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.

A097374 Perfect 4-composites: a perfect 4-composite is a natural number that can be represented in the form a^(a^(a^........(a^(a) ) ) ) for some natural number a and some number b>=1 of up-arrows.

Original entry on oeis.org

4, 16, 27, 256, 3125, 46656, 65536, 823543, 16777216, 387420489, 10000000000, 285311670611, 7625597484987, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979, 104857600000000000000000000
Offset: 1

Views

Author

Ashutosh (ashu(AT)iitk.ac.in), Sep 18 2004

Keywords

Comments

From Natan Arie Consigli, Jan 17 2016: (Start)
Also, natural numbers of the form H_4(a,b) with a,b > 1. See A054871 for definitions and key links.
Let a and b be positive. a is a unit if there exist b such that a*b=1. The only unit is 1 because only 1*1=1.
x = a*b is composite (in hyper-2) if a,b are nonunits.
In hyper-4 context the only unit is 1 since a[4]b = 1 if and only if a=1.
Hyper 4-composites are numbers of the form H_4(a,b) where a,b are nonunits. This is why for 4-composites we have a,b > 1.
1 and 0 are non-4-composites since H_4(a,b) > 1 if a,b are positive nonunits. (End)

Examples

			4-composites include:
H_4(5,2)= 5^5 = 3125;
H_4(3,3) = 3^3^3 = 3^27 = 7625597484987;
H_4(2,4) = 2^2^2^2 = 2^2^4 = 2^16 = 65536;

Crossrefs

Cf. A257309 (nontrivial hyper-4 powers H_4(a,b) with b<>1).

Programs

Formula

a(n) = A257309(n+2).

Extensions

Corrected by Natan Arie Consigli, Jan 17 2016

A257769 Perfect hyper-5 powers: a^^^b, where b <> 1.

Original entry on oeis.org

0, 1, 4, 65536, 7625597484987
Offset: 1

Views

Author

Natan Arie Consigli, May 07 2015

Keywords

Comments

a^^b (a to the hyper-4 power b) is the right associative power tower a^a^...^a^a of height b.
a^^^b (a to the hyper-5 power b) is the right associative hyper-4 power tower a^^a^^...^^a^^a of height b. a^^^-1 = 0, a^^^0 = 1;
We exclude b=1 because otherwise all natural numbers would be in the sequence.

Examples

			Numbers satisfying the properties stated above:
3^^^2 = 3^^3 = 3^3^3 = 3^27 = 7625597484987;
2^^^3 = 2^^2^^2 = 2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65536;
1^^^4 = 1^^1^^1^^1 = 1.

Crossrefs

Cf. A001597, A257309.

A302553 Hyper-4 powers that are not hyper-5 powers.

Original entry on oeis.org

16, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979, 104857600000000000000000000
Offset: 1

Views

Author

Natan Arie Consigli, Jul 08 2018

Keywords

Comments

A term is in this sequence if it is in A257309 but not in A257769.

Crossrefs

Cf. A257309, A257769.

A302554 Powers that are not hyper-4 powers.

Original entry on oeis.org

8, 9, 25, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764
Offset: 1

Views

Author

Natan Arie Consigli, Jul 08 2018

Keywords

Comments

A term is in this sequence if it is in A001597 but not in A257309.

Crossrefs

Cf. A001597, A257309.

A381538 Numbers of the form m^(m^k).

Original entry on oeis.org

1, 4, 16, 27, 256, 3125, 19683, 46656, 65536, 823543, 16777216, 387420489, 4294967296, 10000000000, 285311670611, 7625597484987, 8916100448256, 302875106592253, 11112006825558016, 298023223876953125, 437893890380859375, 18446744073709551616
Offset: 1

Views

Author

Charles L. Hohn, Feb 26 2025

Keywords

Examples

			27 = 3^(3^1) -> a(4).
256 = 2^(2^3) = 4^(4^1) -> a(5).

Crossrefs

Subsequence of A001597; supersequence of A000312 (apart from initial term), A097374, and A257309 (apart from initial term).
Subset of A380760 for a(n)>=16, and of A067688 for prime m.

Programs

  • PARI
    upto(limit)=my(L=List([1])); for(m=2, oo, my(t=logint(limit,m)); if(tAndrew Howroyd, Feb 26 2025
Showing 1-5 of 5 results.

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