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.

Showing 1-5 of 5 results.

A068507 Highly composite numbers sandwiched between twin primes.

Original entry on oeis.org

4, 6, 12, 60, 180, 240, 7560, 55440, 110880, 73329656400, 18632716502400, 130429015516800, 48519593772249600, 149602080797769600, 74377068101903920953600, 927967188666725711881005276648000, 241271469053348685089061371928480000
Offset: 1

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Author

Lekraj Beedassy, Mar 25 2002

Keywords

Comments

Intersection of (A072826 - 1) and (A072828 + 1). - Lekraj Beedassy, Nov 27 2003
The next term, a(18), is A002182(1002), it has 77 digits. - M. F. Hasler, Jun 23 2019
a(22) > 10^17030, if it exists. - Amiram Eldar, Dec 03 2020

Examples

			60 is between 59 and 61.

Links

Crossrefs

Cf. A002182, A072826, A072828, A117825, A321995.
This is also the intersection of A002182 and A014574.

Formula

a(n) = A002182(A321995(n)). - Amiram Eldar, Dec 03 2020

Extensions

Corrected and extended by Lior Manor, Jun 03 2002
More terms from Bill McEachen, May 24 2006
a(18)-a(20) from M. F. Hasler, Jun 23 2019

A072828 Primes p such that p+1 is a highly composite number.

Original entry on oeis.org

3, 5, 11, 23, 47, 59, 179, 239, 359, 719, 839, 1259, 5039, 7559, 10079, 55439, 110879, 166319, 665279, 1081079, 1441439, 6486479, 32432399, 61261199, 698377679, 735134399, 1102701599, 1396755359, 2205403199, 20951330399, 41902660799
Offset: 1

Views

Author

Shyam Sunder Gupta, Jul 21 2002

Keywords

Examples

			47 is a term because it is prime and also 47+1=48 is a highly composite number.

Links

Crossrefs

Cf. A002182, A068507, A072826, A214873, A306588.

Formula

a(n) == 2 (mod 3) for n > 1 (because highly composite numbers > 4 are == 0 (mod 3); see A002182). - Jonathan Sondow, Nov 05 2015
a(n) = A002182(A306588(n)) - 1. - Amiram Eldar, Dec 03 2020

A306588 Numbers k such that A002182(k)-1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 19, 20, 21, 28, 30, 31, 37, 39, 40, 45, 52, 55, 65, 66, 67, 68, 70, 79, 81, 84, 101, 108, 118, 131, 132, 136, 143, 148, 149, 151, 163, 170, 174, 185, 191, 200, 203, 208, 212, 231, 259, 261, 286, 289, 297, 317, 326, 327
Offset: 1

Views

Author

Dmitry Kamenetsky, Mar 02 2019

Keywords

Links

Crossrefs

Cf. A002182 (highly composite numbers), A072826, A306587.

Extensions

More terms from Daniel Suteu, Mar 02 2019

A103199 Primes p such that p-1 has more divisors than any smaller prime-1.

Original entry on oeis.org

2, 3, 5, 7, 13, 31, 37, 61, 181, 241, 421, 1009, 1321, 1801, 2161, 2521, 6301, 7561, 12601, 15121, 20161, 30241, 45361, 55441, 100801, 110881, 196561, 332641, 498961, 786241, 982801, 1108801, 1580041, 1940401, 1995841, 2402401, 3880801, 4324321, 11476081, 11531521
Offset: 1

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Author

Don Reble, Mar 19 2005

Keywords

Comments

There are infinitely many primes p such that d(p-1) > exp(c*log(p)/log(log(p))), where d(k) is the number of divisors of k, and c > 0 is a constant (Prachar, 1955). Therefore, this sequence is infinite. - Amiram Eldar, Apr 16 2024

Links

Crossrefs

Cf. A000005, A008328, A066814, A067573, A072826.

Programs

  • Mathematica
    seq[pmax_] := Module[{d, dm = 0, s = {}, p = 1}, While[p < pmax, p = NextPrime[p]; d = DivisorSigma[0, p-1]; If[d > dm, dm = d; AppendTo[s, p]]]; s]; seq[10^6] (* Amiram Eldar, Apr 16 2024 *)
  • PARI
    lista(pmax) = {my(dm = 0, d); forprime(p = 1, pmax, d = numdiv(p-1); if(d > dm, dm = d; print1(p, ", ")));} \\ Amiram Eldar, Apr 16 2024

Extensions

a(38)-a(40) added and name clarified by Amiram Eldar, Apr 16 2024

A353300 Numbers k such that A004394(k)-1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 19, 20, 24, 25, 26, 30, 32, 41, 47, 48, 49, 51, 57, 59, 76, 82, 83, 92, 104, 105, 117, 119, 131, 134, 137, 139, 143, 154, 166, 170, 172, 180, 209, 214, 215, 216, 217, 227, 231, 234, 247, 265, 269, 271, 284, 317, 327, 348
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2022

Keywords

Comments

First differs from A306588 at n=15.

Examples

			3 is a term since A004394(3)-1 = 4-1 = 3 is prime.

Links

Crossrefs

Cf. A004394, A072826, A306588, A353301, A353302.

Programs

  • Mathematica
    s = {}; abm = 0; k = 0; Do[ab = DivisorSigma[-1, n]; If[ab > abm, abm = ab; k++; If[PrimeQ[n - 1], AppendTo[s, k]]], {n, 1, 10^6}]; s
Showing 1-5 of 5 results.

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