A268236 -id:A268236 - cp's OEIS frontend

A268540 Numbers whose Fouriest transform (see A268236) is 44.

24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 68, 72, 76, 80, 84, 88, 92, 96, 108, 112, 116, 128, 132, 140, 152, 156, 176, 180, 184, 188, 192, 204, 212, 216, 220, 232, 236, 240, 252, 264, 272, 296, 304, 312, 320, 332, 336, 348, 380, 392, 396, 408, 412, 416, 428, 432, 436, 456, 468, 472, 476, 480, 492, 500, 508, 512, 516

Offset: 1

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 27 2016

If we are ever going to understand A268236 then we need to understand this sequence first.
Based on Nathan Fox's extended table in A268236.
Equivalently, numbers 4k (k>5) whose representations in bases 5 through k-2 each contain at most one 4.
Equivalently, numbers 4k (k>5) whose representations in integer bases less than sqrt(4k) each contain at most one 4.
Is this sequence infinite?

Nathan Fox, Table of n, a(n) for n = 1..507

Cf. A268236, A268237, A268238, A268541.

A270030 a(n) is the smallest b for which the base-b representation of n contains at least one 4 (or 0 if no such base exists).

Original entry on oeis.org

0, 0, 0, 5, 0, 0, 0, 0, 5, 6, 7, 8, 9, 5, 11, 6, 13, 7, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 5, 7, 7, 7, 7, 5, 8, 8, 8, 8, 5, 6, 9, 9, 9, 5, 5, 5, 5, 5, 5, 11, 11, 6, 7, 5, 12, 12, 12, 6, 5, 6, 6, 6, 6, 5, 6, 14, 7, 8, 5, 5, 5, 5, 5, 5, 16, 6, 7, 7, 5, 7, 7, 6, 7, 5, 9, 18, 18, 6, 5, 19

Offset: 1

Nathan Fox, Mar 08 2016

a(n) > 0 for n >= 9 since 14 is n written in base n-4.

The only perfect k-th powers (k >= 2) that can appear in this sequence are 2^k, 3^k, or 4^k, with k a prime number.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A268236, A268237, A270027, A270028, A216194, A270029, A270031, A270032, A270033, A270034, A270035, A270040.

Table[SelectFirst[Range[5, 10^3], DigitCount[n, #, 4] > 0 &], {n, 9, 120}] (* Michael De Vlieger, Mar 10 2016, Version 10 *)

a(n) = if ((n<9) && (n!=4), 0, my(b=5); while(!vecsearch(Set(digits(n, b)), 4), b++); b); \\ Michel Marcus, Mar 10 2016

A270040 a(n) = Smallest m >= 9 containing no fours when represented in any base from 5 through n.

Original entry on oeis.org

10, 11, 12, 13, 15, 15, 17, 17, 66, 75, 75, 86, 86, 90, 138, 138, 138, 138, 138, 138, 138, 138, 138, 182, 182, 182, 182, 182, 182, 182, 182, 182, 185, 781817578165, 781817578165, 7826560751018861596150680

Offset: 5

Nathan Fox, Mar 09 2016

It remains to be determined if the sequence is finite.

These numbers are not very Foury, at least not initially. (See A268236.)

Nathan Fox, Table of n, a(n) for n = 5..59

Cf. A268236, A268237, A270030, A270037, A270038, A216192, A270039, A270041, A270042, A270043, A270044, A270045.

Table[SelectFirst[Range[9, 10^3], Total@ Map[Function[k, DigitCount[#, k, 4]], Range[5, n]] == 0 &], {n, 5, 60}] /. n_ /; MissingQ@ n -> Nothing (* Michael De Vlieger, Mar 10 2016, Version 10.2 *)

A268237 From the Fouriest transform of n: write n in that base b >= 4 which maximizes the number of 4's; in case of a tie pick the smallest b; sequence gives the base b.

Original entry on oeis.org

4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 6, 7, 8, 9, 5, 11, 6, 13, 7, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 5, 7, 7, 7, 7, 5, 8, 8, 8, 8, 5, 9, 9, 9, 9, 10, 5, 5, 5, 11, 5, 11, 11, 12, 7, 5, 12, 13, 12, 6, 5, 14, 6, 6, 6, 6, 6, 14, 7, 16, 5, 5, 5, 17, 5, 5, 16, 18, 7, 7, 5, 19, 7, 6, 7, 20, 9, 18, 18, 21, 5, 19, 19, 22, 19, 5, 5, 23, 5, 5, 5, 6

Offset: 0

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 06 2016

If no base b gives any 4's then we take b=4.

a(n) > 4 for n >= 9 since 14 is n written in base n-4. - Chai Wah Wu, Feb 06 2016

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Nathan Fox, First 10000 terms of A268236, A268237, A268238. Square brackets are used to separate the "digits" of A268236, since for n >= 66 these can be greater than 10.
Zach Weinersmith, Fouriest number, SMBC (Saturday Morning Breakfast Cereal) column, Feb 01, 2013.

Cf. A268236 (Fouriest transform of n), A268238 (number of 4's).

A268238 From the Fouriest transform of n: write n in that base b >= 4 which maximizes the number of 4's; in case of a tie pick the smallest b; sequence gives the number of 4's.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2

Offset: 0

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 06 2016

If no base b gives any 4's then we take b=4.
The first occurrence of any value m in this sequence is at position 5^m-1.
a(n) > 0 for n >= 9 since 14 is n written in base n-4. - Chai Wah Wu, Feb 06 2016

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Nathan Fox, First 10000 terms of A268236, A268237, A268238. Square brackets are used to separate the "digits" of A268236, since for n >= 66 these can be greater than 10.
Zach Weinersmith, Fouriest number, SMBC (Saturday Morning Breakfast Cereal) column, Feb 01, 2013.

Cf. A268236 (Fouriest transform of n), A268237 (the base b).

A268541 Base in which the Fouriest transform of A268540(n) is 44.

Original entry on oeis.org

5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 31, 32, 34, 37, 38, 43, 44, 45, 46, 47, 50, 52, 53, 54, 57, 58, 59, 62, 65, 67, 73, 75, 77, 79, 82, 83, 86, 94, 97, 98, 101, 102, 103, 106, 107, 108, 113, 116, 117, 118, 119, 122, 124, 126, 127, 128, 134, 137, 138, 139, 146, 158, 163, 164

Offset: 1

Jake Baron, Patrick Devlin, Nathan Fox, and N. J. A. Sloane, Feb 27 2016

If we are ever going to understand A268236 then we need to understand this sequence first.

Based on Nathan Fox's table in A268236.

Nathan Fox, Table of n, a(n) for n = 1..507

Cf. A268236, A268237, A268238, A268540.

a(n) = A268540(n)/4 - 1

A268544 Numbers whose name in American English has exactly four e's.

Original entry on oeis.org

17, 73, 77, 103, 107, 112, 113, 114, 115, 116, 121, 125, 128, 129, 133, 137, 143, 147, 153, 157, 163, 167, 170, 172, 174, 176, 181, 185, 188, 189, 191, 195, 198, 199, 211, 218, 219, 223, 227, 271, 275, 278, 279, 283, 287, 293, 297, 301, 305, 308, 309, 310, 320

Offset: 1

N. J. A. Sloane, Feb 07 2016

Arose in discussion of A268236.

			17 is a term because seventeen has four e's.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, American English names for the numbers from 0 to 100999 without spaces or hyphens

Cf. A001477, A268236.

from num2words import num2words
def ok(n): return num2words(n).count('e') == 4
print([k for k in range(321) if ok(k)]) # Michael S. Branicky, Mar 25 2025

a(35) and beyond from Michael S. Branicky, Mar 25 2025

A349031 The "Fouriest" numbers: numbers that can be expressed as a string of 4's longer than the original number in some base.

Original entry on oeis.org

624, 3124, 6220, 15624, 37324, 78124, 78432, 223948, 390624, 549028, 1343692, 1953124, 3843200, 8062156, 9586980, 9765624, 26902404, 48372940, 48828124, 76695844, 188316832, 244140624, 290237644, 613566756, 1220703124, 1318217828, 1741425868, 4908534052

Offset: 1

David Consiglio, Jr., Nov 06 2021

Inspired by Saturday Morning Breakfast Cereal comics.

The "Fouriest" numbers. The number shown in the comment, 624, is correctly identified as a term of the sequence.

			624 is a member of this sequence because 624 expressed in base 5 is 4444. 4444 has 4 digits and 624 has only 3.

David Consiglio, Jr., Table of n, a(n) for n = 1..461
Saturday Morning Breakfast Cereal comics, Teaching math was way more fun after tenure, Feb 01 2013.

For the Fouriest transform see A268236-A268238, A268300. - N. J. A. Sloane, Nov 19 2021

from math import log
def A349031(limit):
    super_fours = []
    for length in range(4,int(log(limit,5))+1):
        fours = "4"*length
        for base in range(5, 10):
            keep = 4*(1-base**length)//(1-base)
            if len(str(keep)) < len(fours) and keep < limit:
                super_fours.append(keep)
    return sorted(super_fours)
result = A349031(10**20)

A268360 "Four"ier sequence: there is a unique base b >= 5 such that (n base b) contains the largest number of 4's.

Original entry on oeis.org

9, 10, 11, 12, 13, 15, 17, 24, 28, 32, 36, 40, 44, 48, 49, 52, 56, 60, 68, 72, 74, 76, 80, 81, 84, 88, 92, 96, 99, 108, 109, 112, 114, 116, 119, 122, 123, 124, 128, 130, 132, 140, 149, 152, 154, 156, 166, 170, 171, 172, 173, 174, 176, 178, 179, 180, 184, 188, 192, 199

Offset: 1

Erich Friedman, Feb 03 2016

			130 is in the list because 130 in base 7 is written 244, and this is the only base where 130 contains two 4's.

See also the Fouriest transform of n, A268236.

cp's OEIS Frontend

A268540 Numbers whose Fouriest transform (see A268236) is 44.

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A270030 a(n) is the smallest b for which the base-b representation of n contains at least one 4 (or 0 if no such base exists).

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A270040 a(n) = Smallest m >= 9 containing no fours when represented in any base from 5 through n.

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A268237 From the Fouriest transform of n: write n in that base b >= 4 which maximizes the number of 4's; in case of a tie pick the smallest b; sequence gives the base b.

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A268238 From the Fouriest transform of n: write n in that base b >= 4 which maximizes the number of 4's; in case of a tie pick the smallest b; sequence gives the number of 4's.

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A268541 Base in which the Fouriest transform of A268540(n) is 44.

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A268544 Numbers whose name in American English has exactly four e's.

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A349031 The "Fouriest" numbers: numbers that can be expressed as a string of 4's longer than the original number in some base.

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A268360 "Four"ier sequence: there is a unique base b >= 5 such that (n base b) contains the largest number of 4's.

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