A372053 -id:A372053 - 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

A372034 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 nonincreasing order.

Original entry on oeis.org

4, 8, 9, 22, 32, 33, 44, 55, 64, 77, 88, 93, 99, 422, 633, 775, 844, 933, 993, 4222, 4442, 6333, 6655, 6663, 7533, 7744, 7775, 8444, 8884, 9663, 9993, 44222, 66333, 88444, 99633, 99933, 99993, 933333, 966333, 996663, 999993, 4442222, 6663333, 7777775, 8884444, 9663333, 9666633, 9666663

Offset: 1

Scott R. Shannon, Apr 16 2024

Is it true that no terms end with 1? A separate search on those shows none with < 70 digits. Michael S. Branicky, Apr 23 2024

Testing all products of repunit primes (A004022, A004023), there are no terms ending in 1 less than 10^3000. - Michael S. Branicky, Apr 24 2024

			The initial terms and their factorizations are:
4 = [2, 2]
8 = [2, 2, 2]
9 = [3, 3]
22 = [2, 11]
32 = [2, 2, 2, 2, 2]
33 = [3, 11]
44 = [2, 2, 11]
55 = [5, 11]
64 = [2, 2, 2, 2, 2, 2]
77 = [7, 11]
88 = [2, 2, 2, 11]
93 = [3, 31]
99 = [3, 3, 11]
422 = [2, 211]
633 = [3, 211]
775 = [5, 5, 31]
844 = [2, 2, 211]
933 = [3, 311]
993 = [3, 331]
4222 = [2, 2111]
4442 = [2, 2221]
6333 = [3, 2111]
6655 = [5, 11, 11, 11]
6663 = [3, 2221]
7533 = [3, 3, 3, 3, 3, 31]
7744 = [2, 2, 2, 2, 2, 2, 11, 11]
...

Michael S. Branicky, Table of n, a(n) for n = 1..197 (all terms with <= 21 digits)

Cf. A004022, A004023, A009996, A028867, A372053, A372029, A027746, A372055.

from sympy import factorint, isprime
def ni(s): return sorted(s, reverse=True) == 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 ni(s+f)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Apr 23 2024

# faster for initial segment of sequence
from sympy import factorint, isprime
from itertools import 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("987654321", d))
def agen(): # generator of terms
    for d in range(1, 70):
        out = set()
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if ni(s+"".join(str(p)*e for p, e in factorint(t).items())):
                out.add(t)
        yield from sorted(out)
print(list(islice(agen(), 50))) # Michael S. Branicky, Apr 23 2024

A372249 The smallest number k which, when written in base n, has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

Original entry on oeis.org

297, 440, 1440, 8596, 15552, 121638, 282240, 1039104, 5757696, 24108000

Offset: 7

Scott R. Shannon, Apr 24 2024

The offset begins at 7 as in bases 2..6 no term exists.

Assumes all factors f_i > 1. If f_i = 1 is allowed, then a(7) = 104 = 1*4*26, a(10) = 4830 = 1*2*5*7*69, a(12) = 72240 = 1*4*6*7*430, a(15) = 4244940 = 1*2*3*7*9*10*1123, ... - Chai Wah Wu, Apr 24 2024

			The terms and their factorizations are:
a(7) = 297 = [9, 33] = 603_7 = [12_7, 45_7] = "6031245" which contains all digits 0..6 once.
a(8) = 440 = [2, 4, 5, 11] = 670_8 = [2_8, 4_8, 5_8, 13_8] = "67024513" which contains all digits 0..7 once.
a(9) = 1440 = [3, 4, 5, 24] = 1870_9 = [3_9, 4_9, 5_9, 26_9] = "187034526" which contains all digits 0..8 once.
a(10) = 8596 = [2, 14, 307] = "8596214307" which contains all digits 0..9 once. See also A370970.
a(11) = 15552 = [2, 3, 6, 8, 54] = 10759_11 = [2_11, 3_11, 6_11, 8_11, 4a_11] = "1075923684a" which contains all digits 0..a once.
a(12) = 121638 = [2, 3, 11, 1843] = 5a486_12 = [2_12, 3_12, b_12, 1097_12] = "5a48623b1097" which contains all digits 0..b once.
a(13) = 282240 = [2, 3, 5, 7, 21, 64] = 9b60a_13 = [2_13, 3_13, 5_13, 7_13, 18_13, 4c_13] = "9b60a2357184c" which contains all digits 0..c once.
a(14) = 1039104 = [2, 3, 4, 6, 8, 11, 82] = 1d097a_14 = [2_14, 3_14, 4_14, 6_14, 8_14, b_14, 5c_14] = "1d097a23468b5c" which contains all digits 0..d once.
From _Chai Wah Wu_, Apr 24 2024: (Start)
a(15) = 5757696 = [2, 3, 4, 12, 84, 238] = 78aeb6_15 = [2_15, 3_15, 4_15, c_15, 59_15, 10d_15] = "78aeb6234c5910d" which contains all digits 0..e once.
a(16) = 24108000 = [3, 4, 5, 7, 10, 41, 140] = 16fdbe0_16 = [3_16, 4_16, 5_16, 7_16, a_16, 29_16, 8c_16] = "16fdbe03457a298c" which contains all digits 0..f once. (End)

Cf. A372053, A370970, A371958.

a(15)-a(16) from Chai Wah Wu, Apr 24 2024

Added escape clause to definition at the suggestion of Chai Wah Wu. - N. J. A. Sloane, Apr 25 2024

A380760 Integers k with at least one proper factorization for which the sum of the same fixed integer power >= 2 of the factors equals k.

Original entry on oeis.org

16, 27, 48, 54, 256, 270, 528, 1134, 1755, 2916, 3125, 7216, 7830, 11520, 11934, 15360, 19683, 22464, 30000, 31752, 40095, 40960, 46656, 65536, 69168, 81702, 86436, 93555, 100368, 146880, 200000, 212400, 264654, 273600, 291060, 303030, 317520, 340470, 362880

Offset: 1

Charles L. Hohn, Feb 02 2025

nonn

Superset of A381538 for values >= 16, and it is conjectured that the terms that match multiple nonequivalent factorizations here, such as a(5) = 256 (see Example), are exactly the terms of A381538 that can be produced as m^(m^e) by multiple m.

			a(1) = 16: 2 * 2 * 2 * 2 = 2^2 + 2^2 + 2^2 + 2^2 = 16.
a(5) = 256: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 = 256, and also 4 * 4 * 4 * 4 = 4^3 + 4^3 + 4^3 + 4^3 = 256.
a(8) = 1134: 2 * 3 * 3 * 7 * 9 = 2^3 + 3^3 + 3^3 + 7^3 + 9^3 = 1134.

Sums of squares only: A380902.

Cf. A162247, A372053, A381538.

a380760_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(r==1, my(c=0); for(p=2, oo, my(t=sum(i=1, #f, f[i]^p)); if(tCharles L. Hohn, Mar 09 2025

Edited by Peter Munn, Mar 25 2025

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))

A372294 The smallest number k which, when written in base n, has a factorization k = f_1f_2...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

Original entry on oeis.org

-1, -1, -1, -1, -1, 104, 440, 1440, 4830, 15552, 72240, 282240, 1039104, 4244940, 24108000

Offset: 2

Chai Wah Wu, Apr 25 2024

Similar to A372249, except that here the factors are allowed to be equal to 1. Differs from A372249 at n = 7, 10, 12, 15, ...

			a(7)  =      104 = 1*4*26
a(8)  =      440 = 2*4*5*11
a(9)  =     1440 = 3*4*5*24
a(10) =     4830 = 1*2*5*7*69
a(11) =    15552 = 2*3*6*8*54
a(12) =    72240 = 1*4*6*7*430
a(13) =   282240 = 2*3*5*7*21*64
a(14) =  1039104 = 2*3*4*6*8*11*82
a(15) =  4244940 = 1*2*3*7*9*10*1123
a(16) = 24108000 = 3*4*5*7*10*41*140

Cf. A372249, A372053, A370970, A371958.

a(n) <= A372249(n).

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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A372034 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 nonincreasing order.

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A372249 The smallest number k which, when written in base n, has a factorization k = f1*f2*...*fr where the digits of {k, f1, f2, ..., fr} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

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A380760 Integers k with at least one proper factorization for which the sum of the same fixed integer power >= 2 of the factors equals k.

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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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A372294 The smallest number k which, when written in base n, has a factorization k = f_1*f_2*...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

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A372249 The smallest number k which, when written in base n, has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.

A372294 The smallest number k which, when written in base n, has a factorization k = f_1f_2...*f_r where f_i >= 1 and the digits of {k, f_1, f_2, ..., f_r} together contain the digits 0,1,...,(n-1) exactly once. Set a(n) = -1 if no such k exists.