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A371641
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The smallest composite number which divides the concatenation of its ascending ordered prime factors, with repetition, when written in base n.
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8
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85329, 4, 224675, 4, 1140391, 4, 9, 4, 28749, 4, 841, 4, 9, 4, 239571, 4, 343, 4, 9, 4, 231, 4, 25, 4, 9, 4, 315, 4, 343, 4, 9, 4, 25, 4, 9761637601, 4, 9, 4, 4234329, 4, 715, 4, 9, 4, 609, 4, 49, 4, 9, 4, 195, 4, 25, 4, 9, 4, 1015, 4, 76729, 4, 9, 4, 25, 4, 14332171
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OFFSET
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2,1
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COMMENTS
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The number 4 = 2*2 in any base b = 3 + 2*n, n >= 0, will always divide the concatenation of its prime divisors as 4 = "2"_b + "2"_b = "22"_b = 2*(3 + 2*n) + 2 = 8 + 4*n, which is divisible by 4.
The number 9 = 3*3 in any base b = 8 + 6*n, n >= 0, will always divide the concatenation of its prime divisors as 9 = "3"_b + "3"_b = "33"_b = 3*(8 + 6*n) + 3 = 27 + 18*n, which is divisible by 9.
Theorem: if p is prime, then a(p*(m+2)-1) <= p^2 for all m >= 0. Proof: If p is prime and base b = p*(m+2)-1 for some m >= 0, then b > p and p expressed in base b = "p"_b and thus "pp"_b = p*(b+1) = p^2*(m+2), i.e., divisible by p^2. - Chai Wah Wu, Apr 01 2024
a(36) <= 9761637601. a(40) = 4234329. By the theorem above if n+1 is composite, then a(n) <= p^2 where p = A020639(n+1) is the smallest prime factor of n+1. The first few terms where the inequality is strict are: a(406) = 105, a(766) = 105, a(988) = 195, a(1036) = 105, a(1072) = 231, ... - Chai Wah Wu, Apr 11 2024
Theorem: If n is even, then a(n) >= 9 is odd.
Proof: This is true for n = 2. Let n > 2 be even. Let m be even.
If m=2^k for k>1, then 2 concatenated k times in base n is 2*(n^(k-1)+...+n+1) = 2*(n^k-1)/(n-1).
Since n^k-1 is odd, m does not divide 2*(n^k-1)/(n-1). If m has an odd prime divisor then concatenating the primes in base n will result in an odd number that is not divisible by m.
Finally, a(n) >= 9 since the first odd composite number is 9. (End)
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LINKS
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EXAMPLE
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a(2) = 85329 as 85329 = 3_10 * 3_10 * 19_10 * 499_10 = 11_2 * 11_2 * 10011_2 * 111110011_2 = "111110011111110011"_2 = 255987_10 which is divisible by 85329.
a(10) = 28749 as 28749 = 3_10 * 7_10 * 37_10 * 37_10 = "373737"_10 = 373737_10 which is divisible by 28749. See also A259047.
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PROG
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(Python)
from itertools import count
from sympy.ntheory import digits
from sympy import factorint, isprime
def fromdigits(d, b):
n = 0
for di in d: n *= b; n += di
return n
def a(n):
for k in count(4):
if isprime(k): continue
sf = []
for p, e in factorint(k).items():
sf.extend(e*digits(p, n)[1:])
if fromdigits(sf, n)%k == 0:
return k
(Python)
from itertools import count
from sympy import factorint, integer_log
for m in count(4):
f = factorint(m)
if sum(f.values()) > 1:
c = 0
for p in sorted(f):
a = pow(n, integer_log(p, n)[0]+1, m)
for _ in range(f[p]):
c = (c*a+p)%m
if not c:
(PARI) has(F, n)=my(f=F[2], t); for(i=1, #f~, my(p=f[i, 1], d=#digits(p, n), D=n^d); for(j=1, f[i, 2], t=D*t+p)); t%F[1]==0
a(k, lim=10^6, startAt=4)=forfactored(n=startAt, lim, if(vecsum(n[2][, 2])>1 && has(n, k), return(n[1]))); a(k, 2*lim, lim+1) \\ Charles R Greathouse IV, Apr 11 2024
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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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