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 11-18 of 18 results.

A096175 Numbers k such that k^3-1 is an odd semiprime.

Original entry on oeis.org

6, 8, 12, 14, 20, 24, 38, 54, 62, 80, 90, 110, 138, 150, 164, 168, 192, 194, 272, 278, 314, 332, 348, 398, 402, 434, 500, 572, 642, 644, 720, 728, 762, 798, 812, 860, 864, 878, 920, 992, 1020, 1022, 1070, 1092, 1098, 1118, 1130, 1182, 1202, 1230, 1260, 1308
Offset: 1

Views

Author

Hugo Pfoertner, Jun 22 2004

Keywords

Examples

			a(1)=6 because 6^3 - 1 = 216 - 1 = 215 = 5*43.

Links

Crossrefs

Cf. A096173: k^3+1 is an odd semiprime; A081257: largest prime factor of k^3-1; A096176 (k^3-1)/(k-1) is prime; A046315.

Programs

  • PARI
    forstep (k=2,1310,2,if(bigomega(k^3-1)==2,print1(k,", ")))
\\ Hugo Pfoertner, Nov 28 2017

A096174 Even numbers k such that (k^3+1)/(k+1) is prime.

Original entry on oeis.org

2, 4, 6, 16, 18, 22, 28, 34, 42, 58, 60, 70, 72, 76, 78, 90, 100, 102, 106, 112, 118, 120, 132, 142, 144, 148, 154, 156, 162, 168, 174, 190, 204, 210, 216, 232, 246, 280, 288, 294, 310, 330, 352, 358, 370, 382, 384, 396, 406, 436, 448, 454, 456, 490, 496, 526
Offset: 1

Views

Author

Hugo Pfoertner, Jun 20 2004

Keywords

Examples

			a(1)=2 because (2^3+1)/(2+1)=9/3=3 is prime, a(8)=34: (34^3+1)/(34+1)=39305/35=1123 is prime.

Links

Crossrefs

Cf. A055494.
Cf. A081256, A096173.

Programs

Extensions

Definition corrected by N. J. A. Sloane, Apr 08 2010

A237037 Numbers k such that (2*k)^3 + 1 is a semiprime.

Original entry on oeis.org

1, 2, 3, 8, 9, 11, 14, 21, 29, 30, 35, 36, 39, 50, 51, 53, 56, 74, 78, 81, 95, 105, 116, 140, 155, 165, 176, 179, 191, 198, 224, 228, 245, 284, 303, 336, 378, 393, 410, 413, 414, 428, 429, 438, 464, 485, 491, 504, 506, 515, 534, 546, 554, 575, 596, 611, 638, 641, 648, 659, 680, 683, 711, 714, 725, 744, 765, 774, 791
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Comments

Numbers k such that 2*k+1 and 4*k^2 - 2*k + 1 are both prime.
Same as k/2 such that k^3 + 1 is a semiprime, because then k must be even.

Examples

			(2*1)^3 + 1 = 9 = 3*3 is a semiprime, so a(1) = 1.

Links

Crossrefs

Cf. A001358, A046315, A081256, A096173, A096174, A237038, A237039, A237040.

Programs

  • Mathematica
    Select[Range[800], PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
Select[Range[800],PrimeOmega[(2#)^3+1]==2&] (* Harvey P. Dale, Nov 28 2024 *)

Formula

a(n) = A096173(n)/2 = (1/2)*(A237040(n)-1)^(1/3).

A268043 Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.

Original entry on oeis.org

6, 1092, 1932, 2730, 4158, 6552, 11172, 25998, 30492, 55440, 76650, 79632, 85092, 102102, 150990, 152082, 152418, 166782, 211218, 235662, 236208, 248640, 264600, 298410, 300300, 301182, 317772, 380310, 387198, 441798, 476028, 488418
Offset: 1

Views

Author

Vincenzo Librandi, Jan 25 2016

Keywords

Comments

Obviously, k+1 and k-1 are always prime numbers.
If k is a term then m = (k - 1) * (k^2 + k + 1) is a term of A169635, i.e., A001157(m) = A001157(m+2) (De Koninck, 2002). - Amiram Eldar, Apr 19 2024

Examples

			a(1) = 6 because 6^3-1 = 215 = 5*43 and 6^3+1 = 217 = 7*31, therefore 215, 217 are both semiprimes.

Links

Crossrefs

Subsequence of A014574 and A136242.
Cf. A002384, A055494, A088707, A096173, A096175, A109373.
Cf. A001157, A169635.

Programs

  • Magma
    IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..300000] | IsSemiprime(n^3+1) and IsSemiprime(n^3-1) ];
  • Mathematica
    Select[Range[500000], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 2 &]
Select[Range[10^6], And @@ PrimeQ[{# - 1, # + 1, #^2 - # + 1, #^2 + # + 1}] &] (* Amiram Eldar, Apr 19 2024 *)
  • PARI
    isok(n) = (bigomega(n^3-1) == 2) && (bigomega(n^3+1) == 2); \\ Michel Marcus, Jan 26 2016
  • PARI
    is(n) = isprime(n - 1) && isprime(n + 1) && isprime(n^2 - n + 1) && isprime(n^2 + n + 1); \\ Amiram Eldar, Apr 19 2024

A237038 Primes p such that (2*p)^3 + 1 is a semiprime.

Original entry on oeis.org

2, 3, 11, 29, 53, 179, 191, 491, 641, 659, 683, 1103, 1499, 1901, 2129, 2543, 2549, 3803, 3851, 4271, 4733, 4943, 5303, 5441, 6101, 6329, 6449, 7193, 7211, 8093, 8513, 9059, 9419, 10091, 10271, 10733, 10781, 11321, 12203, 12821, 13451, 14561, 15233, 15803, 17159, 17333, 18131, 19373, 19919
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Comments

Same as Sophie Germain primes p such that 4*p^2 - 2*p + 1 is also prime (because (2*p)^3 + 1 = (2*p + 1)(4*p^2 - 2*p + 1)).
Primes in A237037.
For n>1, 8*a(n)^3 is a solution for the equation phi(x+1) - phi(x) = x/2. - Farideh Firoozbakht, Dec 17 2014

Examples

			11 is prime and (2*11)^3 + 1 = 10649 = 23*463 is a semiprime, so 11 is a member.

Links

Crossrefs

Cf. A000010, A001358, A005384, A046315, A081256, A096173, A096174, A237037, A237039, A237040.

Programs

  • Mathematica
    Select[Range[20000], PrimeQ[#] && PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]
Select[Prime[Range[2500]],PrimeOmega[(2#)^3+1]==2&] (* Harvey P. Dale, Jun 28 2021 *)

Formula

a(n) = (1/2)*(A237039(n)-1)^(1/3).

A237039 Semiprimes of the form (2*p)^3 + 1, where p is prime.

Original entry on oeis.org

65, 217, 10649, 195113, 1191017, 45882713, 55742969, 946966169, 2106997769, 2289529433, 2548895897, 10735357817, 26946035993, 54958685609, 77199941513, 131561576057, 132495001193, 440016501017, 456888832409, 623273556089, 848202406697, 966188398457
Offset: 1

Views

Author

Jonathan Sondow, Feb 02 2014

Keywords

Examples

			(2*2)^3 + 1 = 65 = 5*13 is a semiprime, so a(1) = 2.

Links

Crossrefs

Cf. A001358, A046315, A081256, A096173, A096174, A237037, A237038, A237040.

Programs

  • Mathematica
    L = Select[Range[5000], PrimeQ[#] && PrimeQ[(2 #)^2 - 2 # + 1] && PrimeQ[2 # + 1] &]; (2 L)^3 + 1
Select[Table[(2p)^3+1,{p,Prime[Range[1000]]}],PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 21 2021 *)

Formula

a(n) = (2*A237038(n))^3 + 1.

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, ", ")))

A330508 Numbers k such that k + 6^t is semiprime for t = 0 to 9.

Original entry on oeis.org

61273, 109441, 160213, 274501, 275473, 311593, 360673, 394201, 477181, 486061, 514993, 522085, 617137, 620053, 715477, 725485, 803833, 812677, 847117, 1063585, 1146913, 1182577, 1215865, 1232917, 1409425, 1508113, 1587241, 1768993, 1863073, 1895413, 2085517, 2095177
Offset: 1

Views

Author

K. D. Bajpai, Dec 16 2019

Keywords

Comments

a(2620) = 530079693 is the first multiple of 3 in this sequence; there are no multiples of 2. - Charles R Greathouse IV, Dec 20 2019

Examples

			a(1) = 61273:
  61273 + 6^0  =    61274 =   2 *  30637;
  61273 + 6^1  =    61279 = 233 *    263;
  61273 + 6^2  =    61309 =  37 *   1657;
  61273 + 6^3  =    61489 =  17 *   3617;
  61273 + 6^4  =    62569 =  13 *   4813;
  61273 + 6^5  =    69049 =  29 *   2381;
  61273 + 6^6  =   107929 =  37 *   2917;
  61273 + 6^7  =   341209 =  11 *  31019;
  61273 + 6^8  =  1740889 = 197 *   8837;
  61273 + 6^9  = 10138969 =  89 * 113921;
all ten results are semiprime.

Links

Crossrefs

Subsequence of A076274.
Cf. A001358, A082919, A096173, A104238, A105041.

Programs

  • Magma
    f:=func; [k:k in [1..2100000]|forall{m:m in [0..9]|f(k+6^m)}]; // Marius A. Burtea, Dec 20 2019
  • Mathematica
    fX[n_] = PrimeOmega[n] == 2; Select[Range[2000000], AllTrue[# + 6^{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, fX] &]
  • PARI
    issemi(n)=bigomega(n)==2
is(n)=for(t=0,9, if(!issemi(n+6^t), return(0))); 1 \\ Charles R Greathouse IV, Dec 20 2019
Previous Showing 11-18 of 18 results.

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

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