A376270 -id:A376270 - cp's OEIS frontend

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.

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.

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.

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.

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.

Original entry on oeis.org

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

A376273 a(n) is the smallest elated number of height n.

Original entry on oeis.org

1, 10, 13, 51, 67, 97, 668, 77, 746, 92, 717, 5369, 8888999999

Offset: 0

Michel Marcus, Sep 18 2024

a(13) = 8158 * 10^13888887 - 1, per Fox et al., which is too large to include. - Michael S. Branicky, Sep 18 2024

N. Bradley Fox et al., Elated Numbers, arXiv:2409.09863 [math.NT], 2024.

Cf. A376270 (map), A376272 (elated numbers).

A007013 is the analog for base 2 (with shifted offset).

f(n) = if (n, my(d=digits(n)); d[1]*norml2(d), 0); \\ A376270
g(n) = my(list=List()); listput(list, n); while(1, my(m=f(n)); if (m==1, return(#list)); if (#select(x->(x==m), Vec(list)), return(0)); listput(list, m); n=m); -1;
a(n) = if (n==0, 1, my(k=2); while(g(k) != n, k++); k);

from itertools import count, islice, combinations_with_replacement as mc
def f(n): return (d:=list(map(int, str(n))))[0] * sum(di*di for di in d)
def iters(n):
    if n == 1: return 0
    traj, c = {n}, 0
    while (n:=f(n)) not in traj: traj.add(n); c += 1
    return c if 1 in traj else float('inf')
def bgen():
    yield from (int(f+"".join(m)) for d in count(1) for f in "123456789" for m in mc("0123456789", d-1))
def agen(): # generator of terms
    adict, n = dict(), 0
    for k in bgen():
        v = iters(k)
        if v not in adict:
            adict[v] = k
            while n in adict: yield adict[v]; n += 1
print(list(islice(agen(), 13))) # Michael S. Branicky, Sep 18 2024

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

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

Showing 1-8 of 8 results.

cp's OEIS Frontend

A376272 Elated numbers: numbers whose trajectory under iteration of the A376270 map includes 1.

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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.

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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.

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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.

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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.

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A376273 a(n) is the smallest elated number of height n.

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PARI

Python

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

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Python

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

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Python