A322525 Numbers such that the list of exponents of their factorization is a palindromic list of primes.

2700, 5292, 9000, 13068, 18252, 24300, 24500, 24696, 31212, 38988, 47628, 55125, 57132, 60500, 68600, 84500, 90828, 95832, 103788, 117612, 136125, 144500, 147852, 158184, 164268, 166012, 180500, 181548, 190125, 199692, 218700, 231525, 231868, 238572, 243000, 264500, 266200, 280908, 303372, 325125

Offset: 1

Author

Pierandrea Formusa, Dec 13 2018

Keywords

• nonn

Comments

I mean nontrivial palindrome: more than one number and not all equal numbers.

Factorization is meant to produce p1^e1*...*pk^ek, with pi in increasing order.

Examples

			9000 is a term as 9000=2^3*3^2*5^3 and the correspondent exponents list [3,2,3] is a palindromic list of primes.
		

Crossrefs

Subsequence of A242414.

Programs

Mathematica

aQ[s_] := Length[Union[s]]>1 && AllTrue[s, PrimeQ] && PalindromeQ[s]; Select[Range[1000], aQ[FactorInteger[#][[;;,2]]] &] (* Amiram Eldar, Dec 14 2018 *)

PARI

isok(n) = (ve=factor(n)[,2]~) && (Vecrev(ve)==ve) && (#ve>1) && (#Set(ve)>1) && (#select(x->(!isprime(x)), ve) == 0); \\ Michel Marcus, Dec 14 2018

Python

from sympy.ntheory import factorint,isprime
def all_prime(l):
    for i in l:
        if not(isprime(i)): return(False)
    return(True)
def all_equal(l):
    ll=len(l)
    set_l=set(l)
    lsl=list(set_l)
    llsl=len(lsl)
    return(llsl==1)
def pal(l):
    return(l == l[::-1])
n=350000
r=""
lp=[]
lexp=[]
def calc(n):
    global lp,lexp
    a=factorint(n)
    lp=[]
    for p in a.keys():
        lp.append(p)
    lexp=[]
    for exp in a.values():
        lexp.append(exp)
    return
for i in range(4,n):
    calc(i)
    if len(lexp)>1:
        if all_prime(lexp):
            if not(all_equal(lexp)):
                if pal(lexp):
                    r += ","+str(i)
print(r[1:])