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A114808 The numbers 5^n-1 written in groups of three digits, with leading zeros omitted.

424, 124, 624, 312, 415, 624, 781, 243, 906, 241, 953, 124, 976, 562, 448, 828, 124, 244, 140, 624, 122, 70, 312, 461, 35, 156, 243, 51, 757, 812, 415, 258, 789, 62, 476, 293, 945, 312, 438, 146, 972, 656, 241, 907, 348, 632, 812, 495, 367, 431, 640, 624, 476, 837, 158, 203, 124, 238, 418, 579

Offset: 1

Jonathan Vos Post, Feb 19 2006

			4, 24, 124, 624, 3124, 15624, ...

Cf. A024049, A103455, A114645.

L := [] ;
for n from 1 to 30 do
    dggs := ListTools[Reverse](convert(5^n-1,base,10) );
    L := [op(L),op(dggs)] ;
end do:
for k from 1 to nops(L)-3 by 3 do
    op(k,L)*100+op(k+1,L)*10+op(k+2,L) ;
    printf("%d,",%) ;
end do: # R. J. Mathar, Jun 23 2014

FromDigits[#] & /@ Partition[ Flatten@ IntegerDigits@ Table[5^n - 1, {n, 22}], 3] (* Robert G. Wilson v, Jun 23 2014 *)

Definition and terms realigned with A114645 by Robert G. Wilson v, Jun 23 2014

A115983 Apocalypse primes: 10^665+a(n) has 666 decimal digits and is prime.

Original entry on oeis.org

123, 1837, 6409, 7329, 8569, 8967, 9663, 13213, 14943, 16299, 17053, 17857, 22611, 24601, 25261, 25729, 27847, 30567, 32413, 33321, 33379, 34257, 34557, 34723, 38097, 39387, 39787, 39889, 39973, 43501, 43719, 44889, 48139, 49587, 53683

Offset: 1

Eric W. Weisstein, Feb 09 2006

This sequence contains about 10^662 terms, the last of which is 10^666-1157.

Robert G. Wilson v, Table of n, a(n) for n = 1..6666
Brady Haran and Tony Padilla, Interesting 666-digit Numbers, YouTube Numberphile video, 2024.
Eric Weisstein's World of Mathematics, Apocalypse Number

lst={};a=10^665;Do[If[PrimeQ[a+n], Print[n];AppendTo[lst, n]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 30 2008 *)
Select[Range[55000],IntegerLength[10^665+#]==666&&PrimeQ[10^665+#]&] (* Harvey P. Dale, Jul 30 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.

A132725 A nonsense sequence.

Original entry on oeis.org

6, 9, 11, 1, 4, 9, 12, 2, 4, 17, 11, 2, 4, 6, 9, 1, 4, 6, 8, 11, 4, 7, 9, 11, 2, 6, 9, 11, 1, 4, 9, 12, 2, 4, 17, 11, 2, 4, 6, 9, 1, 4, 6, 8, 11, 4, 7, 9, 11, 2, 6, 9, 11, 1, 4, 9, 12, 2, 4, 17, 11, 2, 4, 6, 9, 1, 4, 6, 8, 11, 4, 7, 9, 11, 2, 6, 9, 11, 1, 4, 9, 12, 2, 4, 17, 11, 2, 4, 6, 9, 1, 4, 6, 8

Offset: 1

Roger L. Bagula, Nov 16 2007

Obtained from the substitution given by the Mathematica program.

Cf. A133451.

s[1] = {2, 5, 7, 9, 12};
s[2] = {3, 6, 8, 10, 1};
s[3] = {4, 7, 9, 11, 2};
s[4] = {5, 8, 10, 12, 3};
s[5] = {6, 9, 11, 1, 4};
s[6] = {7, 10, 12, 2, 5};
s[7] = {8, 11, 1, 3, 6};
s[8] = {9, 12, 2, 4, 17};
s[9] = {10, 1, 3, 5, 8};
s[10] = {11, 2, 4, 6, 9};
s[11] = {12, 3, 5, 7, 10};
s[12] = {1, 4, 6, 8, 11};
t[a_] := Flatten[s /@ a];
p[0] = {4, 4, 4, 4, 9, 9, 4, 4, 11, 11, 4, 4};
p[1] = t[p[0]];
p[n_] := t[p[n - 1]];
p[2]

A133794 Times on a 12-hour digital clock with all digits in {1, 2, 3, 4, 5, 6}.

Original entry on oeis.org

111, 112, 113, 114, 115, 116, 121, 122, 123, 124, 125, 126, 131, 132, 133, 134, 135, 136, 141, 142, 143, 144, 145, 146, 151, 152, 153, 154, 155, 156, 211, 212, 213, 214, 215, 216, 221, 222, 223, 224, 225, 226, 231, 232, 233, 234, 235, 236, 241, 242, 243

Offset: 1

Jonathan Vos Post, Jan 05 2008

Digital clock dice integers. The number of values with 3 digits is 180. The number of values with 4 digits is 60. The number of values with 5 digits is 5400. The number of values with 6 digits is 1800. The total number of values is 7440, to the maximum 125656 equated to "12:56:56." Prime values must end with one of {11, 13, 21, 23, 31, 33, 41, 43, 51, 53}. The number of prime values with 3 digits is 23. The number of prime values with 4 digits is 6, namely 1123, 1151, 1153, 1213, 1223, 1231. Prime values with 5 digits begin 11113, 11131, 11213, 11243, 11251, 11257.

			"151" equated to "1:51"; "123456" equated to "12:34:56".

Nathaniel Johnston, Table of n, a(n) for n = 1..7440 (full sequence)

Cf. A000040, A036960, A052382, A057436, A133783, index for "digital clock".

c:=0: for h from 0 to 12 do for m from 0 to 59 do for s from 0 to 59 do t:=10000*h+100*m+s: d:=convert(t,base,10): if(t>100 and (h>0 or m<=12) and numboccur(d,0)=0 and numboccur(d,7)=0 and numboccur(d,8)=0 and numboccur(d,9)=0)then printf("%d, ", t): c:=c+1: fi: od: if(c>=80)then break: fi: od: od: # Nathaniel Johnston, May 17 2011

FromDigits/@Flatten[Table[{h,m1,m2},{h,6},{m1,5},{m2,6}],2] (* Harvey P. Dale, Mar 13 2023 *)

A057436 INTERSECTION {integers that can appear on a 12-hour digital clock, concatenated from either hours:minutes or hours:minutes:seconds}.

Comments corrected by Nathaniel Johnston, May 17 2011

A253828 Digit of Pi raised to the power of the next digit of Pi.

Original entry on oeis.org

3, 1, 4, 1, 1953125, 81, 64, 7776, 125, 243, 390625, 134217728, 4782969, 40353607, 729, 9, 8, 6561, 4096, 4096, 36, 64, 1296, 64, 27, 6561, 512, 9, 128, 40353607, 59049, 1, 0, 256, 16777216, 4096, 4, 1, 4782969, 7, 1, 10077696, 729, 19683, 387420489, 729, 2187

Offset: 1

Jonathan PP Martin, Jan 16 2015

From Felix Fröhlich, Sep 23 2019: (Start)
The convention 0^0 = 1 was applied in computing the terms.
There are 61 values that can occur in this sequence, namely all numbers of the form x^y for some 0 <= x, y <= 9. (End)

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A000796.

Module[{nn=1000,pidg},pidg=Partition[RealDigits[Pi,10,nn][[1]],2,1];If[ # == {0,0},1,#[[1]]^#[[2]]]&/@pidg] (* Harvey P. Dale, Oct 24 2021 *)

pistring(n) = default(realprecision, n+10); my(x=Pi); floor(x*10^n)
pidigit(n) = pistring(n)-10*pistring(n-1)
a(n) = pidigit(n-1)^pidigit(n) \\ Felix Fröhlich, Sep 23 2019

a(n) = A000796(n)^A000796(n+1). - Felix Fröhlich, Sep 23 2019

More terms from Felix Fröhlich, Sep 23 2019

A270338 Primes whose decimal expansion contains only 3's and 4's, in which every 4 is preceded and followed by a 3.

Original entry on oeis.org

3343, 3433, 33343, 333433, 334333, 343333, 343433, 3333433, 3343343, 3343433, 3433333, 34333333, 333334333, 333343343, 333343433, 333433343, 333434333, 334334333, 3333334343, 3333433343, 3334333333, 3343334333, 3343434343, 3433434343, 3434343433, 33333333343

Offset: 1

Arkadiusz Wesolowski, Mar 15 2016

A sequence related to A054356. These primes look like "EEhEEhEEE" when viewed upside down by rotation of 180 degrees (3343 - "EhEE", 3433 - "EEhE", 33343 - "EhEEE", 333433 - "EEhEEE").

Giorgio Balzarotti, Paolo P. Lava, Centotre curiosità matematiche, Hoepli, 2010, pp. 3-4.

Robert Israel, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!: 47
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!: 3343

Cf. A054356. Subsequence of A020461.

[p: p in [3..33333333343 by 10] | (p mod 100 eq 33 or p mod 100 eq 43) and IsPrime(p) and Position(IntegerToString(p), IntegerToString(3)) eq 1 and Set(Intseq(p)) subset [3, 4] and not IntegerToString(44) in IntegerToString(p)];

S:= {}:
for n from 3 to 16 do
  for k from 1 to floor((n-1)/2) do
     for r in combinat:-choose(n-1-k,k) do
        L:=subsop(seq(t=(3,4),t=r),[3$(n-k)]);
        x:= add(L[i]*10^(n-i),i=1..n);
        if isprime(x) then S:= S union {x} fi
od od od:
sort(convert(S,list)); # Robert Israel, Mar 15 2016

Select[Flatten[Table[FromDigits/@Select[Tuples[{3,4},n],SequenceCount[ #,{3,4,3},Overlaps->True]==Count[#,4]&],{n,3,11}]],PrimeQ]//Sort (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 17 2016 *)

A316599 The nonnegative integers sorted by increasing width of bounding box when printing their decimal representation using the Arial font, and in case of ties, sorted by increasing value.

Original entry on oeis.org

1, 7, 0, 3, 9, 8, 2, 6, 5, 4, 11, 71, 31, 51, 91, 81, 61, 21, 41, 12, 14, 10, 13, 16, 17, 18, 19, 15, 72, 74, 70, 32, 52, 92, 73, 76, 77, 82, 78, 79, 34, 54, 62, 94, 30, 50, 84, 90, 80, 33, 36, 37, 53, 56, 57, 75, 93, 96, 97, 64, 83, 86, 87, 38, 39, 58, 59, 60, 98, 99, 88, 89, 22, 63, 66, 67, 35, 55, 68, 69

Offset: 0

Hugo Pfoertner, Jul 08 2018

The linked xkcd webcomic number 2016 (subtitled "OEIS keeps rejecting my submissions") shows as its second example "SUB[44]: Integers in increasing order of width when printed in Helvetica". To make a sequence involving a font well-defined, the definition can be interpreted using information provided in the corresponding Adobe Font Metrics File. However since Helvetica is a proprietary font, the almost-identical font metrics of the Arial regular font are used here instead. Also, the vague description "width when printed" is interpreted as the size of the bounding box in the printing direction. This takes into account the actual left and right margin of the characters, which is smaller than the fixed width of 556/1000 size units for all digits, as well as the effect of the special kerning data for the character pair "11" (which is printed closer together by -74/1000 size units than other digit pairs).

The sequence is a permutation of the nonnegative integers.

			All dimensions are given in units of 1/1000 of the font size. The relevant dimensions are the constant width of all numerical digits of w=556 units and the lower and upper x-limits llx and urx of the bounding box:
       llx urx
  zero  41 509
  one  108 373
  two   30 504
  three 41 511
  four  12 508
  five  41 517
  six   37 511
  seven 47 511
  eight 40 513
  nine  41 513
Additionally an addend of kpx("1","1") = -74 is applied for each occurrence of this pair.
.
a(0) = 1 because the character "1" has the smallest width of bounding box: 373 - 108 = 265.
a(1) = 7, because 7 is the digit with next smallest bounding box width: 511 - 47 = 464.
a(6) = 2, a(7) = 6 both have a bounding box width of 474 units, but 2 < 6.
The width of the bounding box for a(10) = 11 is calculated as 2*w - llx(1) - (w-urx(1)) + kpx("1","1") = 2*556 - 108  - (556-373) - 74 = 747.
The next term a(11) = 71 has bounding box width 2*w - llx(7) - (w-urx(1)) = 2*556 - 47 - (556-373) = 882.
a(29368) = 111111 is a first notable anomaly, because its bounding box width of 2675 lies between those of a(29367) = 49115, with bounding box width 2655, and a(29369) = 70002, with bounding box width 2681.

Hugo Pfoertner, Table of n, a(n) for n = 0..100000
Adobe, Adobe Font Metrics File Format Specification, Version 4.1, 1998
Randall Munroe, OEIS Submissions, xkcd Web Comic #2016, Jul 06 2018.
Hugo Pfoertner, Example of printed numbers, showing only samples of larger numbers.

Cf. A001477, A316600.

A316600 Integers in increasing order of width when printed in Helvetica.

Original entry on oeis.org

1, 9, 8, 6, 2, 0, 5, 7, 3, 4, 11, 91, 61, 81, 71, 31, 51, 21, 41, 12, 15, 19, 18, 16, 10, 13, 14, 17, 92, 62, 82, 72, 95, 99, 32, 52, 98, 65, 69, 85, 89, 96, 75, 79, 90, 68, 88, 35, 39, 66, 78, 86, 93, 22, 60, 76, 80, 94, 97, 42, 55, 59, 70, 38, 36, 63, 83, 30, 58, 64, 67, 73, 84, 87, 56

Offset: 0

Hugo Pfoertner, Jul 09 2018

The title is copied verbally from the second example "SUB[44]" of the linked xkcd webcomic number 2016 (subtitled "OEIS keeps rejecting my submissions"). To make the definition exact, the full title should be: The nonnegative integers sorted by increasing width of bounding box when printing their decimal representation using the Helvetica font, and in case of ties, sorted by increasing value.
The differences from the corresponding sequence A316599, which uses the Arial font, arise from the different width data for the digit characters and the absence of kerning for the number pair "11" in Helvetica. So the print width becomes a function of the number of printed digits, of the left margin of the bounding box of the most significant digit and of the right margin of the bounding box of the least significant digit.
See the comment in A316599 for more information.

			The relevant part of the applicable Adobe Font Metrics File Helvetica.afm is as follows:
Header information:
StartFontMetrics 4.1
Comment Creation Date: Thu May  1 12:38:23 1997
Comment UniqueID 43054
Version 002.000
Notice Copyright (c) 1985, 1987, 1989, 1990, 1997 Adobe Systems Incorporated.  All Rights Reserved. Helvetica is a trademark of Linotype-Hell AG and/or its subsidiaries.
.
Individual character metrics:
C number Decimal value of default character code,
WX number Character width in x for writing direction 0,
B llx lly urx ury Character bounding box where llx, lly, urx, and ury are all numbers.
.
  C 48 ; WX 556 ; N zero ; B 37 -19 519 703 ;
  C 49 ; WX 556 ; N one ; B 101 0 359 703 ;
  C 50 ; WX 556 ; N two ; B 26 0 507 703 ;
  C 51 ; WX 556 ; N three ; B 34 -19 522 703 ;
  C 52 ; WX 556 ; N four ; B 25 0 523 703 ;
  C 53 ; WX 556 ; N five ; B 32 -19 514 688 ;
  C 54 ; WX 556 ; N six ; B 38 -19 518 703 ;
  C 55 ; WX 556 ; N seven ; B 37 0 523 688 ;
  C 56 ; WX 556 ; N eight ; B 38 -19 517 703 ;
  C 57 ; WX 556 ; N nine ; B 42 -19 514 703 ;
.
The resulting first few sorted print widths w are:
  a(0) = 1, w(1) = 258 = 359 - 101;
  a(1) = 9, w(9) = 472 = 514 - 42;
  a(2) = 8, w(8) = 479 = 517 - 38;
  a(3) = 6, w(6) = 480 = 518 - 38;
  a(4) = 2, w(2) = 481 = 507 - 26;
  a(5) = 0, w(0) = 482 = 519 - 37;
  a(6) = 5, w(5) = 482 = 514 - 32, 0 < 5;
  a(7) = 7, w(7) = 486 = 523 - 37;
  a(8) = 3, w(3) = 488 = 522 - 34;
  a(9) = 4, w(4) = 498 = 523 - 25;
  a(10) = 11, w(11) = 814 = 556 - 101 + 359;
  a(11) = 91, w(91) = 873 = 556 - 42 + 359;
  a(12) = 61, w(61) = 877 = 556 - 38 + 359;
  a(13) = 81, w(81) = 877 = 556 - 38 + 359, 61 < 81;
  a(14) = 71, w(71) = 878 = 556 - 37 + 359

Hugo Pfoertner, Table of n, a(n) for n = 0..100000
Adobe, Adobe Font Metrics File Format Specification, Version 4.1, 1998
Adobe Font technical notes, Font Metrics for PDF Core 14 Fonts (Win), File Helvetica.afm dated 11/10/1999 in ZIP archive.
Randall Munroe, OEIS Submissions, xkcd Web Comic #2016, Jul 06 2018.
Hugo Pfoertner, Illustration of print widths.

Cf. A001477, A316599.

llx=[101, 26, 34, 25, 32, 38, 37, 38, 42, 37];
urx=[359, 507, 522, 523, 514, 518, 523, 517, 514, 519];
lsd(x)=if(x%10==0,10,x%10);
msd(x)=floor(x/10^logint(x,10));
width(x)=556*logint(x,10)-llx[msd(x)]+urx[lsd(x)];
w=vector(100);
large=1000000;
w[1]=large*(urx[10]-llx[10]);
for(k=2,100,w[k]=large*width(k-1)+k-1);
v=vecsort(w);
for(k=1,75,print1(v[k]%large,", ")) \\ Hugo Pfoertner, Jul 12 2018

A325911 Screaming numbers in base 16: numbers whose hexadecimal representation is AAAAAAA...

Original entry on oeis.org

10, 170, 2730, 43690, 699050, 11184810, 178956970, 2863311530, 45812984490, 733007751850, 11728124029610, 187649984473770, 3002399751580330, 48038396025285290, 768614336404564650, 12297829382473034410, 196765270119568550570, 3148244321913096809130

Offset: 1

Eliora Ben-Gurion, Sep 08 2019

In any base b > 10, we may express ten as a digit by using the letter A.

			a(10) = 733007751850_10 = AAAAAAAAAA_16.

Colin Barker, Table of n, a(n) for n = 1..800
Eric Weisstein's World of Mathematics, Hexadecimal
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A131865, A001025, A228774.

10Accumulate[16^Range[0, 31]] (* Alonso del Arte, Sep 17 2019 *)
LinearRecurrence[{17,-16},{10,170},20] (* Harvey P. Dale, Apr 02 2023 *)

a(n)={10*(16^n-1)/15} \\ Andrew Howroyd, Sep 08 2019

Vec(10*x / ((1 - x)*(1 - 16*x)) + O(x^20)) \\ Colin Barker, Sep 16 2019

a = 10
while a:
    a = a*16+10
    print(a)

def a(n): return int("A"*n, 16)
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Jan 17 2022

a(n) = Sum_{i=0..n} 10*16^(i).
a(n) = A131865(n-1)*10.
a(n) = 10*(16^n-1)/15. - Andrew Howroyd, Sep 08 2019
From Colin Barker, Sep 16 2019: (Start)
G.f.: 10*x / ((1 - x)*(1 - 16*x)).
a(n) = 17*a(n-1) - 16*a(n-2) for n>2.
(End)
E.g.f.: (2/3)*exp(x)*(-1 + exp(15*x)). - Stefano Spezia, Sep 17 2019

cp's OEIS Frontend

A114808 The numbers 5^n-1 written in groups of three digits, with leading zeros omitted.

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Maple

Mathematica

Extensions

A115983 Apocalypse primes: 10^665+a(n) has 666 decimal digits and is prime.

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Author

Keywords

Comments

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Programs

Mathematica

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

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Comments

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A132725 A nonsense sequence.

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Mathematica

A133794 Times on a 12-hour digital clock with all digits in {1, 2, 3, 4, 5, 6}.

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Mathematica

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A253828 Digit of Pi raised to the power of the next digit of Pi.

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Mathematica

PARI

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A270338 Primes whose decimal expansion contains only 3's and 4's, in which every 4 is preceded and followed by a 3.

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Magma

Maple

Mathematica

A316599 The nonnegative integers sorted by increasing width of bounding box when printing their decimal representation using the Arial font, and in case of ties, sorted by increasing value.

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