This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361373 #38 Sep 20 2024 14:15:29
%S A361373 0,1,1,2,1,3,1,3,2,4,1,5,1,4,3,4,1,6,1,5,3,5,1,6,2,5,3,5,1,9,1,5,4,6,
%T A361373 3,8,1,6,4,7,1,9,1,6,5,6,1,8,2,7,4,6,1,8,3,7,4,6,1,10,1,6,5,6,3,10,1,
%U A361373 7,4,10,1,9,1,7,5,7,3,10,1,8,4,7,1,12,3,7
%N A361373 Number of prime powers p^m <= n such that p | n.
%C A361373 Let p be prime. The term "prime power" p^m, m > 0, used here is that of A246655 = A000040 U A246547, the union of primes and perfect prime powers. Essentially, 1 is not considered a prime power.
%H A361373 Michael De Vlieger, <a href="/A361373/b361373.txt">Table of n, a(n) for n = 1..10000</a>
%H A361373 Michael De Vlieger, <a href="/A361373/a361373.png">Diagram</a> illustrating a(1440) = 20. Terms are arranged according to prime decomposition and sorted vertically. This sequence counts primes (red) and perfect prime powers (gold).
%F A361373 a(n) = Sum_{p | n} floor(log n / log p).
%F A361373 a(n) = number of prime powers in row n of A162306.
%F A361373 a(n) < A000005(n), since A000005 counts 1.
%F A361373 a(n) < A010846(n), since A010846 counts 1.
%F A361373 Let tau = A000005, rad = A007947, rcf = A010846, and lpf = A020639.
%F A361373 a(p) = tau(p) - 1 = rcf(p) - 1 = 1 since S = row p of both A027750 and A162306 = {1, p} contains the prime power p.
%F A361373 a(p^m) = tau(p^m) - 1 = rcf(p^m) = 1 = m since S = row p^m of both A027750 and A162306 = {1, p, p^2, ..., p^m} contains the prime powers {p, p^2, ..., p^m}.
%F A361373 a(k) = tau(k) - 1 = 3 for squarefree composite k = p*q, p < q < p^2 in A138109 since S = row k of A162306 = {1, p, q, p^2, p*q} contains 3 prime powers {p, q, p^2}.
%F A361373 a(k) < tau(k) for k in A138511 and k in A126706 since m = lpf(k)^(-1 + floor(log k / log lpf(k))) is such that m < k but m does not divide k.
%e A361373 Let S = {k <= n : rad(k) | n} = row n of A162306
%e A361373 a(1) = 0 since S = {1} has 0 prime powers.
%e A361373 a(2) = 1 since S = {1, [2]} has 1 prime power.
%e A361373 a(4) = 2 since S = {1, [2, 4]} has 2 prime powers.
%e A361373 a(6) = 3 since S = {1, [2, 3, 4], 6} has 3 prime powers.
%e A361373 a(10) = 4 since S = {1, [2, 4, 5, 8], 10} has 4 prime powers.
%e A361373 a(12) = 5 since S = {1, [2, 3, 4], 6, [8, 9], 12} has 5 prime powers, etc.
%p A361373 a := n -> add(ilog[p](n), p in NumberTheory:-PrimeFactors(n)):
%p A361373 seq(a(n), n = 1..92); # _Peter Luschny_, Jun 20 2024
%t A361373 {0}~Join~Table[Total@ Map[Floor@ Log[#, n] &, FactorInteger[n][[All, 1]]], {n, 2, 120}]
%o A361373 (PARI) a(n) = if (n==1, 0, my(f=factor(n)[,1]); sum(k=1, #f, logint(n, f[k]))); \\ _Michel Marcus_, Jun 20 2024
%o A361373 (Python)
%o A361373 from sympy import integer_log, primefactors
%o A361373 def A361373(n): return sum(integer_log(n,p)[0] for p in primefactors(n)) # _Chai Wah Wu_, Sep 20 2024
%Y A361373 Cf. A000005, A000040, A007947, A010846, A020639, A126706, A138109, A138511, A162306, A246655, A246547.
%K A361373 nonn,easy
%O A361373 1,4
%A A361373 _Michael De Vlieger_, Jun 17 2024