cp's OEIS Frontend

A353172 a(n) is the least k > 1 such that Omega(n) = Omega(n mod k), where Omega = A001222.

%I A353172 #79 Oct 31 2022 05:36:19
%S A353172 2,3,4,5,3,7,4,9,5,6,3,13,5,5,9,17,3,10,4,12,11,6,3,25,7,10,15,10,3,
%T A353172 11,4,33,9,5,13,20,5,8,5,24,3,15,4,9,25,6,3,49,5,14,9,11,3,19,7,20,12,
%U A353172 6,3,22,7,7,11,65,11,18,4,10,5,25,3,40,5,5,19,16
%N A353172 a(n) is the least k > 1 such that Omega(n) = Omega(n mod k), where Omega = A001222.
%C A353172 It appears that a(m) = m*k/p if m = p*2^n ... . Are these formulas related to some well-known sequence of rational numbers?
%H A353172 David A. Corneth, <a href="/A353172/b353172.txt">Table of n, a(n) for n = 1..10000</a>
%F A353172 a(A029744(n)) = A029744(n) + 1.
%F A353172 a(A003627(n)) = 3.
%F A353172 a(A000040(n)) = A095925(n).
%F A353172 a(A077065(n)) = 6. For n > 2.
%F A353172 If a(n) = 10, then n mod 10 is in most cases 8 and seldom 6.
%F A353172 a(m) = m*3/5 if m = 5*2^n or m = 15. This formula is valid for all positive n because (5*2^n) mod (5*2^n)*(3/5) = 2^(n+1). If the sequence of solutions does not create powers of two in the modulo operation, it will be of finite length. See next two formulas:
%F A353172 a(m) = m*3/11 if m = 11, 22, 33 or 66.
%F A353172 a(m) = m*4/43 if m = 43*2^n for n < 4. This series of solutions terminates because of the next formula which replaces the powers of two:
%F A353172 a(m) = m*41/(43*2^4) if m = 43*2^4*2^n. This formula is valid for all positive n.
%F A353172 a(m) = m*5/9 if m = 9*2^n or m = 27 or 45. This formula is valid for all positive n.
%F A353172 For each k = a(p) if k < p and gcd(k, p) = 1 such a formula, of the form a(m) = m*k/p, if m = p*2^n ..., can be developed.
%e A353172 a(10) = 6 because 10 = 5*2 and 10 mod 6 = 4 = 2*2.
%o A353172 (PARI)
%o A353172 a(n) = my(k=2); while(bigomega(n) != bigomega(max(n%k,1)), k++); k
%o A353172 (Python)
%o A353172 from itertools import count
%o A353172 from sympy.ntheory.factor_ import primeomega
%o A353172 def A353172(n):
%o A353172     a = primeomega(n)
%o A353172     for k in count(2):
%o A353172         if (m := n % k) > 0 and primeomega(m) == a:
%o A353172             return k # _Chai Wah Wu_, Jun 20 2022
%Y A353172 Cf. A000040, A001222, A003627, A029744, A061142, A077065, A095925.
%K A353172 nonn,easy
%O A353172 1,1
%A A353172 _Thomas Scheuerle_, Apr 28 2022