A247011 -id:A247011 - 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

A245363 Numbers n for which A242719(n) = prime(n)*(prime(n)+6) + 1.

Original entry on oeis.org

19, 37, 49, 69, 73, 102, 165, 236, 253, 365, 465, 542, 595, 694, 713, 723, 762, 920, 962, 979, 1119, 1162, 1259, 1334, 1387, 1441, 1706, 1797, 1843, 1906

Offset: 1

Vladimir Shevelev, Sep 09 2014

nonn

These are such numbers n for which prime(n), prime(n)+6, prime(n)*(prime(n)+6)-2 are primes, but prime(n) is not in A062326; besides, if prime(n) is in A001359, then also prime(n+1) is not in A062326.

Prime(n)*(prime(n)+6) + 1 is the third minimal possible value of A242719(n) after prime(n)^2 + 1 and (prime(n)+2)^2 + 1 (cf. A247011 and comment there).

Cf. A242719, A246748, A247011.

prime(n) == 7 (mod 10), A242719(n) == 92 (mod 100).

More terms from Peter J. C. Moses, Sep 09 2014

A247280 Numbers n for which A242720(n) = prime(n)*(prime(n)+4)+3.

Original entry on oeis.org

4, 6, 8, 19, 50, 59, 63, 65, 78, 85, 93, 112, 117, 143, 237, 254, 264, 276, 287, 303, 333, 371, 380, 425, 435, 440, 447, 459, 483, 485, 537, 612, 614, 659, 731, 851, 877, 920, 983, 994, 1025, 1080, 1096, 1182, 1358, 1380, 1468, 1476, 1481, 1582, 1628, 1690

Offset: 1

Vladimir Shevelev, Sep 11 2014

nonn

prime(n)*(prime(n)+4) + 3, such that prime(n)+4 and prime(n)*(prime(n)+4)+2 are primes, is the second minimal possible value of A242720(n) after (prime(n)+1)^2 + 2, n>=3 (cf. A246824).

Cf. A242719, A242720, A242847, A245363, A246748, A246824, A247011.

More terms from Peter J. C. Moses, Sep 11 2014

A247549 Numbers n for which A242720(n) = prime(n)*(prime(n)+8)+3.

Original entry on oeis.org

5, 32, 43, 79, 126, 142, 523, 576, 722, 771, 1026, 1152, 1234, 1402, 1442, 1480, 1623, 1630, 1767, 1814, 1829, 1962, 1995, 2062, 2084, 2353, 2705, 3104, 3174, 3355, 3588, 3718, 4005, 4035, 4126, 4266, 4581, 4616, 4785, 4854, 4859, 5068, 5131, 5145, 5164

Offset: 1

Vladimir Shevelev, Sep 19 2014

nonn

prime(n)*(prime(n)+8) + 3, such that prime(n)+8 and prime(n)*(prime(n)+8)+2 are primes, is the third minimal possible value of A242720(n) after (prime(n)+1)^2 + 2, n>=3 (cf. A246824) and prime(n)*(prime(n)+4) + 3 (cf. A247280).

Cf. A242719, A242720, A242847, A245363, A246748, A246824, A247011, A147280.

If n is in the sequence, then prime(n) == 1 (mod 10), A242720(n) == 12 (mod 100).

More terms from Peter J. C. Moses, Sep 19 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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A245363 Numbers n for which A242719(n) = prime(n)*(prime(n)+6) + 1.

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A247280 Numbers n for which A242720(n) = prime(n)*(prime(n)+4)+3.

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A247549 Numbers n for which A242720(n) = prime(n)*(prime(n)+8)+3.

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