A191648
a(1)=1, a(2)=1. For n > 2, start with n and iterate the map (k -> concatenation of anti-divisors of k) until we reach a prime q; then a(n) = q. If we never reach a prime, a(n) = 0.
Original entry on oeis.org
1, 1, 2, 3, 23, 3
Offset: 1
The anti-divisors of 5 are 2, 3, and 23 is prime, hence a(5) = 23.
The anti-divisors of 7 are 2, 3, 5, and 235 is composite; the anti-divisors of 235 are 2, 3, 7, 10, 67, 94, 157, and 237106794157 = 59*547*7346909 is composite; the anti-divisors of 237106794157 start 2, 3, 5, 15, 118, 1094, 1709, 4519, 61403, 64546, 7722971, 14693818, 104937727, but the others are unknown, hence a(7) is also unknown.
Note that the example speaks of the anti-divisors of 237106794157 being incomplete. As of this date, those are well-established (see code of A066272). It is the primality evaluation chain from anti-divisors for n=7 from 23515118109417094519614036454677229711469381810493772727748015786693526280375184463161423922194842717663158071196105 that is incomplete. - _Bill McEachen_, Dec 14 2022
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antidivisors := proc(n) local a, k; a := {} ; for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a := a union {k} ; end if; end do: a ; end proc:
A130846 := proc(n) digcatL(sort(convert(antidivisors(n),list))) ; end proc:
A191648 := proc(n) if n <=2 then 1; else m := A130846(n) ; while not isprime(m) do m := A130846(m) ; end do: return m; end if; end proc: # R. J. Mathar, Jun 30 2011
A191859
The primes created by concatenation of anti-divisors in A191647.
Original entry on oeis.org
2, 3, 23, 347, 349, 311, 391627, 3471331, 384067, 2310175897, 239111323273399, 23167, 3784097136227, 235983149249, 3428116271, 37111677121283, 23293, 3471949133311, 231314228398154359, 378112153101159371, 2379127163381
Offset: 1
A191647(6) = 16, the anti-divisors of 16 are 3, 11. Hence a(6) = 311.
A191647(8) = 46, the anti-divisors of 46 are 3, 4, 7, 13, 31. Hence a(8) = 3471331.
A308533
Numbers such that the sum of anti-divisors divides the concatenation (in ascending order) of anti-divisors.
Original entry on oeis.org
3, 4, 6, 15, 27, 30, 54, 69, 90, 96, 99, 120, 126, 481, 564, 1050, 1656, 3480, 7680, 9612, 11520, 393216, 612846, 2220864, 5506086, 5579652, 8177664, 18087936, 23711514, 111544794, 440477976, 555176025
Offset: 1
Anti-divisors of 1656 are 7, 11, 16, 43, 48, 77, 144, 301, 368, 473, 1104 and their sum is 2592. Then, 711164348771443013684731104 / 2592 = 274368961717377705896887.
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P:=proc(q) local a,b,k,n; for n from 3 to q do a:=0: b:=0:
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then
a:=a*10^length(k)+k: b:=b+k: fi; od; if frac(a/b)=0 then print(n);
fi; od; end: P(10^6);
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from itertools import islice, count
from sympy.ntheory.factor_ import antidivisors
def A308533gen(): # generator of terms
for n in count(3):
a = antidivisors(n)
if int(''.join(str(s) for s in a)) % sum(a) == 0:
yield n
A308533_list = list(islice(A308533gen(),22)) # Chai Wah Wu, Dec 08 2021
Showing 1-3 of 3 results.
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