A080768 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 k numbers.

2, 3, 14, 35, 3, 22, 242, 23, 253, 13, 155, 12, 3, 77, 5, 4, 65, 860, 3, 10882, 14, 91, 13, 5, 80, 543, 946, 25, 1350, 13, 190, 16, 85, 1575, 865, 82, 8130, 989, 1425, 65, 209, 862, 43608, 26, 192, 5665, 148, 968, 23632, 17, 6, 41, 8, 294, 27, 828, 1359, 6, 4, 11

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).

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 k
        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, A080767, A067739, A067741.

Sequence in context: A355176 A346606 A059188 * A370616 A203578 A329442

Adjacent sequences: A080765 A080766 A080767 * A080769 A080770 A080771

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