login
A244078
Consider a number of k digits n = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + … + d_(2)*10 + d_(1). Sequence lists the numbers n such that n’ = Sum_{i=1..k-1}{Sum_{j=1..i}{d_(j)*10^(j-1)}}’, where n’ is the arithmetic derivative of n (see example below).
1
13, 17, 23, 37, 43, 53, 67, 73, 83, 97, 131, 211, 241, 271, 311, 331, 431, 461, 541, 571, 631, 641, 661, 761, 811, 911, 941, 971, 1601, 3701, 5101, 5701, 6101, 6701, 8101, 9601, 13001, 16138, 18497, 19001, 22879, 24001, 54001, 69001, 93001, 97001, 99361, 270001
OFFSET
1,1
COMMENTS
Mainly primes.
EXAMPLE
If n = 16138, starting from the least significant digit, let us cut the number into the set {8, 38, 138, 6138}. We have:
8’ = 12;
38’ = 21;
138’ = 121;
6138’ = 7917.
Finally, 12 + 21 + 121 + 7917 = 16138’ = 8071.
MAPLE
with(numtheory); P:=proc(q) local a, c, k, n, p;
for n from 10 to q do
a:=0; k:=1; while (n mod 10^k)<n do c:=(n mod 10^k);
a:=a+c*add(op(2, p)/op(1, p), p=ifactors(c)[2]); k:=k+1; od;
if a=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]) then print(n);
fi; od; end: P(10^10);
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jun 19 2014
STATUS
approved