OFFSET
1,2
COMMENTS
The 15x+1 analog of A267703 (which treats the 7x+1 map). Here 15 = 2^4 - 1 is a Mersenne number, and 15*1 + 1 = 16 = 2^4, so 1 lies on the cycle 1 -> 16 -> 8 -> 4 -> 2 -> 1.
"Conjectured" in the same sense as A267703: every term shown provably reaches 1, but it is conjectural that the numbers not shown never do (a trajectory might in principle descend after exceeding the search bound).
Terms were computed by exact backward-tree enumeration from 1 (preimages of v are 2*v, and (v-1)/15 when that is an odd integer > 1) with excursion bound 10^16; the list is unchanged at bounds 10^17 and 10^18 and is certified complete below 10^16 under that bound.
The sequence is infinite: it contains A131865, since 15*(16^(j+1)-1)/15 + 1 = 16^(j+1) is a power of 2. If k is a term, then so is 2*k. All powers of 2 are terms.
Almost all positive integers diverge under this map (Heppner 1978); the present sequence has density 0.
The number of terms <= X appears to grow like X^c with c ~= 0.1310, where c is the maximum over b of (H(b) - b*L)/(1 - b*L) with L = log2(15) and H the binary entropy function (conjectural). There are 57, 87, 123, 182 terms <= 10^4, 10^5, 10^6, 10^7 respectively.
LINKS
Karan Sharma, Table of n, a(n) for n = 1..182
E. Heppner, Eine Bemerkung zum Hasse-Syracuse-Algorithmus, Arch. Math. 31 (1978), 317-320.
EXAMPLE
9 is a term: 9 -> 136 -> 68 -> 34 -> 17 -> 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1.
3 is (conjecturally) not a term: 3 -> 46 -> 23 -> 346 -> 173 -> 2596 -> ... appears to grow without ever reaching 1.
PROG
(Python)
def aupto(lim, bound=10**12): # bound = excursion cap, must far exceed lim
seen, todo = {1}, [1]
for v in todo: # walk the tree of preimages of 1
kids = [2*v]
if v % 15 == 1 and (v // 15) % 2: # (v-1)/15 is an odd integer
kids.append(v // 15)
for u in kids:
if 1 < u <= bound and u not in seen:
seen.add(u); todo.append(u)
return sorted(x for x in seen if x <= lim)
print(aupto(10**4))
CROSSREFS
KEYWORD
nonn
AUTHOR
Karan Sharma, Jun 11 2026
STATUS
approved