A372034 -id:A372034 - cp's OEIS frontend

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

Scott R. Shannon, Apr 16 2024

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

			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.

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

Cf. A372053, A372055, A372034.

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

# 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

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

A372280 Composite numbers k such that the digits of k are in nondecreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nonincreasing order.

Original entry on oeis.org

4, 8, 9, 16, 22, 25, 27, 33, 44, 49, 55, 77, 88, 99, 125, 128, 155, 256, 279, 1477, 1555, 1688, 1899, 2799, 3479, 3577, 14777, 16888, 18999, 22599, 36799, 444577, 455777, 1112447, 1555555, 2555555, 2799999, 3577777, 3799999, 45577777, 124556677, 155555555555, 279999999999

Offset: 1

Scott R. Shannon, Apr 25 2024

A number 155...555 will be a term if it has two prime factors 5 and 3111...111. Therefore 155555555555 and 1555555555555 are both terms. See A056704.

The next term is greater than 10^11.

			444577 is a term as 444577 = 7 * 7 * 43 * 211, and 444577 has nondecreasing digits while its prime factor concatenation "7743211" has nonincreasing digits.

Michael S. Branicky, Table of n, a(n) for n = 1..64 (all terms <= 20 digits)

Cf. A372308, A372034, A056704, A372029, A372055, A027746, A372249.

from sympy import factorint, isprime
from itertools import count, islice, combinations_with_replacement as mc
def ni(s): return s == "".join(sorted(s, reverse=True))
def bgen(d):
    yield from ("".join(m) for m in mc("0123456789", d) if m[0]!="0")
def agen(): # generator of terms
    for d in count(1):
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if ni("".join(str(p)*e for p,e in factorint(t).items())):
                yield t
print(list(islice(agen(), 41))) # Michael S. Branicky, Apr 26 2024

a(42)-a(43) from Michael S. Branicky, Apr 26 2024

A372308 Composite numbers k such that the digits of k are in nonincreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nondecreasing order.

Original entry on oeis.org

4, 6, 8, 9, 10, 20, 21, 30, 32, 40, 42, 50, 54, 60, 63, 64, 70, 72, 74, 75, 80, 81, 84, 90, 92, 94, 96, 98, 100, 111, 200, 210, 222, 300, 320, 333, 400, 420, 432, 441, 444, 500, 531, 540, 553, 554, 600, 611, 630, 632, 640, 666, 700, 711, 720, 750, 752, 800, 810, 840, 851, 864, 871, 875, 882

Offset: 1

Scott R. Shannon, Apr 26 2024

As all the numbers 10,20,...,90,100 are terms, all numbers that are recursively 10 times these values are also terms as they just add an additional 2 and 5 to their parent's prime factor list.

A number 999...9998 will be a term if it has two prime factors 2 and 4999...999. Therefore 999999999999998 and 999...9998 (with 54 9's) are both terms. See A056712.

			42 is a term as 42 = 2 * 3 * 7, and 42 has nonincreasing digits while its prime factor concatenation "237" has nondecreasing digits.

Michael S. Branicky, Table of n, a(n) for n = 1..884 (terms 1..458 from Scott R. Shannon; all terms < 10^18)

Cf. A372295, A372280, A056712, A372034, A372029, A372055, A027746, A372249.

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):
    yield from ("".join(m) for m in mc("9876543210", d) if m[0]!="0")
def agen(): # generator of terms
    for d in count(1):
        out = set()
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if nd("".join(str(p)*e for p,e in factorint(t).items())):
                out.add(t)
        yield from sorted(out)
print(list(islice(agen(), 65))) # Michael S. Branicky, Apr 26 2024

A372295 Composite numbers k such that k's prime factors are distinct, the digits of k are in nonincreasing order while the digits of the concatenation of k's ascending order prime factors are in nondecreasing order.

Original entry on oeis.org

6, 10, 21, 30, 42, 70, 74, 94, 111, 210, 222, 553, 554, 611, 851, 871, 885, 998, 5530, 5554, 7751, 8441, 8655, 9998, 85511, 95554, 99998, 9999998, 77744411, 5555555554, 7777752221, 8666666655, 755555555554, 95555555555554, 999999999999998, 5555555555555554, 8666666666666655, 755555555555555554

Offset: 1

Scott R. Shannon, Apr 25 2024

A number 999...9998 will be a term if it has two prime factors 2 and 4999...999. Therefore 999999999999998 and 999...9998 (with 54 9's) are both terms. See A056712.

The next term is greater than 10^11.

			77744411 is a term as 77744411 = 233 * 333667 which has distinct prime factors, 77744411 has nonincreasing digits while its prime factor concatenation "233333667" has nondecreasing digits.

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

Cf. A372308, A372280, A056712, A372034, A372029, A372055, A027746, A372249.

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):
    yield from ("".join(m) for m in mc("9876543210", d) if m[0]!="0")
def agen(): # generator of terms
    for d in count(1):
        out = set()
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            f = factorint(t)
            if len(f) < sum(f.values()): continue
            if nd("".join(str(p) for p in f)):
                out.add(t)
        yield from sorted(out)
print(list(islice(agen(), 29))) # Michael S. Branicky, Apr 26 2024

a(33)-a(38) from Michael S. Branicky, Apr 26 2024

A372335 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) alternate being larger than and then smaller than the previous digit.

Original entry on oeis.org

14, 15, 78, 161, 591, 1214, 1317, 1318, 1326, 1407, 1418, 1438, 1506, 1509, 1514, 1527, 1538, 1618, 1626, 1646, 1658, 1703, 1714, 1718, 1734, 1739, 1758, 1814, 1834, 1838, 1839, 1857, 1858, 1934, 1938, 2307, 2427, 2509, 2517, 2534, 2535, 2715, 2757, 2758, 2869, 2958, 3419, 3439, 3514, 3523

Offset: 1

Scott R. Shannon, Apr 28 2024

No term can end in 0 or 2; a number ending in 2 would mean the first prime factor is 2, which would disqualify the number, while a number ending in 0 would mean the first 3 distinct prime factors would have to be 2, 3, 5 or 2, 5, either of which would also disqualify the number.

			161 is a term as 161 = 7 * 23 which when concatenated give "161723", the digits of which alternate from being larger than and then smaller than the previous digit.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A372336, A372280, A372308, A372034, A372029, A056704.

A372336 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) alternate being smaller than and then larger than the previous digit.

Original entry on oeis.org

6, 51, 91, 106, 219, 323, 406, 435, 437, 518, 529, 609, 614, 626, 629, 634, 658, 703, 705, 818, 826, 838, 878, 906, 938, 978, 2051, 2093, 2173, 3053, 3241, 4151, 4171, 4281, 5041, 5063, 5141, 5183, 5241, 6251, 6591, 7021, 7081, 7251, 8051, 8121, 8491, 8571, 8781, 9121, 9231, 9291, 9583

Offset: 1

Scott R. Shannon, Apr 28 2024

No term can end in 0 or 2; a number ending in 2 would mean the first prime factor is 2, which would disqualify the number, while a number ending in 0 would mean the first 3 distinct prime factors would have to be 2, 3, 5 or 2, 5, either of which would also disqualify the number.

			106 is a term as 106 = 2 * 53 which when concatenated give "106253", the digits of which alternate from being smaller than and then larger than the previous digit.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A372335, A372280, A372308, A372034, A372029, A056704.

A372151 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 k is either 3, 4 or 5 and the digits of L(k) are in nondecreasing order.

Original entry on oeis.org

35, 333, 335, 445, 33445, 334445, 3333335, 3334445, 3444445, 33333445, 333333335, 334444445, 3333333335, 33333334445, 333333333335, 33333333334445, 33333333444445, 444444444444445, 333333334444444445, 333334444444444445, 444444444444444445, 3333333333333444445

Offset: 1

Chai Wah Wu, Apr 26 2024

Subsequence of A372029. Sequence is inspired by the observation that most terms in A372029 so far contain only the digits 3, 4 and 5.

			   35 = 5*7
  333 = 3*3*37
  335 = 5*67
  445 = 5*89
33445 = 5*6689
333333333333333333333333444444444444444444444445 = 5*66666666666666666666666688888888888888888888889

Chai Wah Wu, Table of n, a(n) for n = 1..71

Cf. A372053, A372055, A372034, A372029.

from itertools import count, islice, combinations_with_replacement
from sympy import isprime, factorint
def A372151_gen(): # generator of terms
    for l in count(1):
        for d in combinations_with_replacement('345',l):
            a, n = d[-1], int(''.join(d))
            if not isprime(n):
                for p in factorint(n,multiple=True):
                    s = str(p)
                    if s[0] < a or sorted(s) != list(s):
                        break
                    a = s[-1]
                else:
                    yield n
A372151_list = list(islice(A372151_gen(),20))

cp's OEIS Frontend

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.

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A372280 Composite numbers k such that the digits of k are in nondecreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nonincreasing order.

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A372308 Composite numbers k such that the digits of k are in nonincreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nondecreasing order.

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A372295 Composite numbers k such that k's prime factors are distinct, the digits of k are in nonincreasing order while the digits of the concatenation of k's ascending order prime factors are in nondecreasing order.

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A372335 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) alternate being larger than and then smaller than the previous digit.

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A372336 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) alternate being smaller than and then larger than the previous digit.

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A372151 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 k is either 3, 4 or 5 and the digits of L(k) are in nondecreasing order.

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