A060365 -id:A060365 - cp's OEIS frontend

A060366 Powers of one thousand which are described by single words in dated British English usage, extended by using "-ard" beyond 10^9.

1, 1000, 1000000, 1000000000, 1000000000000, 1000000000000000, 1000000000000000000, 1000000000000000000000, 1000000000000000000000000, 1000000000000000000000000000, 1000000000000000000000000000000

Offset: 1

Gerard Schildberger

			10^0 = one, 10^3 = thousand, 10^6 = million, 10^9 = milliard, 10^12 = billion, etc.

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 14.

Tanya Khovanova, Non Recursions
More information about A060365 and A060366.

Cf. A060365.

[10^(3*(n-1)) : n in [1..15]]; // Wesley Ivan Hurt, Jan 21 2016

A060366:=n->10^(3*(n-1)): seq(A060366(n), n=1..15); # Wesley Ivan Hurt, Jan 21 2016

10^(3 Range[0, 10]) (* or *) Table[10^(3 (n - 1)), {n, 15}] (* Wesley Ivan Hurt, Jan 21 2016 *)

a(n) = 10^(3*(n-1)).

Description changed by Ivan Panchenko, Nov 10 2019

A127352 Integers less than 10^303 in reverse alphabetical order in U.S. English.

Original entry on oeis.org

0, 2000000000000000000000000002000000000000000000000002000000002202, 2000000000000000000000000002000000000000000000000002000000002222, 2000000000000000000000000002000000000000000000000002000000002223, 2000000000000000000000000002000000000000000000000002000000002226, 2000000000000000000000000002000000000000000000000002000000002227, 2000000000000000000000000002000000000000000000000002000000002221

Offset: 1

Michael B. Porter, Nov 24 2009

Since the use of alphabetic names is rare for numbers greater than 10^15, there is no universal agreement on the naming scheme for large integers, and there is some question whether this sequence would well-defined without the "less than 10^303" clause.
The Wikipedia article compares 8 dictionary sources and has names for the powers of 1000 up to 10^63 and for 10^303. These are also in the Mathworld link.
There are several conflicting schemes for extending the dictionary definitions. If we assume that the system of alphabetic names greater than 10^63 defines a word for every power of 1000 and that word comes before "vigintillion" alphabetically, the sequence can include all integers. However, many of the extension schemes listed do not meet that standard - some have multiple words and some have words that are alphabetically after "vigintillion".
For the powers of 1000 between 10^66 and 10^303, one source (http://www.mrob.com/pub/math/largenum.html) coins the name "vigintinonillion" for 10^90, but this format is inconsistent with other names listed in the same source, e.g. "duovigintillion", "sexoctogintillion". The name "novemvigintillion" seems to be more common. Otherwise, all sources have "vigintillion" as alphabetically last of all the powers of 1000 up to 10^303.
The terms are from Andrew Weimholt.

			zero,
two vigintillion two undecillion two trillion two thousand two hundred two,
two vigintillion two undecillion two trillion two thousand two hundred twenty two, etc.

Michael B. Porter, Table of n, a(n) for n = 1..15
Robert P. Munafo, Large numbers
Landon Curt Noll, How high can you count?
John J. G. Savard, Some notes on big numbers
Gerard Schildberger, Lists of Words for the Numbers in A060365 and A060366
Eric Weisstein's World of Mathematics, Large Number
Wikipedia, Names of large numbers

Cf. A004740, A019440, A060365, A060366.

See A026081 for another version.

A271527 a(n) = 1000^n + 1.

Original entry on oeis.org

2, 1001, 1000001, 1000000001, 1000000000001, 1000000000000001, 1000000000000000001, 1000000000000000000001, 1000000000000000000000001, 1000000000000000000000000001, 1000000000000000000000000000001, 1000000000000000000000000000000001

Offset: 0

Ilya Gutkovskiy, Apr 09 2016

All terms in this sequence are palindromes (A002113).
Also, A062395 written in base 2 (see example).
a(n) minus one gives the number of nodes at n-th level of a 1000-ary tree.
More generally, the ordinary generating function for sequences of the form k^n + m, is (1 + m - (1 + k*m)*x)/((1 - x)*(1 - k*x)), and the exponential generating function is exp(k*x) + m*exp(x).

			a(n), n>0, is the binary representation of A062395(n)
n  ------------------------------------------
0  2........................................2
1  1001.....................................9
2  1000001.................................65
3  1000000001.............................513
4  1000000000001.........................4097
5  1000000000000001.....................32769
6  1000000000000000001.................262145
7  1000000000000000000001.............2097153
8  1000000000000000000000001.........16777217
9  1000000000000000000000000001.....134217729

Ilya Gutkovskiy, Examples of the ordinary generating function for the sequences of the form k^n + m
Index entries for linear recurrences with constant coefficients, signature (1001,-1000)

Cf. A000035, A000533, A007088, A007395, A007953, A057427, A060365, A062395, A152756.

Table[1000^n + 1, {n, 0, 11}]
LinearRecurrence[{1001, -1000}, {2, 1001}, 12]

x='x+O('x^99); Vec((2-1001*x)/((1-x)*(1-1000*x))) \\ Altug Alkan, Apr 09 2016

for n in range(0,10**4):print(1000**n+1)
# Soumil Mandal, Apr 10 2016

G.f.: (2 - 1001*x)/((1 - x)*(1 - 1000*x)).
E.g.f.: exp(1000*x) + exp(x).
a(n) = 1001*a(n-1) - 1000*a(n-2).
a(n) = A060365(n) + 1.
a(n) = A000533(3n), n>0.
a(n) = A007088(A062395(n)).
A007953(a(n)) = A007395(n).
A000035(a(n)) = A057427(n).
Sum_{n>=0} 1/a(n) = 0.501000001999002...
Lim_{n->infinity} a(n + 1)/a(n) = 1000.

cp's OEIS Frontend

A060366 Powers of one thousand which are described by single words in dated British English usage, extended by using "-ard" beyond 10^9.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A127352 Integers less than 10^303 in reverse alphabetical order in U.S. English.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A271527 a(n) = 1000^n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula