OFFSET
1,2
COMMENTS
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".
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
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Robert Dougherty-Bliss, The Number 2 Does Not Exist and other p-removed primes
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(PARI) isp(n) = (isprime(n) && (n%2)) || (n==8) || (!(n%2) && isprime(n/2)); \\ A232803
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ali Sada, Jan 07 2020
STATUS
approved