A080767 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 s numbers.

1, 3, 12, 20, 144, 220, 240, 5060, 5520, 5520, 10800, 11520, 8928, 15120, 31680, 33984, 56576, 60372, 39168, 65280, 80640, 149760, 149760, 169920, 281600, 398200, 664092, 669600, 940896, 1235520, 2232576, 3953664, 4787200, 6586272, 10444356, 11545600, 13170600

Offset

1,2

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. - Sean A. Irvine, Sep 29 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
    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 s
        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, A080766, A080768, A067739, A067741.

Sequence in context: A374072 A193558 A256131 * A043465 A074276 A055041

Adjacent sequences: A080764 A080765 A080766 * A080768 A080769 A080770

Keyword

nonn

Author

Yasutoshi Kohmoto

Extensions

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

Offset corrected and ordering defined by Sean A. Irvine, Sep 29 2025

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

Status

approved