A242719 -id:A242719 - cp's OEIS frontend

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

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

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

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

A246824 Numbers k for which A242720(k) = (prime(k)+1)^2 + 2.

Original entry on oeis.org

3, 35, 41, 52, 57, 81, 104, 209, 215, 343, 373, 398, 473, 477, 584, 628, 768, 774, 828, 872, 1117, 1145, 1189, 1287, 1324, 1435, 1615, 1634, 1653, 1704, 1886, 1925, 2070, 2075, 2123, 2171, 2193, 2425, 2449, 2605, 2633, 2934, 2948, 3019, 3194, 3273, 3533, 3552, 3685, 3758

Offset: 1

Vladimir Shevelev, Sep 04 2014

nonn

By a comment in A246748, A242720(k) >= (prime(k)+1)^2 + 2, and equality is attained in this sequence.

Prime(a(n)) >= 5 and is in the intersection of A001359 and A157468.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001359, A157468, A242719, A242720, A246748, A246819, A246821.

lpf[n_] := FactorInteger[n][[1, 1]]; aQ[n_] := Module[{k=6}, While[PrimeQ[k-3] && PrimeQ[k-1] || lpf[k-1]<=lpf[k-3] || lpf[k-3]Amiram Eldar, Dec 10 2018 *)

lpf(k) = factorint(k)[1, 1];
f(n) = my(k=6); while((isprime(k-3) && isprime(k-1)) || lpf(k-1)<=lpf(k-3) || lpf(k-3)A242720
isok(n) = f(n) == (prime(n)+1)^2 + 2; \\ Michel Marcus, Dec 10 2018

from sympy import prime, isprime, factorint
A246824_list = [a for a, b in ((n, prime(n)+1) for n in range(3,10**3)) if (not (isprime(b**2-1) and isprime(b**2+1)) and (min(factorint(b**2+1)) > min(factorint(b**2-1)) >= b-1))] # Chai Wah Wu, Jun 03 2019

a(40)-a(50) from b-file by Robert Price, Sep 08 2019

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

A246821 Max _{2<=k<=n} floor(A242720(k)/prime(k)) - prime(n).

Original entry on oeis.org

1, 2, 4, 8, 6, 8, 6, 14, 14, 16, 10, 14, 12, 26, 26, 20, 18, 12, 26, 37, 31, 27, 50, 42, 38, 36, 32, 30, 41, 27, 23, 27, 25, 15, 16, 22, 16, 26, 20, 14, 29, 19, 34, 30, 40, 40, 28, 24, 22, 18, 12, 10, 20, 20, 14, 8, 16, 10, 26, 41, 31, 17, 13, 11, 45, 31, 47

Offset: 2

Vladimir Shevelev, Sep 04 2014

nonn

Conjecture: a(n) = o(prime(n)), as n goes to infinity.

If the conjecture is true, then A242720(n) ~ prime(n)^2. Indeed, A242720(n) >= prime(n)^2 + 2*prime(n) + 3; on the other hand, by the conjecture, we have A242720(n)/prime(n) <= a(n) + 1 + prime(n) = prime(n)*(1+o(1)).

Peter J. C. Moses, Table of n, a(n) for n = 2..2501

Cf. A242719, A242720, A246748, A246819.

More terms from Peter J. C. Moses, Sep 04 2014

A243803 Number of numbers k in interval [(p_n)^2+1, (p_n)^4] for which lpf(k-3)>lpf(k-1)>=p_n, where p_n=prime(n) and lpf = A020639.

Original entry on oeis.org

40, 85, 393, 625, 1557, 2106, 4069, 9558, 11476, 22060, 31530, 35998, 49142, 76678, 113799, 125010, 176824, 216378, 234064, 313511, 372054, 481764, 668344, 768307, 811635, 926452, 975785, 1105924, 1751993, 1949976, 2299392, 2394921, 3130534, 3250605, 3751262, 4306910, 4683674, 5332960

Offset: 3

Vladimir Shevelev and Peter J. C. Moses, Jun 10 2014

nonn

a(n) and A243804(n) approximate each other with a small relative error.

Positions n for which a(n) < A243804(n) are 11, 13, 14, 17, 18, 19, 20 (...).

Cf. A020639 (lpf), A242719, A242720, A242847, A242758, A243804.

A243804 Number of numbers k in interval [(p_n)^2+1, (p_n)^4] for which lpf(k-1)>lpf(k-3)>=p_n, such that {k-3, k-1} is not a pair of twin primes, where p_n=prime(n) and lpf = A020639.

Original entry on oeis.org

36, 84, 382, 593, 1526, 2070, 4023, 9536, 11535, 22050, 31552, 36034, 49032, 76464, 113887, 125138, 176940, 216419, 233932, 313011, 371787, 480984, 666608, 767403, 811022, 925567, 974900, 1104796, 1749737, 1948447, 2298322, 2393928, 3129862, 3248932, 3750166, 4305141, 4682343, 5332158

Offset: 3

Vladimir Shevelev and Peter J. C. Moses, Jun 10 2014

nonn

a(n) and A243803(n) approximate each other with the relative error tending to zero with growth of n.

Cf. A020639, A242719, A242720, A242847, A242758, A243803.

A244343 Least even k such that sfdf(k-3) > sfdf(k-1) >= A050376(n), where sfdf(n) is the smallest Fermi-Dirac factor of n (A223490).

Original entry on oeis.org

16, 46, 46, 64, 100, 254, 326, 392, 392, 590, 776, 776, 1190, 1520, 1814, 2420, 2624, 3764, 3764, 3764, 4454, 4454, 4892, 5752, 6400, 6400, 7210, 9380, 9524, 11414, 11414, 13190, 13190, 13190, 18272, 18272, 19940, 20414, 20414, 21824, 24614, 24614, 25592

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Jun 26 2014

nonn

a(n) is a Fermi-Dirac analog of A242719.

			If k is even such that k-1 is either 1 or in A050376, then k cannot be required. Thus, if n=2, then k=2,4,6,8,10,12,14 are not required, while for k=16 we have sfdf(16-3) = 13 > sfdf(16-1) = 3 = A050376(2). So a(2)=16.

Cf. A050376, A223490, A242719.

cp's OEIS Frontend

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

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

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A246824 Numbers k for which A242720(k) = (prime(k)+1)^2 + 2.

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

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A246821 Max _{2<=k<=n} floor(A242720(k)/prime(k)) - prime(n).

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A243803 Number of numbers k in interval [(p_n)^2+1, (p_n)^4] for which lpf(k-3)>lpf(k-1)>=p_n, where p_n=prime(n) and lpf = A020639.

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