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

1,1

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.

Links

Table of n, a(n) for n=1..40.

Example

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.

Mathematica

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

Prog

(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:])
(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

Crossrefs

Subsequence of A242414.

Sequence in context: A254477 A115930 A109994 * A254805 A254798 A253814

Adjacent sequences: A322522 A322523 A322524 * A322526 A322527 A322528

Keyword

nonn

Author

Pierandrea Formusa, Dec 13 2018

Status

approved