A232803 -id:A232803 - cp's OEIS frontend

A331025 Products of terms of A232803.

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, 24, 64, 54, 14, 60, 17, 243, 28, 30, 30, 144, 19, 33, 32, 135, 22, 72, 23, 63, 80, 39, 26, 324, 36, 75, 40, 72, 29, 192, 35, 162, 44, 42, 31, 180, 34, 51, 96, 729

Offset: 1

Ali Sada, Jan 07 2020

nonn

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

			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.

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

Cf. A232803.

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 *)

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

A343926 a(n) is the least k such that A343443(k) = n or 0 if there is no such k.

Original entry on oeis.org

1, 0, 2, 4, 8, 16, 32, 64, 6, 256, 512, 12, 2048, 4096, 24, 36, 32768, 48, 131072, 72, 96, 1048576, 2097152, 144, 216, 16777216, 30, 288, 134217728, 432, 536870912, 576, 1536, 4294967296, 864, 60, 34359738368, 68719476736, 6144, 1728, 549755813888, 2592, 2199023255552

Offset: 1

Michel Marcus, May 04 2021

nonn

The indices for which a(n) = 2^(n-2) appear to be A232803. - Michel Marcus, May 05 2021

This is true. We can check it for n <= 10. For n > 10 there are only primes and twice primes in A232803. Any number k > 10 not in A232803 can be factored as k = m*p where m, p > 2 and m >= p. We then have A343443(2^(m-2)*3^(p-2)) = m*p = k. But 2^(k-2) = 2^(m*p-2) > 2^(m-2)*3^(p-2). As m, p > 2 we have 2^(m-2)*3^(p-2) not in A232803. - David A. Corneth, May 05 2021

David A. Corneth, Table of n, a(n) for n = 1..3325 (first 52 terms from Michel Marcus)

Cf. A025487, A232803, A343443.

a(n) <= 2^(n-2) for n >= 3.

A336429 First location of n in A331025 or 0 if number is absent.

Original entry on oeis.org

1, 0, 2, 3, 5, 7, 11, 13, 4, 17, 19, 6, 23, 29, 10, 9, 31, 14, 37, 15, 22, 41, 43, 21, 25, 47, 8, 33, 53, 34, 59, 39, 38, 61, 55, 12, 67, 71, 46, 51, 73, 58, 79, 57, 20, 83, 89, 18, 121, 85, 62, 69, 97, 28, 95, 87, 74, 101, 103, 30, 107, 109, 44, 27, 115, 82

Offset: 1

J. Lowell, Jul 21 2020

nonn

26 is the smallest number not in this sequence; A331025(21) and A331025(26) both equal 24; the smallest number to occur in A331025 at least twice.

			a(9) = 4 because A331025(4) = 9 and no earlier term of A331025 is 9.

Cf. A232803, A331025.

Block[{nn = 66, s}, s = Select[Range[3 nn], And[# != 2, Or[Log2[#] == 3, PrimeQ@ #, PrimeQ[#/2]]] &]; Insert[#, 0, 2][[1 ;; nn]] &@ Values[KeySort@ PositionIndex@ Array[Times @@ Map[If[#[[1]] == 1, 1, # /. {p_, e_} :> s[[PrimePi@ p]]^e] &, FactorInteger[#]] &, Prime@ Length@ s]][[All, 1]]] (* Michael De Vlieger, Aug 21 2020 *)

isp(n) = (isprime(n) && (n%2)) || (n==8) || (!(n%2) && isprime(n/2)); \\ A232803
vA331025(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;}
lista(nn) = {my(vall = vA331025(nn)); my(vr = vector(nn)); for (n=1, nn, my(vs = select(x->(x==n), vall, 1)); if (#vs == 0, vr[n] = 0, vr[n] = vs[1]);); my(vz = select(x->(x==0), vr, 1)); if (#vz > 1, vr = vector(vz[2]-1, k, vr[k]);); vr;} \\ Michel Marcus, Sep 14 2020

More terms from Rémy Sigrist, Jul 21 2020

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A331025 Products of terms of A232803.

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A343926 a(n) is the least k such that A343443(k) = n or 0 if there is no such k.

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A336429 First location of n in A331025 or 0 if number is absent.

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