A372029 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nondecreasing order.

12, 25, 35, 111, 112, 125, 222, 245, 333, 335, 445, 1225, 2225, 11125, 33445, 334445, 3333335, 3334445, 3444445, 33333445, 333333335, 334444445, 3333333335, 33333334445, 333333333335, 33333333334445, 33333333444445, 444444444444445, 2222222222222225, 11111111111111125

Offset: 1

Author

Scott R. Shannon, Apr 16 2024

Keywords

• nonn
• base

Comments

Terms cannot end in 4, 6, 8, or 9 because 2 would be a factor and no prime consists entirely of 9's. - Michael S. Branicky, Apr 22 2024

Examples

			The initial terms and their factorizations are:
   12 = [2, 2, 3]
   25 = [5, 5]
   35 = [5, 7]
   111 = [3, 37]
   112 = [2, 2, 2, 2, 7]
   125 = [5, 5, 5]
   222 = [2, 3, 37]
   245 = [5, 7, 7]
   333 = [3, 3, 37]
   335 = [5, 67]
   445 = [5, 89]
   1225 = [5, 5, 7, 7]
   2225 = [5, 5, 89]
   11125 = [5, 5, 5, 89]
   33445 = [5, 6689]
   334445 = [5, 66889]
   3333335 = [5, 666667]
   3334445 = [5, 666889]
   3444445 = [5, 688889]
   33333445 = [5, 6666689]
   333333335 = [5, 66666667]
   334444445 = [5, 66888889]
   ...
12 is a term since the list L(12) is [12,2,2,3], in which the digits 1,2,2,2,3 are in nondecreasing order.
121 is not a term since L(121) = [121,11,11], and the digits 1,2,1,1,1,1,1 are not in nondecreasing order.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..48
• Michael S. Branicky, Table of n, a(n), and the prime factorization of a(n) for n = 1..48

Crossrefs

Cf. A372053, A372055, A372034.

Programs

Python

from sympy import factorint, isprime
def nd(s): return sorted(s) == list(s)
def ok(n):
    if n < 4 or isprime(n): return False
    s, f = str(n), "".join(str(p)*e for p, e in factorint(n).items())
    return nd(s+f)
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Apr 22 2024

Python

# faster for initial segment of sequence
from sympy import factorint, isprime
from itertools import count, islice, combinations_with_replacement as mc
def nd(s): return s == "".join(sorted(s))
def bgen(d): # can't end in 8 or 9
    yield from ("".join(m) for m in mc("1234567", d))
def agen(): # generator of terms
    for d in count(2):
        for s in bgen(d):
            t = int(s)
            if any(s[-1] > c and t%int(c) == 0 for c in "2357"): continue
            if isprime(t): continue
            if nd(s+"".join(str(p)*e for p, e in factorint(t).items())):
                yield t
print(list(islice(agen(), 25))) # Michael S. Branicky, Apr 22 2024

Extensions

a(25) and beyond from Michael S. Branicky, Apr 22 2024