A074736 Goedel encoding of the prime factors of n, in increasing order and repeated according to multiplicity.

1, 4, 8, 36, 32, 108, 128, 900, 216, 972, 2048, 4500, 8192, 8748, 1944, 44100, 131072, 13500, 524288, 112500, 17496, 708588, 8388608, 308700, 7776, 6377292, 27000, 2812500, 536870912, 337500, 2147483648, 5336100, 1417176, 516560652, 69984, 1543500, 137438953472

Offset: 1

Author

Joseph L. Pe, Sep 28 2002

Keywords

• nonn

Comments

For irregular triangle T(n,k) at A027746, a(n) = Product_{1..A001222(n)} pi(k)^T(n,k). - Michael De Vlieger, May 04 2020.

Examples

			The prime factors of 12 in increasing order and repeated according to multiplicity are 2, 2, 3. Hence a(12) = 2^2 * 3^2 * 5^3 = 4500.
		

References

• K. Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", Dover Publications, 1992.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3322
• Wikipedia, Gödel's encoding

Crossrefs

Cf. A001222, A027746.

Programs

Maple

a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
        sort(map(i-> i[1]$i[2], ifactors(n)[2]))):
seq(a(n), n=1..40);

Mathematica

Array[Times @@ MapIndexed[Prime[First[#2]]^#1 &, Apply[Join, ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, 34, 2] (* Michael De Vlieger, May 04 2020 *)

PARI

for(n=2,50,m=factor(n):s=1:c=1:for(k=1,matsize(m)[1], for(l=1,m[k,2],s=s*prime(c)^m[k,1]:c=c+1)):print1(s",")) [Does not compile. - Robert C. Lyons, Nov 04 2024]

Python

from math import prod
from sympy import prime, factorint
def A074736(n): return prod(prime(i)**j for i, j in enumerate(factorint(n,multiple=True),1)) # Chai Wah Wu, Nov 04 2024

Formula

a(n) = prime(1)^p_1 * prime(2)^p_2 * ... * prime(k)^p_k, where p_1 <= ... <= p_k are the prime factors of n, repeated according to multiplicity.

Extensions

More terms from Ralf Stephan, Mar 22 2003

a(1)=1 prepended by Alois P. Heinz, Nov 04 2024