A350046 Numbers m such that, in the prime factorization of m! (with the primes in ascending order), no two successive exponents differ by a composite number.

2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 15, 32, 35, 162, 163

Offset: 1

Author

Devansh Singh, Dec 11 2021

Keywords

• nonn

Comments

Is this sequence finite?

After 163, there are no more terms through 10^10. - Jon E. Schoenfield, Dec 15 2021

Sequence is the same as numbers m such that, in the prime factorization of m! (with the exponents in ascending order), no two successive exponents differ by a composite number. This is due to the fact that in the factorization of m! with the primes in ascending order, the corresponding exponents are in descending order. - Chai Wah Wu, Jan 10 2022

Examples

			15! = 2^11 * 3^6 * 5^3 * 7^2 * 11^1 * 13^1; the exponents are 11, 6, 3, 2, 1, 1, and no two successive/neighboring exponents differ by a composite number, so 15 is a term of the sequence.
		

Crossrefs

Cf. A000142, A330706.

Programs

Mathematica

q[n_] := AllTrue[Differences @ Reverse[FactorInteger[n!][[;; , 2]]], !CompositeQ[#] &]; Select[Range[2, 200], q] (* Amiram Eldar, Dec 11 2021 *)

PARI

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=my(o=valp(n,2),e); forprime(p=3,, e=valp(n,p); if(o-e<4, return(1)); if(isprime(o-e), o=e, return(0))) \\ Charles R Greathouse IV, Dec 15 2021

Python

import sympy
def last_exponent(c,i):
    if sympy.nextprime(i//2)<=i:
        if c==1 or sympy.isprime(c-1):
            return(True)
        else: return(False)
    else:
        if c==1 or sympy.isprime(c):
            return(True)
        else: return(False)
A350046_n=[2,3]
for i in range(4,1001):
    p_expo=True
    x = list(sympy.primerange(2,i//2+1))
    prime_expo=[]
    for j in (x):
        c=i//j
        s=0
        while c!=0:
            s=s+c
            c=c//j
        prime_expo.append(s)
    c=prime_expo[0]
    l=len(prime_expo)
    for j in range(1,l):
        c=c-prime_expo[j]
        if c!=0:
            if c!=1 and not sympy.isprime(c):
                p_expo=False
                break
        c=prime_expo[j]
    prime_expo=last_exponent(c,i)
    if p_expo==True:
       A350046_n.append(i)
print(A350046_n)

Python

from collections import Counter
from itertools import count, islice
from sympy import factorint, isprime
def A350046_gen(): # generator of terms
    f = Counter()
    for m in count(2):
        f += Counter(factorint(m))
        e = sorted(f.items())
        if all(d <= 1 or isprime(d) for d in (abs(e[i+1][1]-e[i][1]) for i in range(len(e)-1))):
            yield m
A350046_list = list(islice(A350046_gen(),15)) # Chai Wah Wu, Jan 10 2022