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.

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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

Views

Author

Vladimir Shevelev, Sep 11 2014

Keywords

Comments

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).

Crossrefs

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

Extensions

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

A242974 Let M_n = A002110(n) (the n-th primorial), let N*(n)(N**(n), respectively) be the number of numbers k in [1, M_n] for which lpf(k-3) > lpf(k-1) >= prime(n) (lpf(k-1) > lpf(k-3) >= prime(n), respectively) such that k-3, k-1 are not twin primes, where lpf=least prime factor. Then a(n) = N*(n) - N**(n).

Original entry on oeis.org

1, 1, 3, 25, 67, 131, 1556, -1671
Offset: 3

Views

Author

Vladimir Shevelev, Jun 13 2014

Keywords

Comments

Small values of |a(n)| with respect to N*(n) + N**(n) (cf. A243867) clearly demonstrate the fact of statistical closeness of N*(n) and N**(n). See also comment in A243867.
If we don't exclude twin primes in the definition then, instead of this sequence, we would obtain -3, -14, -66, -443, -4569, -57422, -894506, -18465384, ... (cf. A000882). Thus twin primes strongly destroy the statistical closeness of N*(n) and N**(n).

Links

Crossrefs

Cf. A020639, A243867, A242719, A242720, A242758, A243803, A243804.

Programs

  • PARI
    lpf(k) = factorint(k)[1, 1];
a(n) = {my(p=prime(n), r=1, s=2, t, u=0); for(k=4, prod(i=1, n, prime(i)), if((t=lpf(k-1))>r, if(r>=p&&(r=p, u++)); r=s; s=t); u; } \\ Jinyuan Wang, Mar 13 2020

Extensions

More terms from Peter J. C. Moses, Jun 13 2014

A243867 Sum of the numbers N*(n) and N**(n) in A242974.

Original entry on oeis.org

1, 7, 97, 1289, 20611, 365775, 7813466, 212149365
Offset: 3

Views

Author

Vladimir Shevelev, Jun 13 2014

Keywords

Links

Crossrefs

Cf. A242974, A242719, A242720, A242758, A243803, A243804.

Formula

Let B(n) be the number of twin primes pairs not exceeding the n-th primorial M_n = A002110(n). Then we know that B(n) = O(M_n/(log(M_n))^2) = o(M_n/log((p_(n-1)))^2. For sufficiently large n, a(n) + B(n) >= 0.416...*M_n/(log(prime(n-1)))^2 (cf. Shevelev link) and thus for large n, for example, we have a(n) >= 0.4*M_n/(log(prime(n-1)))^2.

Extensions

More terms from Peter J. C. Moses, Jun 13 2014

A246502 a(n) is the smallest term of A242720 that is repeated exactly n times, or 0 if there is no such term.

Original entry on oeis.org

12, 440, 8054, 129554, 227432, 7986230, 62015624, 280729964
Offset: 1

Views

Author

Vladimir Shevelev, Aug 27 2014

Keywords

Comments

We conjecture that a(n)>0.

Crossrefs

Cf. A246501, A242719, A242720.

Extensions

More terms from Peter J. C. Moses, Aug 27 2014

A243200 Let A242720(n)-3 = prime(l)*prime(m), l<=m; a(n)=m-l.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 3, 4, 4, 3, 2, 2, 6, 6, 5, 1, 1, 5, 2, 2, 2, 10, 5, 5, 3, 3, 3, 2, 2, 2, 1, 1, 1, 3, 2, 2, 5, 1, 1, 3, 3, 3, 3, 6, 7, 4, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 2, 2, 2, 2, 2, 2, 8, 6, 6, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 4, 2, 2
Offset: 2

Views

Author

Vladimir Shevelev, Jun 01 2014

Keywords

Comments

Since for a prime p>3, p^2 == 1 (mod 3), then for n>2, A242720(n) is not equal to p^2 + 3 (otherwise, lpf(A242720(n) - 1) = 3). So, the sequence has only zero term a(2).

Crossrefs

Cf. A242720, A242719, A243154, A243158.

Extensions

More terms from Peter J. C. Moses, Jun 01 2014

A243990 Primes which do not divide the numbers of the form A242720(n)-3.

Original entry on oeis.org

2, 41, 61, 103, 113, 157, 227, 263, 283, 337, 349, 373, 383, 389, 431, 433, 449, 457, 557, 563, 577, 601, 631, 641, 677, 683, 691, 733, 751, 857, 881, 883, 911, 929, 953, 967, 983, 991, 1009, 1039, 1093, 1097, 1151, 1181, 1279, 1303, 1373, 1427, 1451, 1481
Offset: 1

Views

Author

Vladimir Shevelev, Jun 17 2014

Keywords

Comments

If a prime p=prime(k) does not divide A242720(i)-3 for i=2,3,...,k, then it is in the sequence.

Crossrefs

Cf. A242720, A242719, A243960.

Extensions

More terms from Peter J. C. Moses, Jun 18 2014

A244412 Least even k such that sfdf(k-1) > sfdf(k-3) >= A050376(n), where sfdf(n) is the smallest Fermi-Dirac factor of n (A223490), and k-3 is not the lesser of a pair of Fermi-Dirac twin primes (A229064).

Original entry on oeis.org

18, 38, 38, 80, 102, 212, 224, 440, 440, 440, 578, 728, 1250, 1460, 1742, 2012, 2282, 3434, 3482, 4190, 4664, 4760, 4760, 6890, 7212, 7212, 7212, 8054, 10772, 12830, 12830, 13592, 13592, 14282, 17402, 17402, 17402, 18212, 22502, 22502, 22502, 25220, 28202
Offset: 2

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jun 27 2014

Keywords

Comments

A Fermi-Dirac analog of A242720.

Examples

			If k>=4 is even such that k-3 is either 1 or in A050376, then k cannot be a solution. Thus, if n=2, then k=4,6,8,10,12,14 are not allowed; for k=16 we have sfdf(16-1) = 3 < sfdf(16-3) = 13; finally, for k=18 we have sfdf(18-1) = 17 > sfdf(18-3) = 3 = A050376(2). Since 15 is not in A229064,  then a(2)=18.

Crossrefs

Cf. A050376, A223490, A242719, A244343, A242720, A229064.

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

Views

Author

Vladimir Shevelev, Sep 19 2014

Keywords

Comments

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).

Crossrefs

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

Formula

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

Extensions

More terms from Peter J. C. Moses, Sep 19 2014
Previous Showing 21-28 of 28 results.

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