Nathan Fox - cp's OEIS frontend

User: Nathan Fox

Nathan Fox has authored 176 sequences. Here are the ten most recent ones:

A375212 a(n) is the product of the leading base-3 digit of n and the sum of the squares of its base-3 digits.

1, 8, 1, 2, 5, 8, 10, 16, 1, 2, 5, 2, 3, 6, 5, 6, 9, 8, 10, 16, 10, 12, 18, 16, 18, 24, 1, 2, 5, 2, 3, 6, 5, 6, 9, 2, 3, 6, 3, 4, 7, 6, 7, 10, 5, 6, 9, 6, 7, 10, 9, 10, 13, 8, 10, 16, 10, 12, 18, 16, 18, 24, 10, 12, 18, 12, 14, 20, 18, 20, 26, 16, 18, 24, 18, 20

Offset: 1

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 16 2024

Nathan Fox, Table of n, a(n) for n = 1..100000
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A006287, A376270

A377089 Numbers that are both happy (A007770) and elated (A376272).

Original entry on oeis.org

1, 10, 13, 97, 100, 103, 130, 226, 262, 319, 356, 365, 391, 556, 565, 907, 970, 1000, 1003, 1030, 1122, 1177, 1188, 1212, 1221, 1222, 1277, 1300, 1339, 1393, 1448, 1478, 1484, 1487, 1557, 1575, 1717, 1727, 1748, 1755, 1771, 1772, 1784, 1818, 1844, 1847, 1874

Offset: 1

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

Every power of 10 is in this sequence, as both the sum of squared digits map (A003132) and the map A376270 map powers of 10 to 1.

Nathan Fox, Table of n, a(n) for n = 1..12205

Cf. A003132, A007770, A376270, A376272.

def ssd(n): return sum(int(d)**2 for d in str(n))
def f(n): return (d:=list(map(int, str(n))))[0] * sum(di*di for di in d)
def happy(n):
    if n == 1: return True
    s = list(map(int, str(n)))
    while n not in [1, 4]: n = ssd(n) # iterate until fixed point or cycle
    return n == 1
def elated(n):
    if n == 1: return True
    traj = {n}
    while (n:=f(n)) not in traj: traj.add(n)
    return 1 in traj
def ok(n): return happy(n) and elated(n)
print([k for k in range(1, 2001) if ok(k)]) # Michael S. Branicky, Oct 16 2024

A377088 Number of attractors under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.

Original entry on oeis.org

1, 5, 2, 3, 8, 6, 11, 4, 16, 14, 23, 18, 42, 7, 24, 34, 26, 58, 98, 51, 99, 88, 51, 57, 103, 72, 89, 60, 69, 35, 78, 146, 39, 90, 73, 11, 109, 113, 71, 156, 220, 93, 176, 101, 132, 172, 187, 10, 160, 95, 221, 226, 69, 55, 163, 110, 137, 287, 168, 69, 260, 194, 208

Offset: 2

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

If b>=2 and a>=b^3 then E(a,2,b)

			In the decimal system all integers go to (1), (298), (46, 208, 136), (26, 80, 512, 150), or (33, 54, 205, 58, 445, 228, 144) under iteration of the map A376270, hence there are two fixed points, one 3-cycle, one 4-cycle, and one 7-cycle. Therefore a(10) = 1 + 1 + 3 + 4 + 7 = 16.

Nathan Fox, Table of n, a(n) for n = 2..100
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A376270, A376272, A377086, A377087.

A193586 is the analog for happy numbers.

A377087 Number of cycles under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 0, 3, 2, 5, 2, 4, 1, 3, 3, 3, 2, 3, 4, 1, 5, 3, 7, 2, 3, 3, 3, 4, 4, 5, 3, 6, 12, 2, 1, 3, 2, 6, 10, 4, 8, 6, 3, 4, 2, 3, 1, 3, 4, 9, 3, 2, 2, 5, 7, 4, 8, 7, 5, 6, 6, 6, 1, 8, 7, 4, 6, 6, 2, 5, 7, 5, 5, 4, 5, 3, 4, 3, 5, 2, 4, 7, 8, 3, 7, 7

Offset: 2

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

If b>=2 and a>=b^3 then E(a,2,b)

			In the decimal system all integers go to (1), (298), (46, 208, 136), (26, 80, 512, 150), or (33, 54, 205, 58, 445, 228, 144) under iteration of the map A376270, hence there are two fixed points and three cycles. Therefore a(10) = 3.

Nathan Fox, Table of n, a(n) for n = 2..100
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A376270, A376272, A377086, A377088.

A193585 is the analog for happy numbers.

A377086 Number of fixed points under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.

Original entry on oeis.org

1, 2, 2, 1, 1, 4, 3, 4, 2, 3, 1, 3, 5, 2, 4, 4, 2, 4, 1, 3, 3, 3, 1, 5, 2, 3, 5, 4, 4, 9, 2, 1, 1, 4, 2, 6, 4, 1, 2, 5, 3, 6, 3, 1, 2, 3, 1, 7, 2, 1, 3, 3, 1, 5, 4, 6, 5, 4, 2, 8, 3, 2, 7, 3, 1, 4, 4, 2, 3, 6, 3, 9, 2, 3, 4, 9, 3, 7, 3, 2, 6, 5, 1, 7, 3, 3, 3

Offset: 2

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

If b>=2 and a>=b^3 then E(a,2,b)

Unlike the related sequence A193583, which contains only odd numbers, this sequence contains even numbers and odd numbers.

			In the decimal system all integers go to (1), (298), (46, 208, 136), (26, 80, 512, 150), or (33, 54, 205, 58, 445, 228, 144) under iteration of the map A376270, hence there are two fixed points and three cycles. Therefore a(10) = 2.

Nathan Fox, Table of n, a(n) for n = 2..100
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A376270, A376272, A377087, A377088.

A193583 is the analog for happy numbers.

A377085 a(n) = floor(A377084(n)/(n-1)^2).

Original entry on oeis.org

0, 1, 1, 5, 5, 6, 5, 6, 8, 9, 7, 11, 12, 11, 11, 14, 15, 16, 14, 16, 18, 18, 15, 22, 23, 20, 18, 27, 24, 23, 22, 28, 27, 27, 23, 31, 30, 28, 24, 35, 33, 32, 27, 34, 34, 34, 30, 40, 39, 36, 32, 40, 39, 40, 35, 44, 43, 42, 35, 45, 46, 43, 38, 51, 47, 47, 41, 49, 50

Offset: 3

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

nonn

Nathan Fox, Table of n, a(n) for n = 3..100
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A377084.

A377084 a(n) is the maximum integer for which some minimum-length sum equaling a(n) of perfect squares less than n^2 excludes (n-1)^2.

Original entry on oeis.org

3, 16, 31, 128, 191, 324, 368, 561, 891, 1200, 1104, 1993, 2535, 2692, 2896, 4321, 4880, 5832, 5776, 7485, 9144, 9680, 8903, 14353, 15576, 14872, 14527, 22736, 21697, 22500, 22587, 30537, 31451, 33524, 30076, 42768, 43664, 43097, 39317, 59200, 58835, 59893

Offset: 3

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

nonn

			561 cannot be written as a sum of fewer than nine perfect squares less than 10^2. 561 can be written as a sum of nine of these numbers in five ways:
561 = 1^2 + 5^2 + 7^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2
561 = 3^2 + 6^2 + 8^2 + 8^2 + 8^2 + 9^2 + 9^2 + 9^2 + 9^2
561 = 3^2 + 7^2 + 7^2 + 7^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2
561 = 5^2 + 5^2 + 5^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2 + 9^2
561 = 7^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2 + 8^2
The last sum here does not include 9^2, so a(10) >= 561. In fact, a(10) = 561, as every number larger than 561 has 9^2 in every shortest decomposition of this form.

Nathan Fox, Table of n, a(n) for n = 3..100
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A377085.

A377083 Number of iterations required for elated number A376272(n) to converge to 1.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 4, 7, 4, 9, 5, 1, 2, 4, 3, 2, 3, 4, 4, 2, 2, 5, 4, 3, 5, 3, 4, 5, 4, 3, 3, 3, 3, 5, 2, 2, 4, 4, 3, 3, 3, 3, 3, 3, 7, 9, 7, 4, 5, 9, 5, 6, 4, 6, 9, 4, 7, 10, 5, 5, 8, 10, 8, 6, 8, 8, 7, 10, 6, 4, 5, 6, 7, 6, 2, 5, 7, 2, 7, 4, 7, 9, 5, 9, 5, 5

Offset: 1

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

			21 is the 4th elated number and iterating the map A376270 yields 10 then 1, so a(4)=2.

Nathan Fox, Table of n, a(n) for n = 1..7832
N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A376270, A376272.

A090425 is the analog for happy numbers, with a different convention used.

from itertools import count, islice
def f(n): return (d:=list(map(int, str(n))))[0] * sum(di*di for di in d)
def ok_count(n):
    if n == 1: return True, 0
    traj, c = {n}, 0
    while (n:=f(n)) not in traj: traj.add(n); c += 1
    return 1 in traj, c
def agen(): # generator of terms
    for n in count(1):
        elated, iterations = ok_count(n)
        if elated: yield iterations
print(list(islice(agen(), 90))) # Michael S. Branicky, Oct 16 2024

A377082 a(0) = 1, a(1) = 3, a(2) = 13; for n >= 2, a(n+1) = 2*3^((a(n)-1)/4) - 1.

Original entry on oeis.org

1, 3, 13, 53, 3188645

Offset: 0

N. Bradley Fox, Nathan Fox, Helen Grundman, Rachel Lynn, Changningphaabi Namoijam, Mary Vanderschoot, Oct 15 2024

nonn

The next term is too large to include.
a(n) is the least positive integer that requires n steps to reach 1 under iteration of the sum of squares of digits function in base 3 (A006287).
a(n) is also the least positive integer that requires n steps to reach 1 under iteration of the map A375212.

N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.
G. Lapointe, On finding the smallest happy numbers of any heights, arXiv:1904.12032 [math.NT], 2019.

Cf. A006287, A007013, A375212.

A373239 Relative of Hofstadter Q-sequence: a(n) = max(0, n+118) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

Original entry on oeis.org

6, 119, 120, 121, 9, 122, 123, 124, 12, 125, 126, 127, 15, 128, 129, 17, 131, 18, 131, 133, 134, 22, 21, 247, 241, 9, 18, 256, 259, 127, 22, 148, 153, 131, 27, 36, 155, 246, 122, 39, 156, 162, 126, 42, 158, 165, 16, 157, 165, 145, 40, 157, 55, 260, 134, 46, 167, 178, 25, 38, 58, 523, 250, 122, 61, 71, 299, 238, 116, 72, 190, 192

Offset: 1

Nathan Fox, May 28 2024

Sequences like this are more naturally considered with the first nonzero term in position 1. But this sequence would then match A000027 for its first 118 terms.
This sequence has exactly 127 terms (of positive index). a(127) = 0, so an attempt to calculate a(128) would refer to itself.
Without the convention that a(n) = 0 for n <= -118, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-222).
If 118 in this sequence's definition is replaced by any larger number congruent to 6 mod 7, the behavior is essentially the same, though the quasilinear part (see Formula section) lasts longer.

Nathan Fox, Table of n, a(n) for n = 1..127

Cf. A005185, A267501, A278055, A373234, A373235, A373236, A373237, A373238, A274058.

If the index is between 67 and 118 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+120, a(7n+2) = 7n+122, a(7n+3) = 7, a(7n+4) = 2n+281, a(7n+5) = n+229, a(7n+6) = 116.

cp's OEIS Frontend

User: Nathan Fox

A375212 a(n) is the product of the leading base-3 digit of n and the sum of the squares of its base-3 digits.

Views

Author

Keywords

Links

Crossrefs

A377089 Numbers that are both happy (A007770) and elated (A376272).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

A377088 Number of attractors under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A377087 Number of cycles under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A377086 Number of fixed points under iteration of the map sending a positive integer to the product of its leading base-n digit and the sum of the squares of its base-n digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A377085 a(n) = floor(A377084(n)/(n-1)^2).

Views

Author

Keywords

Links

Crossrefs

A377084 a(n) is the maximum integer for which some minimum-length sum equaling a(n) of perfect squares less than n^2 excludes (n-1)^2.

Views

Author

Keywords

Examples

Links

Crossrefs

A377083 Number of iterations required for elated number A376272(n) to converge to 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

A377082 a(0) = 1, a(1) = 3, a(2) = 13; for n >= 2, a(n+1) = 2*3^((a(n)-1)/4) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

A373239 Relative of Hofstadter Q-sequence: a(n) = max(0, n+118) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula