A340594 -id:A340594 - cp's OEIS frontend

A340595 a(n) is the least k for which A340594(k) = n.

2, 4, 8, 21, 42, 65, 80, 217, 488, 721, 2120, 2349, 2796, 9214, 16043, 23287, 28626, 43588, 58176, 116982, 213435, 444329, 640673, 967248, 1399895, 1449156, 1528785, 2768054, 2915135, 3631071, 3673118, 5032731, 12977420

Offset: 0

J. M. Bergot and Robert Israel, Jan 13 2021

nonn

a(n) is the least k >= 2 from which it takes exactly n iterations of A340592 to reach 0, 1 or a prime.

			Starting from 21, it takes 3 iterations of A340592 to reach 0,1 or a prime: 21 -> 16 -> 14 -> 13.  Since this is the first case where 3 iterations are required, a(3) = 21.

Cf. A340592, A340594.

dcat:= proc(L) local i, x;
  x:= L[-1];
  for i from nops(L)-1 to 1 by -1 do
    x:= 10^(1+ilog10(x))*L[i]+x
  od;
  x
end proc:
f:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a, b) -> a[1] < b[1]);
  dcat(map(t -> t[1]$t[2], F)) mod n;
end proc:
g:= proc(n) option remember;
     if isprime(n) then 0 else 1 + procname(f(n)) fi
end proc:
g(0):= 0: g(1):= 0:
V:= Array(0..30): count:= 0:
for n from 2 while count < 31 do
  v:= f(n);
  if v::integer and v <= 100 and V[v] = 0 then
      count:= count+1; V[v]:= n;
    fi
od:
convert(V,list);

A340592 a(n) is the concatenation of the prime factors (with multiplicity) of n mod n.

Original entry on oeis.org

0, 0, 2, 0, 5, 0, 6, 6, 5, 0, 7, 0, 13, 5, 14, 0, 17, 0, 5, 16, 13, 0, 15, 5, 5, 9, 3, 0, 25, 0, 14, 14, 13, 22, 1, 0, 29, 1, 25, 0, 27, 0, 11, 20, 39, 0, 47, 28, 5, 11, 29, 0, 11, 16, 43, 34, 55, 0, 15, 0, 45, 22, 14, 58, 1, 0, 41, 47, 47, 0, 57, 0, 15, 55, 15, 18, 51, 0, 65, 12, 77, 0, 53, 7

Offset: 2

J. M. Bergot and Robert Israel, Jan 12 2021

a(n) = 0 if n is prime.
The first composite n for which a(n)=0 is 28749. Are there others?
There are no other composite n terms for which a(n)=0 up to 5 million. - Harvey P. Dale, Jul 17 2023

			For n = 20 = 2*2*5, a(20) = 225 mod 20 = 5.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A037276, A340594, A340595.

dcat:= proc(L) local i,x;
  x:= L[-1];
  for i from nops(L)-1 to 1 by -1 do
    x:= 10^(1+ilog10(x))*L[i]+x
  od;
  x
end proc:
f:= proc(n) local F;
  F:= sort(ifactors(n)[2],(a,b) -> a[1] < b[1]);
  dcat(map(t -> t[1]$t[2], F)) mod n;
end proc:
map(f, [$2..100]);

Table[Mod[FromDigits[Flatten[IntegerDigits/@Table[#[[1]],#[[2]]]&/@FactorInteger[n]]],n],{n,2,100}] (* Harvey P. Dale, Jul 17 2023 *)

from sympy import factorint
def a(n):
    if n == 1: return 0
    return int("".join(str(f) for f in factorint(n, multiple=True)))%n
print([a(n) for n in range(2, 86)]) # Michael S. Branicky, Jan 18 2022

a(n) = A037276(n) mod n.

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A340595 a(n) is the least k for which A340594(k) = n.

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A340592 a(n) is the concatenation of the prime factors (with multiplicity) of n mod n.

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Maple

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