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 1-7 of 7 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

A072826 Primes p such that p-1 is a highly composite number.

Original entry on oeis.org

2, 3, 5, 7, 13, 37, 61, 181, 241, 2521, 7561, 15121, 20161, 45361, 55441, 110881, 332641, 498961, 4324321, 14414401, 43243201, 110270161, 183783601, 367567201, 4655851201, 13967553601, 73329656401, 293318625601, 1606268664001
Offset: 1

Views

Author

Shyam Sunder Gupta, Jul 21 2002

Keywords

Examples

			13 is a term because it is a prime such that 13-1=12 is a highly composite number.

Links

Crossrefs

Cf. A002182 (highly composite numbers), A072828 (with p+1 instead), A306587.

Formula

a(n) = A002182(A306587(n)) + 1. - Amiram Eldar, Dec 03 2020

A306587 Numbers k such that A002182(k)+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 18, 20, 22, 23, 26, 28, 30, 34, 35, 44, 49, 54, 57, 60, 63, 74, 78, 84, 91, 97, 102, 104, 108, 111, 112, 114, 116, 118, 126, 134, 143, 145, 149, 159, 162, 167, 173, 177, 179, 188, 204, 208, 214, 230, 236, 238, 247, 249, 280, 294, 298
Offset: 1

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Author

Dmitry Kamenetsky, Mar 02 2019

Keywords

Links

Crossrefs

Cf. A002182 (highly composite numbers), A072828, A306588.

Extensions

More terms from Daniel Suteu, Mar 02 2019

A353301 Numbers k such that A004394(k)+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 18, 21, 24, 25, 28, 35, 45, 46, 50, 56, 70, 73, 76, 78, 79, 82, 89, 94, 105, 113, 116, 118, 121, 123, 124, 138, 139, 153, 157, 159, 164, 197, 201, 203, 210, 217, 253, 261, 273, 280, 283, 287, 342, 352, 356, 381, 396, 437, 450, 471
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2022

Keywords

Comments

First differs from A306587 at n=11.

Examples

			1 is a term since A004394(1)+1 = 1+1 = 2 is prime.

Links

Crossrefs

Cf. A004394, A072828, A306587, A353300, 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

A214873 Primes p such that 2*p + 1 is also prime and p + 1 is a highly composite number (definition 1).

Original entry on oeis.org

3, 5, 11, 23, 179, 239, 359, 719, 5039, 55439, 665279, 6486479, 32432399, 698377679, 735134399, 1102701599, 20951330399, 3212537327999, 149602080797769599, 299204161595539199, 2718551763981393634806325317503999
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jul 30 2012

Keywords

Comments

An equivalent definition of this sequence: odd Sophie Germain primes that differ from a highly composite number by 1.
Intersection of A005384 (Sophie Germain primes) and A072828.
With the exception of 5, a subsequence of A002515 (Lucasian primes).
Except for first two terms, this is a subsequence of A054723.
Except for n = 2, 2*a(n) + 1 is a prime factor of A000225(a(n)) (i.e., 2*23 + 1 divides 2^23 - 1).
Conjecture: for n >= 5, GCD(A000032(a(n)), A000225(a(n))) = 2*a(n) + 1.

Examples

			23 is a term because both 23 and 47 are primes and also 24 is a highly composite number.

Links

Crossrefs

Cf. A054723.

Programs

  • Mathematica
    lst = {}; a = 0; Do[b = DivisorSigma[0, n + 1]; If[b > a, a = b; If[PrimeQ[n] && PrimeQ[2*n + 1], AppendTo[lst, n]]], {n, 1, 10^6, 2}]; lst

A227387 Numbers n such that sigma(n)*7 < sigma(n-1) + sigma(n+1).

Original entry on oeis.org

41902660799, 41902660801, 48886437599, 53542288799, 55870214401, 62853991199, 73329656401, 80313433201, 96376119841, 97772875199, 97772875201, 101264763599, 107084577599, 107221514401, 118356638399, 118356638401, 125707982401, 128501493121, 135019684799
Offset: 1

Views

Author

Alex Ratushnyak, Jul 09 2013

Keywords

Comments

Conjecture: the sequence is infinite.

Links

Crossrefs

Cf. A000203, A072828.

Extensions

a(7)-a(19) from Giovanni Resta, Jul 15 2013

A352634 Highly composite numbers that are one more than a prime number.

Original entry on oeis.org

4, 6, 12, 24, 48, 60, 180, 240, 360, 720, 840, 1260, 5040, 7560, 10080, 55440, 110880, 166320, 665280, 1081080, 1441440, 6486480, 32432400, 61261200, 698377680, 735134400, 1102701600, 1396755360, 2205403200, 20951330400, 41902660800, 73329656400, 3212537328000
Offset: 1

Views

Author

J. Lowell, Mar 28 2022

Keywords

Comments

Intersection of A002182 and A008864.

Examples

			36 is not a term because 36 - 1 = 35 = 5*7.

Crossrefs

Cf. A002182, A008864, A072828.

Formula

a(n) = A072828(n) + 1.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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