A242720 -id:A242720 - cp's OEIS frontend

A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).

10, 26, 50, 170, 170, 362, 362, 842, 842, 1370, 1370, 1850, 1850, 2210, 3722, 3722, 3722, 4892, 5042, 7082, 7922, 7922, 7922, 10610, 10610, 10610, 11450, 13844, 16130, 16130, 17162, 19322, 19322, 24614, 24614, 25592, 29504, 29930, 29930, 36020, 36020

Offset: 2

Vladimir Shevelev, May 21 2014

nonn

The sequence is connected with a sufficient condition for the existence of an infinity of twin primes. In contrast to A242489, this sequence is nondecreasing.
All even numbers of the form A062326(n)^2 + 1 are in the sequence. All a(n)-1 are semiprimes. - Vladimir Shevelev, May 24 2014
a(n) <= A242489(n); a(n) >= prime(n)^2+1. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014
Conjecture: all numbers a(n)-3 are primes. Peter J. C. Moses verified this conjecture up to a(2001) (cf. with conjecture in A242720). - Vladimir Shevelev, Jun 09 2014

Jinyuan Wang, Table of n, a(n) for n = 2..5000 (terms 2..2001 from Peter J. C. Moses).

Cf. A001359, A006512, A062326, A137291, A242489, A242490, A242847, A243960, A245363, A246501, A246748, A246819, A247011.

lpf[k_] := FactorInteger[k][[1, 1]];
a[n_] := a[n] = For[k = If[n == 2, 10, a[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
Array[a, 50, 2] (* Jean-François Alcover, Nov 06 2018 *)

lpf(k) = factorint(k)[1,1];
vector(50, n, k=6; while(lpf(k-3)<=lpf(k-1) || lpf(k-1)Colin Barker, Jun 01 2014

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246819). - Vladimir Shevelev, Sep 02 2014

a(n) = prime(n)^2 + 1 for and only for numbers n>=2 which are in A137291. - Vladimir Shevelev, Sep 04 2014

A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).

Original entry on oeis.org

6, 8, 14, 14, 20, 20, 32, 32, 32, 44, 44, 44, 62, 62, 62, 62, 74, 74, 74, 104, 104, 104, 104, 104, 104, 110, 110, 140, 140, 140, 140, 140, 152, 152, 182, 182, 182, 182, 182, 182, 194, 194, 200, 200, 230, 230, 230, 230, 242, 242, 242, 272, 272, 272, 272, 272

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

This is a version of A242720 with the absolute minima of k in the definition. The sequence is nondecreasing. Hypothetically, every pair {a(n)-3, a(n)-1} is a pair of twin primes.
If there exist infinitely many n such that a(n) < A242719(n) < a(n)^2, then from the result in the Shevelev link, it follows that for such n the set of numbers {even k: lpf(k-1) > lpf(k-3) >= prime(n)} either attains the absolute minimum of a(n) only in the case when {a(n)-3, a(n)-1} are twin primes, or does not attain it at all. Therefore, if there is only a finite number of twin primes, we have a contradiction. Thus the above condition is sufficient for infinity of twin primes.
Note also that, if there is only a finite number of twin primes, then after the last pair of them, this sequence will coincide with A242720. Then, in order to avoid a contradiction (again according to the Shevelev link), we should accept that there exists a number N_0 such that, for every n >= N_0, the following inequality holds: max(A242719(n),A242720(n)) > (min(A242719(n),A242720(n)))^2. - Vladimir Shevelev, May 24 2014
It is easy to prove that min(A242719(n), A242720(n)) >= prime(n)^2+1, while we conjecture that max(A242719(n), A242720(n)) <= prime(n)^4. Thus this conjecture implies there are infinitely many twin primes. - Vladimir Shevelev, Jun 01 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..10001
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A001359, A006512, A242489, A242490, A242720.

lpf[k_] := FactorInteger[k][[1, 1]];
a[n_] := a[n] = For[k = If[n == 2, 2, a[n-1]], True, k = k+2, If[lpf[k-1] > lpf[k-3] >= Prime[n], Return[k]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Nov 03 2018 *)

lpf(k) = factorint(k)[1,1];
vector(100, n, k=6; while(lpf(k-1)<=lpf(k-3) || lpf(k-3)Colin Barker, Jun 01 2014

A243937 Even numbers n>=6 for which lpf(n-1) > lpf(n-3), where lpf = least prime factor.

Original entry on oeis.org

6, 8, 12, 14, 18, 20, 24, 30, 32, 36, 38, 42, 44, 48, 54, 60, 62, 66, 68, 72, 74, 78, 80, 84, 90, 96, 98, 102, 104, 108, 110, 114, 120, 122, 126, 128, 132, 138, 140, 144, 150, 152, 156, 158, 162, 164, 168, 174, 180, 182, 186, 188, 192, 194, 198, 200, 204, 210

Offset: 1

Vladimir Shevelev, Jul 10 2014

nonn

Complement of A245024 over even n >= 6.
Conjecture: All differences are 2, 4 or 6 such that there are no two consecutive terms 2 (..., 2, 2, ...), no two consecutive terms 4, while consecutive terms 6 occur 1, 2, 3 or 4 times; also consecutive pairs of terms 2, 4 appear 1, 2, 3 or 4 times. The conjecture is verified up to n = 2.5*10^7. - Vladimir Shevelev and Peter J. C. Moses, Jul 11 2014
Divisibility by 3 means 6m is in the sequence for all m > 0, and 6m + 4 never is, while 6m + 2 is undetermined. Divisibility by 5 means 30m + 8 is always in the sequence, and 30m + 26 never is. This proves the above conjecture. - Jens Kruse Andersen, Aug 19 2014
Note that,
1) Since numbers of the form 6*k evidently are in the sequence, then the counting function of the terms not exceeding x is not less than x/6.
2) Sequence {a(n)-1} contains all primes greater than 3 in the natural order. The subsequence of other terms of {a(n)-1} is 35, 65, 77, 95, ... - Vladimir Shevelev, Jul 15 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A242719, A242720, A245024.

select(n->factor(n-1)[1,1]>factor(n-3)[1,1], vector(200, x, 2*x+4)) \\ Jens Kruse Andersen, Aug 19 2014

More terms from Peter J. C. Moses, Jul 10 2014

A245024 Even numbers n for which lpf(n-1) < lpf(n-3), where lpf = least prime factor.

Original entry on oeis.org

10, 16, 22, 26, 28, 34, 40, 46, 50, 52, 56, 58, 64, 70, 76, 82, 86, 88, 92, 94, 100, 106, 112, 116, 118, 124, 130, 134, 136, 142, 146, 148, 154, 160, 166, 170, 172, 176, 178, 184, 190, 196, 202, 206, 208, 214, 220, 226, 232, 236, 238, 244, 250, 254, 256, 260

Offset: 1

Vladimir Shevelev, Jul 10 2014

nonn

By the definition, either a(n)==1 (mod 3) or, for every pair of primes (p,q), p>q>=3, a(n)==1 (mod p) and a(n) not==3 (mod q).
Conjecture: All differences are 2,4 or 6 such that no two consecutive terms 2 (...,2,2,...), no two consecutive terms 4, while consecutive terms 6 occur 1,2,3 or 4 times; also consecutive pairs of terms 4,2 appear 1,2,3 or 4 times.
Conjecture is verified up to n = 2.5*10^7. - Vladimir Shevelev and Peter J. C. Moses, Jul 11 2014
The first comment is wrong as stated. This would fix it: for every pair of primes (p,q), p>q>=3, if a(n)==1 (mod p) then a(n) not==3 (mod q). Divisibility by 3 means 6m+4 is in the sequence for all m>0, and 6m never is, while 6m+2 is undetermined. Divisibility by 5 means 30m+26 is always in the sequence, and 30m+8 never is. This proves the above conjecture. - Jens Kruse Andersen, Jul 13 2014
Note that the sequence {a(n)-3} contains all odd primes, except for lesser primes in twin primes pairs (A001359). Other terms of {a(n)-3} are 25,49,55,85,91,... - Vladimir Shevelev,  Jul 15 2014

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

Cf. A020639, A242719, A242720, A243937.

lpf:= n -> min(numtheory:-factorset(n)):
select(n -> lpf(n-1) < lpf(n-3),[seq(2*k,k=3..1000)]); # Robert Israel, Jul 15 2014

lpf[n_] := FactorInteger[n][[1, 1]];
Reap[For[n = 6, n <= 300, n += 2, If[lpf[n-1] < lpf[n-3], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 25 2019 *)

More terms from Peter J. C. Moses, Jul 10 2014

A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

Original entry on oeis.org

4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840

Offset: 1

Labos Elemer, Apr 23 2002

Essentially the same as A129293. - R. J. Mathar, Jun 14 2008
Solutions to the equation: A000005(n^4-1) = 8. - Enrique Pérez Herrero, May 03 2012
Terms > 6 are multiples of 30. Subsequence of A070689. - Zak Seidov, Nov 12 2012
{a(n)-1} is a subsequence of A157468; for n>1, {a(n)^2+2} is a subsequence of A242720. - Vladimir Shevelev, Aug 31 2014

			150 is a term since 149, 151 and 22501 are all primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000720, A001359, A006512, A014574, A002496, A005574, A070156, A070689, A129293, A157468, A242720.

select(n -> isprime(n-1) and isprime(n+1) and isprime(n^2+1), [seq(2*i,i=1..10000)]); # Robert Israel, Sep 02 2014

Do[s=n; If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[1+s^2], Print[n]], {n, 1, 1000000}]
Select[Range[22000],AllTrue[{#+1,#-1,#^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 19 2014 *)

is(k) = isprime(k-1) && isprime(k+1) && isprime(k^2+1); \\ Amiram Eldar, Apr 15 2024

For n>1, a(n)^2 = A242720(pi(a(n)-2)) - 2, where pi(n) is the prime counting function (A000720). - Vladimir Shevelev, Sep 02 2014

A242767 Numbers of repetitions of terms in A242758.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

If {pA242758. If this number occurs k times in A242758, then we say that k is the index of the pair of twin primes {p,q} with p in A001359.

Is this the same as A027833 shifted by two indices? - R. J. Mathar, May 23 2014

Peter J. C. Moses, Table of n, a(n) for n = 2..5001

Cf. A001359, A006512, A242489, A242490, A242720, A242758.

From the construction of A242758, in supposition of an infinity of twin primes, we have a(2)=1; for n>=3, a(n) = A027833(n-2). Otherwise, A027833 is finite, while A242758 will coincide with A242720 after the last pair of twin primes. - Vladimir Shevelev, May 26 2014

More terms from Peter J. C. Moses, May 22 2014

A137291 Numbers m such that prime(m)^2-2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 18, 20, 24, 27, 28, 31, 32, 34, 40, 43, 47, 48, 51, 52, 55, 62, 65, 68, 72, 82, 86, 87, 91, 94, 99, 100, 104, 107, 111, 119, 123, 128, 129, 130, 132, 133, 134, 135, 139, 141, 150, 152, 170, 172, 177, 180, 182, 191, 200, 202, 209, 211

Offset: 1

Ctibor O. Zizka, Apr 05 2008

For m>=1, for these and only these numbers m, A242719(m) = prime(m)^2 + 1. Since A242719(m) >= prime(m)^2 + 1, then the equality is obtained on this and only this sequence. - Vladimir Shevelev, Sep 04 2014

			prime(24)*prime(24)-2 = 89*89-2 = 7919 is prime, so n=24 belongs to the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A062326, A242719, A242720, A246824, A010051, A049001, A103960, A210481.

a137291 n = a137291_list !! (n-1)
a137291_list = filter ((== 1) . a010051' . a049001) [1..]
-- Reinhard Zumkeller, Jul 30 2015

Select[Range[211],PrimeQ[Prime[#]^2-2]&] (* James C. McMahon, May 28 2025 *)

is(n,p=prime(n))=isprime(p^2-2) \\ Charles R Greathouse IV, Feb 17 2017

A103960(a(n)) - A210481(a(n)) = 1. - Reinhard Zumkeller, Jul 30 2015

a(n) = A049084(A049002(n)). - R. J. Mathar, Apr 09 2008

More terms from R. J. Mathar, Apr 09 2008

Offset corrected by Reinhard Zumkeller, Jul 30 2015

A242768 Indices of twin primes in {A242758(n)-3}.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 6, 6, 6, 6, 6, 6, 2, 2, 5, 5, 5, 5, 5, 2, 2, 6, 6, 6, 6, 6, 6, 2, 2, 2, 2, 4, 4, 4, 4, 3, 3, 3, 5, 5, 5, 5, 5, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 2, 2, 6

Offset: 2

Vladimir Shevelev, May 22 2014

nonn

For the definition of the index of a twin primes pair, see the comment in A242767.
The sequence is constructed as follows. Consider the sequence {A242758(n)-3}. It begins 3,5,11,11,17,17,29,29,29,...
These numbers occur in A001359 (lesser of twin primes) at the indices 1,2,3,3,4,4,5,5,5,...
We add 1: 2,3,4,4,5,5,6,6,6,... (since in A001359 n>=1, while in A242767 n>=2). Now A242767{2,3,4,4,5,5,6,6,6,...} = {1,1,2,2,2,2,3,3,3,...}: we obtain this sequence.

Peter J. C. Moses, Table of n, a(n) for n = 2..5001

Cf. A001359, A006512, A242489, A242490, A242720, A242758, A242767.

More terms from Peter J. C. Moses, May 23 2014

A244570 Smallest prime p >= prime(n) such that p*prime(n)+2 is prime.

Original entry on oeis.org

3, 7, 11, 19, 17, 37, 23, 37, 43, 47, 47, 61, 53, 73, 109, 79, 89, 71, 109, 227, 113, 97, 139, 137, 127, 149, 127, 131, 283, 137, 139, 181, 173, 151, 167, 191, 173, 193, 193, 181, 239, 199, 233, 223, 239, 251, 239, 241, 233, 313, 241, 251, 271, 277, 373, 271

Offset: 2

Vladimir Shevelev, Jun 30 2014

nonn

See comment in A244572.

Jens Kruse Andersen, Table of n, a(n) for n = 2..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014. Sections 10,14.

Cf. A242719, A242720.

a[n_] := For[p = Prime[n], True, p = NextPrime[p], If[PrimeQ[p Prime[n] + 2], Return[p]]];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Nov 18 2018 *)
sp[n_]:=Module[{p=n},While[CompositeQ[p*n+2],p=NextPrime[p]];p]; Table[ sp[k],{k,Prime[Range[2,60]]}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 19 2020 *)

s=[]; for(n=2, 100, p=q=prime(n); while(!isprime(p*q+2), p=nextprime(p+1)); s=concat(s, p)); s \\ Colin Barker, Jun 30 2014

More terms from Peter J. C. Moses, Jun 30 2014

A244571 Smallest prime p >= prime(n) such that p*prime(n)-2 is prime.

Original entry on oeis.org

3, 5, 7, 23, 13, 23, 19, 41, 29, 61, 37, 59, 43, 47, 71, 107, 61, 73, 71, 97, 109, 113, 89, 157, 113, 103, 107, 127, 167, 127, 131, 149, 139, 179, 163, 163, 181, 227, 173, 251, 199, 191, 223, 257, 229, 211, 223, 233, 271, 233, 239, 271, 281, 257, 443, 389, 373

Offset: 2

Vladimir Shevelev, Jun 30 2014

nonn

See comment in A244572.

Jens Kruse Andersen, Table of n, a(n) for n = 2..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014. Sections 10, 14.

Cf. A244570, A242719, A242720.

a[n_] := For[p = Prime[n], True, p = NextPrime[p], If[PrimeQ[p Prime[n] - 2], Return[p]]];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Nov 18 2018 *)

s=[]; for(n=2, 100, p=q=prime(n); while(!isprime(p*q-2), p=nextprime(p+1)); s=concat(s, p)); s \\ Colin Barker, Jun 30 2014

More terms from Peter J. C. Moses, Jun 30 2014

cp's OEIS Frontend

A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).

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A242758 Smallest even k such that lpf(k-1) > lpf(k-3) >= prime(n), where lpf=least prime factor (A020639).

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A243937 Even numbers n>=6 for which lpf(n-1) > lpf(n-3), where lpf = least prime factor.

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A245024 Even numbers n for which lpf(n-1) < lpf(n-3), where lpf = least prime factor.

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A070155 Numbers k such that k-1, k+1 and k^2+1 are prime numbers.

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A242767 Numbers of repetitions of terms in A242758.

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A137291 Numbers m such that prime(m)^2-2 is prime.

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A242768 Indices of twin primes in {A242758(n)-3}.