrimeOmega[# - 3] == PrimeOmega[# + 3] &]
Showing 1-10 of 10 results.

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