A088878 -id:A088878 - cp's OEIS frontend

A091181 A091180 indexed by A000040.

4, 6, 8, 11, 12, 19, 29, 31, 34, 37, 42, 46, 47, 63, 68, 75, 80, 93, 95, 100, 105, 106, 115, 133, 136, 138, 141, 145, 151, 157, 159, 167, 169, 175, 187, 197, 210, 211, 217, 221, 222, 232, 233, 247, 257, 263, 274, 275, 279, 306, 308, 327, 335, 337, 339, 355, 365

Offset: 1

Ray Chandler, Dec 27 2003

Cf. A088878, A091179, A091180.

a(n) = k such that A000040(k) = A091180(n).

a(n) = A000720(A091180(n)). - Michel Marcus, Aug 06 2021

Offset changed to 1 by Jinyuan Wang, Aug 06 2021

A175566 a(n) = the least prime p such that (2k + 1)p - 2*k, k=1..n are all prime.

Original entry on oeis.org

3, 3, 7, 1171, 64591, 25153801, 25153801, 4747505071

Offset: 1

Zak Seidov, Jul 11 2010

nonn

First primes p such that (2k+1)p-2k k=1..7 are
25153801,507079861,610817971,942962791,1040241511,1223511871,1797884761.
First primes p such that (2k+1)p-2k k=1..8 are
4747505071,10817047081,21071155561,41279978041,44304317821,
49710893611,58845917971,79925475841,96466884361,106120099471,107001847261.

Cf. A088878 Prime numbers p such that 3p-2 is a prime.

A177336 Greater of twin primes p such that 3*p-2 is also greater of twin primes.

Original entry on oeis.org

5, 7, 61, 271, 1951, 3001, 6361, 11491, 11551, 14551, 18541, 19891, 21841, 31081, 32911, 32971, 33331, 33601, 42571, 42841, 50461, 53551, 58111, 68881, 70201, 74611, 79231, 80911, 93811, 96331, 98911, 104311, 109141, 114601, 121021, 125791

Offset: 1

Juri-Stepan Gerasimov, May 07 2010

nonn

			a(1) = 5 because 5 is the greater of the twin primes (3, 5) and 3*5 - 2 = 13 is the greater of the twin primes (11, 13).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006512, A038869, A088878.

Cf. A132929, A174920.

[p:p in PrimesInInterval(3,130000)| IsPrime(p-2) and IsPrime(3*p-2) and IsPrime(3*p-4)]; // Marius A. Burtea, Dec 23 2019

Select[Range[3, 126000], And @@ PrimeQ[{#, # - 2, 3# - 2, 3# - 4}] &] (* Amiram Eldar, Dec 23 2019 *)

From Wesley Ivan Hurt, May 03 2022: (Start)
a(n) = A132929(n) + 1.
a(n) = A174920(n) + 2. (End)

Definition corrected, 1231 and 1483 inserted, and all values above 3000 corrected by R. J. Mathar, May 10 2010
Terms corrected to match definition by D. S. McNeil, May 10 2010
Name corrected by Amiram Eldar, Dec 23 2019

A265763 Numerators of primes-only best approximates (POBAs) to 3; see Comments.

Original entry on oeis.org

7, 5, 17, 13, 23, 19, 31, 41, 37, 53, 59, 71, 67, 89, 113, 109, 131, 127, 139, 157, 179, 181, 199, 211, 239, 251, 269, 293, 311, 307, 337, 383, 379, 409, 419, 449, 491, 487, 503, 499, 521, 541, 571, 577, 593, 599, 631, 683, 701, 719, 751, 773, 769, 787, 809

Offset: 1

Clark Kimberling, Dec 18 2015

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.

			The POBAs for 3 start with 7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.

Cf. A000040, A023208, A088878, A091180, A094525, A222565.

x = 3; 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, A265763/A265764 *)
Numerator[tL]   (* A091180 *)
Denominator[tL] (* A088878 *)
Numerator[tU]   (* A094525 *)
Denominator[tU] (* A023208 *)
Numerator[y]    (* A265763 *)
Denominator[y]  (* A265764 *)

A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.

Original entry on oeis.org

2, 2, 5, 5, 7, 7, 11, 13, 13, 17, 19, 23, 23, 29, 37, 37, 43, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 103, 113, 127, 127, 137, 139, 149, 163, 163, 167, 167, 173, 181, 191, 193, 197, 199, 211, 227, 233, 239, 251, 257, 257, 263, 269, 271, 277, 293

Offset: 1

Clark Kimberling, Dec 18 2015

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.

			The POBAs for 3 start with  7/2, 5/2, 17/5, 13/5, 23/7, 19/7, 31/11, 41/13, 37/13, 53/17. For example, if p and q are primes and q > 13, then 41/13 is closer to 3 than p/q is.

Cf. A000040, A023208, A088878, A091180, A094525, A265763.

x = 3; 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, A265763/A265764 *)
Numerator[tL]   (* A091180 *)
Denominator[tL] (* A088878 *)
Numerator[tU]   (* A094525 *)
Denominator[tU] (* A023208 *)
Numerator[y]    (* A265763 *)
Denominator[y]  (* A265764 *)

A114828 Numbers k such that the k-th octagonal number has 9 prime factors counted with multiplicity.

Original entry on oeis.org

64, 96, 128, 144, 162, 182, 198, 216, 224, 234, 246, 270, 278, 288, 304, 310, 320, 324, 352, 390, 414, 416, 432, 438, 480, 504, 528, 544, 550, 558, 584, 594, 600, 646, 648, 654, 662, 684, 694, 702, 710, 729, 750, 752, 756, 798, 810, 834, 850, 870, 888, 900

Offset: 1

Jonathan Vos Post, Feb 19 2006

k has at most 8 prime factors counted with multiplicity.

			a(1) = 64 because OctagonalNumber(64) = Oct(64) = 64*(3*64-2) = 12160 = 2^7 * 5 * 19 has exactly 9 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 96 because Oct(96) = 96*(3*96-2) = 27456 = 2^6 * 3 * 11 * 13 is 9-almost prime [also 27456 = Oct(96) = Oct(Oct(6)) is an iterated octagonal number].
a(3) = 128 because Oct(128) = 128*(3*128-2) = 48896 = 2^8 * 191.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A088878, A114606, A114618, A114621, A114634, A114635, A114636.

A000567:=func< n | n*(3*n-2) >; Is9almostprime:=func< n | &+[k[2]: k in Factorization(n)] eq 9 >; [ n: n in [2..1000] | Is9almostprime(A000567(n)) ]; // Klaus Brockhaus, Dec 22 2010

Select[Range[900],PrimeOmega[PolygonalNumber[8,#]]==9&] (* James C. McMahon, Jul 30 2024 *)

isok(k) = bigomega(k*(3*k-2)) == 9; \\ Michel Marcus, Aug 02 2024

Integers k such that k*(3*k-2) has exactly nine prime factors (with multiplicity).
Integers k such that A000567(k) is a term of A046312.
Integers k such that A001222(A000567(k)) = 9.
Integers k such that A001222(k) + A001222(3*k-2) = 9.
Integers k such that (3*k-2)*(3*k-1)*(3*k)/((3*k-2)+(3*k-1)+(3*k)) is in A046310.

Missing terms inserted by R. J. Mathar, Dec 22 2010
a(40)-a(52) from James C. McMahon, Jul 30 2024
Name edited by David A. Corneth, Jul 31 2024

A163628 Integers such that the two adjacent integers are a prime and three times a prime.

Original entry on oeis.org

8, 10, 14, 16, 20, 22, 32, 38, 40, 52, 58, 68, 70, 88, 110, 112, 128, 130, 140, 158, 178, 182, 200, 212, 238, 250, 268, 292, 308, 310, 338, 380, 382, 410, 418, 448, 488, 490, 500, 502, 520, 542, 572, 578, 592, 598, 632, 682, 700, 718, 752, 770, 772, 788, 808

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2009

nonn

Union[3*A023208 + 1, 3*A088878 - 1]. [Zak Seidov, Aug 07 2009]

			a(1)=8 which lies between 7=A000040(4) and 9 = A001748(2).
a(2)=10 which lies between 9=A001748(2) and 11 = A000040(5).

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000040, A163492.

n = 1; A023208 = {}; Do[If[PrimeQ[(Prime[k] - 2 n)/(2 n + 1)], AppendTo[A023208, (Prime[k] - 2 n)/(2 n + 1)]], {k, 1, 1000}]; A023208;
A088878 = {}; Do[p = Prime[n]; If[PrimeQ[3*p - 2], AppendTo[A088878, p]], {n, 5!}]; A088878; Union[3*A023208 + 1, 3*A088878 - 1] (* G. C. Greubel, Jul 30 2017 *)

Many terms like 44, 46, 62 etc. removed by R. J. Mathar, Aug 06 2009

A173269 2prime(prime(n))-3 and 3prime(prime(n))-2 are both primes.

Original entry on oeis.org

1, 2, 3, 8, 11, 14, 15, 19, 23, 24, 28, 39, 44, 47, 54, 62, 63, 81, 85, 101, 121, 122, 124, 136, 152, 159, 180, 218, 219, 241, 247, 253, 274, 290, 298, 307, 323, 324, 341, 361, 371, 376, 381, 403, 410, 413, 441, 443, 479, 487, 499, 552, 554, 556, 562, 582, 622

Offset: 1

Juri-Stepan Gerasimov, Feb 14 2010

nonn

			a(1)=1 because 2*p(p(1))-3=2*p(2)-3=2*3-3=3=prime and 3*p(p(1))-2=7=prime; a(2)=2 because 2*p(p(2))-3=2*p(3)-3=2*5-3=7=prime and 3*p(p(2))-2=13=prime; a(3)=3 because 2*p(p(3))-3=2*p(5)-3=2*11-3=19=prime and 3*p(p(3))-2=31=prime; a(4)=8 because 2*p(p(8))-3=2*p(19)-3=2*67-3=131=prime and 3*p(p(8))-2=199=prime.

Cf. A063908, A088878.

Inserted 23 and 24, removed 34, extended the sequence - R. J. Mathar, Mar 01 2010

A173286 2prime(prime(prime(n)))-3 and 3prime(prime(prime(n)))-2 are both primes.

Original entry on oeis.org

1, 2, 5, 8, 9, 15, 26, 53, 63, 86, 92, 93, 95, 116, 137, 152, 233, 254, 281, 303, 329, 334, 352, 386, 392, 415, 423, 460, 470, 476, 508, 565, 570, 601, 660, 673, 680, 725, 748, 898, 907, 942, 948, 952, 958, 1045, 1119, 1126, 1138, 1140, 1259, 1314, 1360

Offset: 1

Juri-Stepan Gerasimov, Feb 15 2010

nonn

			a(1) = 1 because 2*p(p(p(1)))-3 = 7 = prime and 3*p(p(p(1)))-2 = 13 = prime;
a(2) = 2 because 2*p(p(p(2)))-3 = 19 = prime and 3*p(p(p(2)))-2 = 31 = prime;
a(3) = 5 because 2*p(p(p(5)))-3 = 379 = prime and 3*p(p(p(5)))-2 = 251 = prime;
a(4) = 8 because 2*p(p(p(8)))-3 = 991 = prime and 3*p(p(p(8)))-2 = 659 = prime;
a(5) = 9 because 2*p(p(p(9)))-3 = 1291 = prime and 3*p(p(p(9)))-2 = 859 = prime;
a(6) = 15 because 2*p(p(p(15)))-3 = 3889 = prime and 3*p(p(p(15)))-2 = 2591 = prime.

Cf. A038580, A063908, A088878.

pppQ[n_]:=Module[{p=Prime[Prime[Prime[n]]]},AllTrue[{2p-3,3p-2},PrimeQ]]; Select[Range[1400],pppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 25 2016 *)

isok(n) = isprime(2*prime(prime(prime(n)))-3) && isprime(3*prime(prime(prime(n)))-2); \\ Michel Marcus, Sep 02 2013

Extended beyond 15 by R. J. Mathar, Mar 01 2010

A120637 Primes such that their triple is 2 away from a prime number.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 103, 113, 127, 137, 139, 149, 163, 167, 173, 181, 191, 193, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 271, 277, 293, 307, 313, 317, 331, 337, 347, 349, 353, 367, 373, 383

Offset: 1

Cino Hilliard, Aug 17 2006

This sequence is a variation of the sequence in the reference. However, this sequence should have an infinite number of terms.

			19 is a prime and 19*3 = 57 which is two away from 59 which is prime.
31 is not in the table because 31*3 = 93 which is 2 away from 91 and 95, both not prime.

R. Crandall and C. Pomerance, Prime Numbers A Computational Perspective, Springer Verlag 2002, p. 49, exercise 1.18.

Cf. A125272.

Select[Prime[Range[200]],PrimeQ[3#+2]||PrimeQ[3#-2]&] (* Harvey P. Dale, Aug 10 2011 *)

primepm2(n,k) { local(x,p1,p2,f1,f2,r); if(k%2,r=2,r=1); for(x=1,n, p1=prime(x); p2=prime(x+1); if(isprime(p1*k+r)||isprime(p1*k-r), print1(p1",") ) ) }

Union of A023208 and A088878.

cp's OEIS Frontend

A091181 A091180 indexed by A000040.

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A175566 a(n) = the least prime p such that (2*k + 1)*p - 2*k, k=1..n are all prime.

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A177336 Greater of twin primes p such that 3*p-2 is also greater of twin primes.

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A265763 Numerators of primes-only best approximates (POBAs) to 3; see Comments.

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A265764 Denominators of primes-only best approximates (POBAs) to 3; see Comments.

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A114828 Numbers k such that the k-th octagonal number has 9 prime factors counted with multiplicity.

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A163628 Integers such that the two adjacent integers are a prime and three times a prime.

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A173269 2*prime(prime(n))-3 and 3*prime(prime(n))-2 are both primes.

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A173286 2*prime(prime(prime(n)))-3 and 3*prime(prime(prime(n)))-2 are both primes.

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A175566 a(n) = the least prime p such that (2k + 1)p - 2*k, k=1..n are all prime.

A173269 2prime(prime(n))-3 and 3prime(prime(n))-2 are both primes.

A173286 2prime(prime(prime(n)))-3 and 3prime(prime(prime(n)))-2 are both primes.