A008536 -id:A008536 - cp's OEIS frontend

A008520 Numbers whose American English name contains the letter 'e'.

0, 1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 63, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

N. J. A. Sloane

A085513(a(n)) > 0. - Reinhard Zumkeller, Jan 23 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Roel and Bas van Dijk, Numerals package, Hackage (Haskell packages)
Index entries for sequences related to number of letters in n

Cf. A006933 (complement), A085513.

Cf. A008519 (o), A008522 (t), A008536 (n), A008538 (s), A008540 (f), A008553 (y).

import Data.Maybe (fromJust)
import Data.Text (Text); import qualified Data.Text as T (any)
import Text.Numeral.Grammar.Reified (defaultInflection)
import qualified Text.Numeral.Language.EN as EN  -- see link
a008520 n = a008520_list !! (n-1)
a008520_list = filter (T.any (== 'e') . numeral) [0..] where
   numeral :: Integer -> Text
   numeral = fromJust . EN.gb_cardinal defaultInflection
-- Reinhard Zumkeller, Jan 23 2015

A008520Q[n_]:=StringContainsQ[IntegerName[n,"Words"],"e"];Select[Range[0,200],A008520Q] (* Paolo Xausa, Aug 11 2023 *)

Name edited by Michael De Vlieger, Aug 11 2023

A008522 Numbers whose American English name contains the letter 't'.

Original entry on oeis.org

2, 3, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

N. J. A. Sloane

			8 = eigh{t}.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008519 (o), A008520 (e), A008536 (n), A008538 (s), A008540 (f), A008553 (y).

A008522Q[n_]:=StringContainsQ[IntegerName[n,"Words"],"t"];Select[Range[0,200],A008522Q] (* Paolo Xausa, Aug 12 2023 *)

Name edited by Paolo Xausa, Aug 12 2023

A008519 Numbers whose American English name contains the letter 'o'.

Original entry on oeis.org

0, 1, 2, 4, 14, 21, 22, 24, 31, 32, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 54, 61, 62, 64, 71, 72, 74, 81, 82, 84, 91, 92, 94, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123

Offset: 1

N. J. A. Sloane

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008520 (e), A008522 (t), A008536 (n), A008538 (s), A008540 (f), A008553 (y).

A008519Q[n_]:=StringContainsQ[IntegerName[n,"Words"],"o"];Select[Range[0,200],A008519Q] (* Paolo Xausa, Aug 10 2023 *)

Name edited by Paolo Xausa, Aug 12 2023

A008538 Numbers whose American English name contains the letter 's'.

Original entry on oeis.org

6, 7, 16, 17, 26, 27, 36, 37, 46, 47, 56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 86, 87, 96, 97, 106, 107, 116, 117, 126, 127, 136, 137, 146, 147, 156, 157, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173

Offset: 1

N. J. A. Sloane

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008519 (o), A008520 (e), A008522 (t), A008536 (n), A008540 (f), A008553 (y).

A008538Q[n_]:=StringContainsQ[IntegerName[n,"Words"],"s"];Select[Range[0,200],A008538Q] (* Paolo Xausa, Aug 12 2023 *)

Name edited by Paolo Xausa, Aug 12 2023

A008540 Numbers whose American English name contains the letter 'f'.

Original entry on oeis.org

4, 5, 14, 15, 24, 25, 34, 35, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 64, 65, 74, 75, 84, 85, 94, 95, 104, 105, 114, 115, 124, 125, 134, 135, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158

Offset: 1

N. J. A. Sloane

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A008519 (o), A008520 (e), A008522 (t), A008536 (n), A008538 (s), A008553 (y).

A008540Q[n_]:=StringContainsQ[IntegerName[n,"Words"],"f"];Select[Range[0,200],A008540Q] (* Paolo Xausa, Aug 12 2023 *)

Name edited by Paolo Xausa, Aug 12 2023

A008553 Numbers whose American English name contains the letter 'y'.

Original entry on oeis.org

20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

N. J. A. Sloane

Paolo Xausa, Table of n, a(n) for n = 1..10000
Tanya Khovanova, Non Recursions

Cf. A008519 (o), A008520 (e), A008522 (t), A008536 (n), A008538 (s), A008540 (f).

A008553Q[n_]:=StringContainsQ[IntegerName[n,"Words"],"y"];Select[Range[0,200],A008553Q] (* Paolo Xausa, Aug 12 2023 *)

Name edited by Paolo Xausa, Aug 12 2023

A348692 Triangle whose n-th row lists the integers m such that A000178(n) / m! is a square, where A000178(n) = n$ = 1!2!...*n! is the superfactorial of n; if there is no such m, then n-th row = 0.

Original entry on oeis.org

1, 2, 0, 2, 0, 0, 0, 3, 4, 0, 0, 0, 6, 0, 8, 9, 0, 8, 9, 0, 7, 0, 10, 0, 0, 0, 12, 0, 0, 0, 14, 0, 0, 0, 15, 16, 0, 18, 0, 18, 0, 0, 0, 20, 0, 0, 0, 22, 0, 0, 0, 24, 25, 0, 0, 0, 26, 0, 0, 0, 28, 0, 0, 0, 30, 0, 32, 0, 32, 0, 0, 0, 34, 0, 0, 0, 35, 36, 0, 0, 0, 38, 0, 0, 0, 40

Offset: 1

Bernard Schott, Oct 30 2021

This sequence is the generalization of a problem proposed during the 17th Tournament of Towns (Spring 1996) and also during the first stage of the Moscow Mathematical Olympiad (1995-1996); the problem asked the question for n = 100 (see Andreescu-Gelca reference, Norman Do link, and Examples section).
Exhaustive results coming from Mabry-McCormick's link and adapted for OEIS:
-> n$ (A000178) is never a square if n > 1.
-> There is no solution if n is odd > 1, hence row(2q+1) = 0 when q > 0.
-> When n is even and there is a solution, then m belongs to {n/2 - 2, n/2 - 1, n/2, n/2 + 1, n/2 + 2}.
-> If 4 divides n (A008536), m = n/2 is always a solution because
    (n$) / (n/2)! = ( 2^(n/4) * Product_{j=1..n/2} ((2j-1)!) )^2.
-> For other cases, see Formula section.
-> When n is even, there are 0, 1 or 2 solutions, so, the maximal length of a row is 2.
-> It is not possible to get more than three consecutive 0 terms, and three consecutive 0 terms correspond to three consecutive rows such that (n, n+1, n+2) = (4u+1, 4u+2, 4u+3) for some u >= 1.

			For n = 4, 4$ / 3! = 48, 4$ / 4! = 12 but 4$ / 2! = 12^2, hence, m = 2.
For n = 8, 8$ / 2! is not a square, but m_1 = 3 because 8$ / 3! = 29030400^2 and m_2 = 4 because 8$ / 4! = 14515200^2.
For n = 14, m_1 = 8 because 14$ / 8! = 1309248519599593818685440000000^2 and m_2 = 9 because 14$ / 9! = 436416173199864606228480000000^2.
For n = 16, m_1 = 8 because 16$ / 8! = 6848282921689337839624757371207680000000000^2 and m_2 = 9 because 16$ / 9! = 2282760973896445946541585790402560000000000^2.
For n = 18, m = 7 because 18$ / 7! = 29230177671473293820176594405114531928195727360000000000000^2 and there is no other solution.
For n = 100, m = 50, unique solution to the Olympiad problems.
Triangle begins:
    1;
    2;
    0;
    2;
    0;
    0;
    0;
    8,  9;
    0;
    ...

Titu Andreescu and Rǎzvan Gelca, Putnam and Beyond, New York, Springer, 2007, problem 725, pp. 253 and 686.
Peter J. Taylor and A. M. Storozhev, Tournament of Towns 1993-1997, Book 4, Tournament 17, Spring 1996, O Level, Senior questions, Australian Mathematics Trust, 1998, problem 3, p. 96.

Diophante, A1963 - Le vilain petit canard (in French).
Norman Do, Factorial fun, Puzzle Corner 13, Gaz. Aust. Math. Soc. 36, 2009, 176-179, page 178.
Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009, pages 346-352.
Tournament of Towns, Tournament 17, 1995-1996, Spring 1996, O Level, Senior questions, question 3 (in Russian).
Index to sequences related to Olympiads and other Mathematical competitions.

Cf. A000178, A001541, A008536, A035008, A060626, A139098.

sf(n)=prod(k=2, n, k!); \\ A000178
row(n) = my(s=sf(n)); Vec(select(issquare, vector(n, k, s/k!), 1));
lista(nn) = {my(list = List()); for (n=1, nn, my(r=row(n)); if (#r, for (k=1, #r, listput(list, r[k])), listput(list, 0));); Vec(list);} \\ Michel Marcus, Oct 30 2021

When there are two such integers m, then m_1 < m_2.
If n = 8*q^2 (A139098), then m_1 = n/2 - 1 = 4q^2-1 (see example for n=8).
If n = 8q*(q+1) (A035008), then m_2 = n/2 + 1 = (2q+1)^2 (see example for n=16).
if n = 4q^2 - 2 (A060626), then m_1 = n/2 + 1 = 2q^2 (see example for n=14).
If n = 2q^2, q>1 in A001541, then m = n/2 - 2 = q^2-2 (see example for n=18).
If n = 2q^2-4, q>1 in A001541, then m_2 = n/2 + 2 = q^2 (see example for n=14).

A095790 Numbers whose name in English contains an "r".

Original entry on oeis.org

3, 4, 13, 14, 23, 24, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 53, 54, 63, 64, 73, 74, 83, 84, 93, 94, 103, 104, 113, 114, 123, 124, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148

Offset: 1

Michael Joseph Halm, Jul 10 2004

A008520 are numbers which contain an "e", A008540 an "f", A011538 a "g", A008536 an "n", A008519 an "o", A008538 an "s", A008522 a "t", A011534 a "u", A011532 a "w", A011536 an "x" and A008553 a "y"

			a(1) = 3 because "three" contains an "r", 0, 1 and 2 do not

Cf. A008520, A008540, A011538, A008536, A008519, A008538, A008522, A011534, A011532, A011536, A008553.

A095798 Numbers whose name in English contains a "v".

Original entry on oeis.org

5, 7, 11, 12, 17, 25, 27, 35, 37, 45, 47, 55, 57, 65, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 85, 87, 95, 97, 105, 107, 111, 112, 117, 125, 127, 135, 137, 145, 147, 155, 157, 165, 167, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 185, 187, 195, 197, 205, 207

Offset: 1

Michael Joseph Halm, Jul 10 2004

A008520 are numbers which contain an "e", A008540 an "f", A011538 a "g", A008536 an "n", A008519 an "o", A008538 an "s", A008522 a "t", A011534 a "u", A011532 a "w", A011536 an "x" and A008553 a "y"

			a(3) = 11 because "eleven" contains a "v" and it is the third number to do so (after "five" and "seven").

Cf. A008520, A008540, A011538, A008536, A008519, A008538, A008522, A011534, A011532, A011536, A008553.

Corrected by Rick L. Shepherd, Jul 10 2004

cp's OEIS Frontend

A008520 Numbers whose American English name contains the letter 'e'.

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A008522 Numbers whose American English name contains the letter 't'.

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A008519 Numbers whose American English name contains the letter 'o'.

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A008538 Numbers whose American English name contains the letter 's'.

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A008540 Numbers whose American English name contains the letter 'f'.

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A008553 Numbers whose American English name contains the letter 'y'.

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A348692 Triangle whose n-th row lists the integers m such that A000178(n) / m! is a square, where A000178(n) = n$ = 1!*2!*...*n! is the superfactorial of n; if there is no such m, then n-th row = 0.

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A095790 Numbers whose name in English contains an "r".

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A095798 Numbers whose name in English contains a "v".

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A348692 Triangle whose n-th row lists the integers m such that A000178(n) / m! is a square, where A000178(n) = n$ = 1!2!...*n! is the superfactorial of n; if there is no such m, then n-th row = 0.