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.

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

Original entry on oeis.org

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617
Offset: 1

Views

Author

Clark Kimberling, Dec 15 2015

Keywords

Comments

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...).
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x Lower POBA Upper POBA POBA
1 A001359/A006512 A006512/A001359 A265759/A265760
3/2 A104163/A158708 A162336/A158709 A265761/A222565
2 A005383/A005382 A005385/A005384 A079149/A120628
3 A091180/A088878 A094525/A023208 A265763/A265764
4 A162857/A062737 A090866/A023212 A265765/A120639
5 A265766/A158318 A265767/A023217 A265768/A265769
6 A227756/A158015 A051644/A007693 A265770/A265771
sqrt(2) A265772/A265773 A265774/A265775 A265776/A265777
sqrt(3) A265778/A265779 A265780/A265781 A265782/A265783
sqrt(5) A265784/A265785 A265786/A265787 A265788/A265789
sqrt(8) A265790/A265791 A265792/A265793 A265794/A265795
tau A265796/A265797 A265798/A265799 A265800/A265801
1/tau A265799/A265798 A265797/A265796 A265806/A265807
pi A265808/A265809 A265810/A265811 A265812/A265813
e A265814/A265815 A265816/A265817 A265818/A265819

Examples

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Crossrefs

Cf. A000040, A001359, A006512, A265759, A265760.

Programs

  • Mathematica
    x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A245365 Semiprimes of the form n*(3*n-1)/2.

Original entry on oeis.org

22, 35, 51, 145, 247, 287, 1247, 1717, 2147, 2501, 3151, 4187, 5017, 7957, 11051, 13207, 15251, 16801, 17767, 20827, 26867, 33227, 49051, 63551, 68587, 71177, 76501, 81317, 96647, 112477, 118301, 128627, 147737, 159251, 182527, 232657, 237407, 241001, 250717
Offset: 1

Views

Author

K. D. Bajpai, Jul 19 2014

Keywords

Comments

Semiprimes among pentagonal numbers A000326 = { (3*n^2-n)/2; n >= 0 }.
We can have an odd prime n = 2k + 1 and (3n - 1)/2 = 3k + 1 also prime, i.e., k in A130800, or n = 2p with p prime and 3n - 1 = 6p - 1 also prime, i.e., p in A158015. Considering the ratio of the two prime factors, the two possibilities are mutually exclusive, so this is the disjoint union of {A033570(n)=(2n+1)(3n+1); n in A130800} = A255584 and {p*(6p-1); p in A158015}. - M. F. Hasler, Dec 13 2019

Examples

			n=6: (3*n^2-n)/2 = 51 = 3 * 17 which is semiprime. Hence, 51 appears in the sequence.
n=10: (3*n^2-n)/2 = 145 = 5 * 29 which is semiprime. Hence, 145 appears in the sequence.

Links

Crossrefs

Cf. A001358, A000326.

Programs

  • Mathematica
    Select[Table[(3*n^2 - n)/2, {n, 500}], PrimeOmega[#] == 2 &]
  • PARI
    select(n->bigomega(n)==2, vector(1000, n, (3*n^2-n)/2)) \\ Colin Barker, Jul 20 2014

A158017 Primes p such that 10*p-1 is also prime.

Original entry on oeis.org

2, 3, 11, 23, 41, 71, 83, 101, 107, 113, 149, 167, 179, 227, 239, 269, 311, 317, 347, 353, 389, 479, 491, 521, 557, 569, 587, 647, 653, 683, 809, 821, 827, 839, 863, 911, 977, 983, 1091, 1229, 1259, 1283, 1289, 1301, 1367, 1373, 1439, 1487, 1493, 1607, 1619
Offset: 1

Views

Author

Roger L. Bagula, Mar 11 2009

Keywords

Comments

The family of prime sequences that generate primes k*p-1 for k = 2, 4, 6, 8, ... also comprises A005382 (k=2), A062737 (k=4), A158015 (k=6), and A158016 (k=8).

Links

Crossrefs

Cf. A005382, A062737.

Programs

  • Magma
    [p: p in PrimesUpTo(3000)|IsPrime(10*p-1)] // Vincenzo Librandi, Jan 29 2011
  • Mathematica
    Flatten[Table[If[PrimeQ[n] && PrimeQ[10*n - 1], n, {}], {n, 1, 10000}]]
Select[Prime[Range[600]], PrimeQ[(10 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

A158016 Primes p such that 8*p-1 is also prime.

Original entry on oeis.org

3, 13, 19, 61, 79, 103, 163, 181, 193, 223, 229, 313, 331, 349, 409, 433, 439, 541, 571, 613, 619, 691, 751, 769, 853, 859, 919, 991, 1021, 1033, 1039, 1321, 1381, 1423, 1483, 1543, 1549, 1621, 1699, 1759, 1801, 1861, 1873, 1879, 1951, 1999, 2011, 2029, 2113
Offset: 1

Views

Author

Roger L. Bagula, Mar 11 2009

Keywords

Links

Crossrefs

Cf. A005382 for the type 2p-1, A062737 for 4p-1, A158015 for 6p-1, A158017 for 10p-1.

Programs

  • Magma
    [p: p in PrimesUpTo(2200) | IsPrime(8*p - 1)]; // Vincenzo Librandi, Apr 14 2013
  • Mathematica
    Select[Prime[Range[600]], PrimeQ[(8 # - 1)]&] (* Vincenzo Librandi, Apr 14 2013 *)

Extensions

Edited by the Associate Editors of the OEIS, Apr 22 2009

A265770 Numerators of primes-only best approximates (POBAs) to 6; see Comments.

Original entry on oeis.org

13, 11, 19, 17, 31, 29, 43, 41, 67, 79, 103, 101, 113, 139, 137, 173, 223, 257, 283, 281, 317, 353, 367, 401, 439, 499, 607, 619, 617, 643, 641, 653, 677, 761, 787, 823, 821, 907, 941, 977, 1039, 1087, 1181, 1193, 1361, 1373, 1399, 1433, 1447, 1543, 1579
Offset: 1

Views

Author

Clark Kimberling, Dec 20 2015

Keywords

Comments

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

Examples

			The POBAs to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.

Crossrefs

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265771.

Programs

  • Mathematica
    x = 6; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265770/A265771 *)
Numerator[tL]   (* A227756 *)
Denominator[tL] (* A158015 *)
Numerator[tU]   (* A051644 *)
Denominator[tU] (* A007693 *)
Numerator[y]    (* A222570 *)
Denominator[y]  (* A265771 *)

A265771 Denominators of primes-only best approximates (POBAs) to 6; see Comments.

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 7, 7, 11, 13, 17, 17, 19, 23, 23, 29, 37, 43, 47, 47, 53, 59, 61, 67, 73, 83, 101, 103, 103, 107, 107, 109, 113, 127, 131, 137, 137, 151, 157, 163, 173, 181, 197, 199, 227, 229, 233, 239, 241, 257, 263, 269, 271, 277, 283, 283, 293, 311
Offset: 1

Views

Author

Clark Kimberling, Dec 20 2015

Keywords

Comments

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

Examples

			The POBAs to 6 start with 13/2, 11/2, 19/3, 17/3, 31/5, 29/5, 43/7, 41/7, 67/11, 79/13, 103/17, 101/17. For example, if p and q are primes and q > 17, then 103/17 (and 101/17) is closer to 6 than p/q is.

Crossrefs

Cf. A000040, A265759, A227756, A158015, A051644, A007693, A265770.

Programs

  • Mathematica
    x = 6; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265770/A265771 *)
Numerator[tL]   (* A227756 *)
Denominator[tL] (* A158015 *)
Numerator[tU]   (* A051644 *)
Denominator[tU] (* A007693 *)
Numerator[y]    (* A222570 *)
Denominator[y]  (* A265771 *)

A223881 Denominators in the expression m!/(prime(m-1)+1) for m > 1 such that this expression is not an integer.

Original entry on oeis.org

3, 2, 19, 31, 37, 79, 41, 97, 53, 139, 71, 157, 83, 199, 211, 229, 131, 271, 137, 307, 331, 337, 173, 367, 379, 197, 439, 227, 499, 263, 547, 281, 577, 293, 197, 199, 601, 607, 619, 661, 227, 229, 691, 239, 727, 383, 269, 811, 829, 283, 431, 877, 467, 937, 313
Offset: 1

Views

Author

Alexander R. Povolotsky, May 01 2013

Keywords

Comments

It appears that all terms are primes.
From Alexander R. Povolotsky, Apr 26 2025: (Start)
The scatter plot reveals four distinct, well-separated, monotonically increasing curves. It became possible to extract the integers (all conjectured to be primes) corresponding to each of the four subsets.
Additionally, the approximation formulas for each of the four subsets were derived.
These four approximation formulas, given in the exponential form y=C_k*x^m were found to have a common slope: m=1.197311990 while their displacement coefficients are: C_1≈6.86845, C_2≈3.42058, C_3≈2.28335, C_4≈1.70460.
Notably, these displacement coefficients values exhibit a clear pattern: C_2≈C_1/2, C_3≈C_1/3, C_4≈C_1/4. (For instance, 3.42058≈6.86845/2, and so on.)
Above approximations were derived using general separation and approximation methods and do not specifically account for the fact that these values correspond to the prime numbers.
It appears that all primes in the groups 4, 2 and 1 are generated by the 6*k+1 formula, and so primes in the above groups constitute three subsets of A002476 terms, while all primes in the group 3 are generated by the 2*k+1 formula, and so primes in that group constitute a subset of the terms presented in A000040.
Also it appears that:
1. The first group constitutes a sequence, such that for n>=1, a(n) = A005382(n+6).
2. The third group constitutes a sequence, such that for n>1, a(n) = A158015(n+20).
3. The fourth group constitutes a sequence, such that for n>=1, a(n) = A158016(n+32).
The text files containing the primes, corresponding to the above discussed four groups, where primes are indexed against their position in the complete primes listing (see OEIS's A000040), are viewable and downloadable at the below links section. (End)

Links

Crossrefs

Cf. A007528, A024898, A002476, A091178.

Programs

  • Mathematica
    Denominator[Select[Table[m!/(Prime[m - 1] + 1), {m, 2, 300}], ! IntegerQ[#] &]] (* T. D. Noe, May 03 2013 *)
  • PARI
    m=M=1;forprime(p=2,1e5,M*=m++;t=denominator(M/(p+1)); if(t>1, print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

A289109 Primes p that remain prime through 3 iterations of function f(x) = 6x - 1.

Original entry on oeis.org

239, 269, 439, 569, 599, 829, 1429, 3389, 6379, 7159, 7649, 8779, 8969, 10799, 10939, 12919, 13729, 13879, 15649, 17159, 18149, 19379, 21649, 22669, 23929, 24799, 25679, 26849, 28219, 30389, 30689, 33749, 34759, 36109, 36209, 36899, 40759, 47659, 49639, 52369
Offset: 1

Views

Author

K. D. Bajpai, Jun 24 2017

Keywords

Comments

All the terms are congruent to 9 (mod 10). The iteration of f(x) on a term of this sequence then produces primes congruent to 3, 7, 1 (mod 10), followed by a nontrivial multiple of 5.

Examples

			239 is prime and 6 * 239 - 1 = 1433, which is also prime. 6 * 1433 - 1 = 8597, which is also prime. 6 * 8597 = 51581, which is also prime. 6 * 51581 - 1 = 309485 = 5 * 11 * 17 * 331, which is composite, but the previous three primes are enough for 239 to be in the sequence.
241 is not in the sequence because 6 * 241 - 1 = 1445 = 5 * 17^2, which is composite.

Links

Crossrefs

Cf. A057326, A057327, A057328, A057329, A057330, A158015.

Programs

  • Maple
    filter:= x -> andmap(isprime, [x,6*x-1,36*x-7,216*x-43]):
select(filter, [seq(i,i=9..60000,10)]); # Robert Israel, May 10 2020
  • Mathematica
    Select[Prime[Range[15000]], And @@ PrimeQ[NestList[6 # - 1 &, #, 3]] &]
  • PARI
    forprime(p= 1, 100000, if(isprime(6*p-1) && isprime(36*p-7) && isprime(216*p-43) , print1(p, ", ")));

A330409 Semiprimes of the form p(6p - 1).

Original entry on oeis.org

22, 51, 145, 287, 1717, 2147, 3151, 5017, 11051, 13207, 16801, 20827, 26867, 63551, 68587, 71177, 76501, 96647, 112477, 147737, 159251, 232657, 237407, 308947, 314417, 342487, 433897, 480251, 587501, 602617, 722107, 772927, 834401, 861467, 879751, 907537, 945257, 1155887, 1177051
Offset: 1

Views

Author

M. F. Hasler, Dec 13 2019

Keywords

Examples

			A158015(1) = 2 is the smallest prime p such that 6p - 1 = 12 - 1 = 11 is also prime, whence a(1) = A049452(2) = 2*(6*2 - 1) = 22.
prime(5) = 11 is the smallest prime not in A024898 or A158015, because 6p - 1 is not a prime, therefore A049452(11) = 11*(6*11 - 1) is not in the sequence, and idem for A049452(13).
But prime(7) = 17 is in A024898 and A158015, so a(5) = A024898(A158015(5)) = A024898(17) = 17*(6*17 - 1).

Links

Crossrefs

Cf. A024898 (6n-1 is prime), A158015 (primes), A049452 = {n(6n-1)}.
Complement of A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)) in A245365 (primes of the form n(3n-1)/2).

Programs

  • Mathematica
    Select[Table[p(6p-1),{p,500}],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 27 2022 *)
  • PARI
    [p*(6*p-1) | p<-primes(99), isprime(6*p-1)]

Formula

a(n) = A049452(A158015(n)) = p(6p - 1) with p = A158015(n).

A330410 a(n) = 6*prime(n) - 1.

Original entry on oeis.org

11, 17, 29, 41, 65, 77, 101, 113, 137, 173, 185, 221, 245, 257, 281, 317, 353, 365, 401, 425, 437, 473, 497, 533, 581, 605, 617, 641, 653, 677, 761, 785, 821, 833, 893, 905, 941, 977, 1001, 1037, 1073, 1085, 1145, 1157, 1181, 1193, 1265, 1337, 1361, 1373, 1397, 1433, 1445
Offset: 1

Views

Author

M. F. Hasler, Dec 13 2019

Keywords

Comments

Composite terms are a(k) with k in {5, 6, 11, 12, 13, 18, 20, 21, ...} = indices of primes missing in A158015. Primes are A016969(A158015 - 1).

Crossrefs

Cf. A000040 (primes), A016969 (6n+5), A024898 (6n-1 is prime), A158015 (primes in A024898), A049452 = {n(6n-1)}, A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)), A245365 (primes of the form n(3n-1)/2).

Programs

  • PARI
    apply( a(n)=6*prime(n)-1, [1..99])
  • PARI
    apply( n->6*n-1, primes(99))

Formula

a(n) = A016969(A000040(n)-1) = 6p - 1 with p = A000040(n) = prime(n).
Showing 1-10 of 10 results.

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