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

1,1

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

Links

Table of n, a(n) for n=1..15.

Example

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.

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A000142, A330706.

Sequence in context: A178338 A048097 A130843 * A087087 A050742 A350572

Adjacent sequences: A350043 A350044 A350045 * A350047 A350048 A350049

Keyword

nonn

Author

Devansh Singh, Dec 11 2021

Status

approved