cp's OEIS Frontend

A331025 Products of terms of A232803.

%I A331025 #35 Sep 14 2020 11:19:57
%S A331025 1,3,4,9,5,12,6,27,16,15,7,36,8,18,20,81,10,48,11,45,24,21,13,108,25,
%T A331025 24,64,54,14,60,17,243,28,30,30,144,19,33,32,135,22,72,23,63,80,39,26,
%U A331025 324,36,75,40,72,29,192,35,162,44,42,31,180,34,51,96,729
%N A331025 Products of terms of A232803.
%C A331025 If 2 were not a prime factor, the prime numbers sequence would change. 4,8, and twice odd primes would become "primes". The new "prime numbers" sequence would be 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, ...  (A232803). The products of the terms of A232803 would become the new "natural numbers".
%C A331025 In order to compute a(n), one must write the prime factorization of n and replace each prime(k) with A232803(k). - _Michel Marcus_, Sep 14 2020
%H A331025 Michael De Vlieger, <a href="/A331025/b331025.txt">Table of n, a(n) for n = 1..10000</a>
%H A331025 Robert Dougherty-Bliss, <a href="https://rwdb.xyz/files/prime_removal.pdf">The Number 2 Does Not Exist and other p-removed primes</a>
%e A331025 In the natural numbers sequence, a(15)=prime(2)*prime(3). If we use the terms of A232803 as prime factors, then prime(2)=4 and prime(3)=5. So, a(15) will be 4*5 = 20.
%t A331025 With[{s = Select[Range[37], And[# != 2, Or[Log2[#] == 3, PrimeQ@#, PrimeQ[#/2]]] &]}, Array[Times @@ Map[If[#[[1]] == 1, 1, # /. {p_, e_} :> s[[PrimePi@ p]]^e] &, FactorInteger[#]] &, Prime@ Length@ s]] (* _Michael De Vlieger_, Aug 21 2020 *)
%o A331025 (PARI) isp(n) = (isprime(n) && (n%2)) || (n==8) || (!(n%2) && isprime(n/2)); \\ A232803
%o A331025 lista(nn) = {my(vall = [1..nn]); my(vp = select(x->isp(x), vall)); for (n=2, nn, my(f=factor(n)); for (k=1, #f~, f[k,1] = vp[primepi(f[k,1])]); vall[n] = factorback(f);); vall;} \\ _Michel Marcus_, Sep 14 2020
%Y A331025 Cf. A232803.
%K A331025 nonn
%O A331025 1,2
%A A331025 _Ali Sada_, Jan 07 2020