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A386499 a(n) is the 5-adic valuation of A386252(n).

%I A386499 #15 Jul 31 2025 17:52:46
%S A386499 1,1,2,1,1,1,2,1,1,3,2,2,3,1,1,2,1,4,1,4,6,2,1,3,3,3,6,5,3,3,2,2,6,7,
%T A386499 5,9,7,3,8,4,8,4,6,5,6,2,3,6,4,10,9,2,4,6,3,2,3,9,8,2,6,1,11,2,5,3,9,
%U A386499 1,1,3,10,3,3,8,2,2,7,2,8,8,5,7,11,3,5,14
%N A386499 a(n) is the 5-adic valuation of A386252(n).
%H A386499 Ken Clements, <a href="/A386499/b386499.txt">Table of n, a(n) for n = 1..4000</a>
%F A386499 a(n) = A112765(A386252(n)).
%e A386499 a(1) = 1 because A386252(1) = 2^1 * 3^1 * 5^1
%e A386499 a(2) = 1 because A386252(2) = 2^2 * 3^1 * 5^1
%e A386499 a(3) = 2 because A386252(3) = 2^1 * 3^1 * 5^2
%e A386499 a(4) = 1 because A386252(4) = 2^2 * 3^2 * 5^1
%t A386499 seq[max_] := IntegerExponent[Select[Table[2^i*3^j*5^k, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}, {k, 1, Log[5, max/(2^i*3^j)]}] // Flatten // Sort, And @@ PrimeQ[# + {-1, 1}] &], 5]; seq[10^12] (* _Amiram Eldar_, Jul 24 2025 *)
%o A386499 (Python)
%o A386499 from math import log10
%o A386499 from gmpy2 import is_prime
%o A386499 l2, l3, l5 = log10(2), log10(3), log10(5)
%o A386499 upto_digits = 100
%o A386499 sum_limit = 3 + int((upto_digits - l3 - l5)/l2)
%o A386499 def TP_pi_3_upto_sum(limit): # Search all partitions up to the given exponent sum.
%o A386499     unsorted_result = []
%o A386499     for exponent_sum in range(3, limit+1):
%o A386499         for i in range(1, exponent_sum -1):
%o A386499              for j in range(1, exponent_sum - i):
%o A386499                 k = exponent_sum - i - j
%o A386499                 log_N = i*l2 + j*l3 + k*l5
%o A386499                 if log_N <= upto_digits:
%o A386499                     N = 2**i * 3**j * 5**k
%o A386499                     if is_prime(N-1) and is_prime(N+1):
%o A386499                         unsorted_result.append((k, log_N))
%o A386499     sorted_result = sorted(unsorted_result, key=lambda x: x[1])
%o A386499     return sorted_result
%o A386499 print([k for k, _ in TP_pi_3_upto_sum(sum_limit) ])
%Y A386499 Cf. A112765, A386252, A386498, A386500, A027856, A384530, A080185.
%K A386499 nonn
%O A386499 1,3
%A A386499 _Ken Clements_, Jul 23 2025