A080766 A unitary phi reciprocal amicable number: consider two different numbers r, s which satisfy the following equation for some integer k: uphi(r) = uphi(s) = (1/k) * r * s / (r-s); or equivalently, 1/uphi(r) = 1/uphi(s) = k * (1/s - 1/r); sequence gives r numbers.

2, 6, 14, 21, 240, 240, 242, 5520, 5566, 6578, 10881, 13056, 14880, 15444, 46200, 47908, 57600, 60480, 65280, 65292, 91392, 152320, 169728, 239540, 285696, 399168, 665280, 702000, 941625, 1405404, 2246400, 4285440, 4856832, 6591375, 10463040, 11713536, 13172544

Offset

1,1

Comments

Here uphi(n) = A047994(n) is the unitary totient function: if n = Product p_i^e_i, uphi(n) = Product (p_i^e_i - 1).

Solutions sorted by increasing r, then increasing s, so repeated values are possible here; they occur at a(5) = a(6), a(59) = a(60), etc. - Michael S. Branicky, Sep 30 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 1..66

Prog

(Python)
from math import prod
from sympy import factorint
from itertools import count, islice
def uphi(n): return prod(p_i**e_i-1 for p_i, e_i in factorint(n).items())
def agen(): # generator of terms (increasing r, then increasing s)
    uphi_level_sets = dict()
    for r in count(1):
        v = uphi(r)
        if v in uphi_level_sets:
            for s in uphi_level_sets[v]:
                k, rem = divmod(r*s, v*(r-s))
                if rem == 0:
                    yield r # use s, k for A080767, A080768, resp.
        else:
            uphi_level_sets[v] = []
        uphi_level_sets[v].append(r)
print(list(islice(agen(), 25))) # Michael S. Branicky, Sep 30 2025

Crossrefs

Cf. A047994, A080767, A080768, A067739, A067741.

Sequence in context: A071122 A173724 A101572 * A262506 A392092 A228649

Adjacent sequences: A080763 A080764 A080765 * A080767 A080768 A080769

Keyword

nonn

Author

Yasutoshi Kohmoto

Extensions

Kohmoto found 2nd, 6th, 13th, 25th terms. Dean Hickerson calculated the other terms.

Offset corrected by Sean A. Irvine, Sep 29 2025

a(31) and beyond from Michael S. Branicky, Sep 29 2025

Status

approved