A244077 Integers m such that m' = Sum_{i=1..k-1} (Sum_{j=1..i} d_(k-j+1)*10^(i-j))', where m' is the arithmetic derivative of m and the digits of m are given by d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + ... + d_(2)*10 + d_(1).

23, 29, 31, 37, 53, 59, 71, 73, 79, 113, 131, 137, 139, 173, 179, 191, 193, 197, 199, 6437, 8339, 14473, 60827, 95611, 107813, 321773, 495407, 1154383, 3001331, 5707209, 6208373, 12898591, 13481347, 18895997, 29223791, 39111253, 99058793, 175113571, 317624885

Offset

1,1

Comments

From 23 to 199 only primes, then composites.

Links

Table of n, a(n) for n=1..39.

Example

If m = 14473, let us cut the number into the set {1, 14, 144, 1447}. We have:
  1' = 0;
  14' = 9;
  144' = 384;
  1447' = 1.
Finally, 0 + 9 + 384 + 1 = 14473' = 394.

Maple

with(numtheory); P:=proc(q) local a, c, k, n, p;
for n from 10 to q do
a:=0; k:=1; while trunc(n/10^k)>0 do c:=trunc(n/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);

Prog

(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = my(d=digits(k)); if (#d>1, sum(i=1, #d-1, ad(fromdigits(Vec(d, i)))) == ad(k)); \\ Michel Marcus, Dec 05 2025
(Python)
from sympy import factorint
from functools import cache
@cache
def ad(n): return 0 if n < 2 else sum(e*n//p for p, e in factorint(n).items())
def ok(n): return len(s:=str(n)) > 1 and ad(n) == sum(ad(int(s[:k])) for k in range(1, len(s)))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Dec 05 2025

Crossrefs

Cf. A003415, A240894-A240903, A241207, A241502, A241503, A244068, A244069, A244078.

Sequence in context: A155913 A227919 A240898 * A277206 A214754 A344375

Adjacent sequences: A244074 A244075 A244076 * A244078 A244079 A244080

Keyword

nonn,base

Author

Paolo P. Lava, Jun 19 2014

Extensions

a(26)-a(37) and new name from Michel Marcus, Dec 05 2025

a(38)-a(39) from Michael S. Branicky, Dec 05 2025

Status

approved