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0, 1, 2, 3, 5, 6, 7, 128, 2401, 8192, 78125, 524288, 823543, 33554432, 282475249, 1220703125, 2147483648, 96889010407, 137438953472, 8796093022208, 19073486328125, 33232930569601, 562949953421312, 11398895185373143, 36028797018963968, 298023223876953125, 2305843009213693952
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internal format)
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OFFSET
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1,3
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COMMENTS
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Theorem: k is a term if and only if k is 0, 1, 3, 6 or of the form 2^(6*m+1), 5^(6*m+1), or 7^(3*m+1), m >= 0.
Proof: 0, 1, 3, 6 can be shown to be terms by direct computation. Since 1 <= A010888(k) <= 9 for k > 0, all terms > 0 are of the form q^m, where 2 <= q <= 9. This implies that all terms > 1 are either 6^m or of the form p^m, where p is a prime <= 7. If k = 6^m for m > 1, then k is divisible by 9 and A010888(k) = 9, whereas A052410(k) = 6 and thus k is not a term. Similarly, if k = 3^m for m > 1, then k is divisible by 9 and A010888(k) = 9 whereas A052410(k) = 3 and thus k is not a term. Lastly, for p = 2, 5 or 7, A010888(p^m) = 1 + ((p^m-1) mod 9), and since p^m and 9 are coprime, this implies that A010888(p^m) = p^m mod 9 whereas A052410(p^m) = p, thus m must satisfy p^m mod 9 = p mod 9. Since 6, 6, 3 are the multiplicative orders of 2, 5, 7 modulo 9 respectively, implying 2^6 == 5^6 == 7^3 == 1 (mod 9), the result follows.
(End)
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LINKS
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MATHEMATICA
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A052409[n_]:=GCD@@Last/@FactorInteger[n]; A010888[n_]:=If[n==0, 0, n-9 Floor[(n-1)/9]]; a={}; kmax = 10^9; For[k=0, k<=kmax, k++, If[k^(1/A052409[k])==A010888[k], AppendTo[a, k]]]; a
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PROG
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(Python)
from itertools import count, islice
from math import gcd
from sympy import factorint, integer_nthroot
def A370688_gen(startvalue=0): # generator of terms >= startvalue
if startvalue <=0: yield 0
if startvalue <=1: yield 1
for k in count(max(startvalue, 2)):
r = 1 + (k - 1) % 9
if r>1:
kmin, kmax = 0, 1
while r**kmax <= k:
kmax <<= 1
while True:
kmid = kmax+kmin>>1
if r**kmid > k:
kmax = kmid
else:
kmin = kmid
if kmax-kmin <= 1:
break
if r**kmin==k:
m = integer_nthroot(k, gcd(*factorint(k).values()))[0]
if m == r:
yield k
(Python)
# faster program based on theorem
from itertools import islice
def A370688_gen(): # generator of terms
kmax, mlist, dlist = 10, [7, 7, 4], [6, 6, 3]
yield from (0, 1, 2, 3, 5, 6, 7)
while True:
klist = []
for i, p in enumerate((2, 5, 7)):
while (k:=p**mlist[i]) <= kmax:
klist.append(k)
mlist[i] += dlist[i]
yield from sorted(klist)
kmax *= 10
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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