A268238 -id:A268238 - cp's OEIS frontend

A268236 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 n in base b.

0, 1, 2, 3, 4, 11, 12, 13, 20, 14, 14, 14, 14, 14, 24, 14, 24, 14, 24, 34, 40, 41, 42, 43, 44, 41, 42, 43, 44, 104, 42, 43, 44, 45, 114, 43, 44, 45, 46, 124, 44, 45, 46, 47, 44, 140, 141, 142, 44, 144, 46, 47, 44, 104, 204, 47, 44, 49, 134, 214, 44, 141, 142, 143, 144, 145

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.
For n>65 "digits" greater than 9 appear in a(n) - see the first link. This explains why this sequence has no b-file: the OEIS restriction to decimal digits means that a(66) cannot be written as a single base-10 number (it would be "4,10").
The Fouriest transform pun suggests (by analogy with shaky, shakier, shakiest) investigating the Foury, Fourier, and Fouriest numbers. Three obvious candidates for the Foury numbers are A011534, A019764, and A268544, which are all "Foury" in different ways.
With respect to a fixed base b, we could say that n is Fourier than m (in base b) if the fraction [or number?] of 4's in the representation of n (base b) is greater than the analogous quantity for m. But it is not clear which definition is to be preferred. In base 10, which is Fourier, 440 or 439454?
This sequence and its companions were created during a dinner following the Experimental Mathematics Seminar at Rutgers University on Feb 04 2016.

			For n=24, the base-5 representation of 24 is 44. So the Fouriest transform of 24 is a(24) = 44, which uses base b = A268237(24) = 5 and contains A268238(24) = 2 4's.
The Fouriest transform of n=66 is 4,10 in base b=14 (note the non-decimal digit) and contains a single 4.

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. A268237 (the base b), A268238 (number of 4's).
See A268540 and A268541 for the "44" entries.
See also the "Foury" numbers A011534, A019764, and A268544.
A268360 and A349031 are other Foury sequences.

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).

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

Original entry on oeis.org

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.

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

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)

cp's OEIS Frontend

A268236 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 n in base b.

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

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

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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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Python