A371958 -id:A371958 - cp's OEIS frontend

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.

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

A372309 The smallest number whose prime factor concatenation, when written in base n, contains all digits 0,1,...,(n-1).

Original entry on oeis.org

2, 6, 38, 174, 2866, 11670, 135570, 1335534, 15618090, 155077890, 5148702870, 31771759110, 774841780230, 11924858870610, 253941409789410, 3867805835651310

Offset: 2

Scott R. Shannon, Apr 26 2024

Up to a(12) all terms have prime factors whose concatenation length in base n is n, the minimum possible value. Is this true for all a(n)?
a(13) <= 31771759110 = 2*3*5*7*13*61*190787 whose prime factors in base 13 are: 2, 3, 5, 7, 10, 49, 68abc. Sequence is a subsequence of A058760. - Chai Wah Wu, Apr 28 2024
From Chai Wah Wu, Apr 29 2024: (Start)
a(14) <= 1138370792790 = 2*3*5*7*11*877*561917 whose prime factors in base 14 are: 2, 3, 5, 7, b, 469, 108acd.
a(15) <= 23608327052310 = 2*3*5*7*11*13*233*3374069 whose prime factors in base 15 are: 2, 3, 5, 7, b, d, 108, 469ace. (End)
a(14) <= 774841780230, a(15) <= 11924858870610, a(16) <= 256023548755170, a(17) <= 4286558044897590. - Daniel Suteu, Apr 30 2024
For n <= 36, all terms have prime factors whose concatenation length in base n is n, the minimum possible value. - Dominic McCarty, Jan 07 2025

			The factorizations to a(12) are:
a(2) = 2 = 10_2, which contains all digits 0..1.
a(3) = 6 = 2 * 3 = 2_3 * 10_3, which contain all digits 0..2.
a(4) = 38 = 2 * 19 = 2_4 * 103_4, which contain all digits 0..3.
a(5) = 174 = 2 * 3 * 29 = 2_5 * 3_5 * 104_5, which contain all digits 0..4.
a(6) = 2866 = 2 * 1433 = 2_6 * 10345_6, which contain all digits 0..5.
a(7) = 11670 = 2 * 3 * 5 * 389 = 2_7 * 3_7 * 5_7 * 1064_7, which contain all digits 0..6.
a(8) = 135570 = 2 * 3 * 5 * 4519 = 2_8 * 3_8 * 5_8 * 10647_8, which contain all digits 0..7.
a(9) = 1335534 = 2 * 3 * 41 * 61 * 89 = 2_9 * 3_9 * 45_9 * 67_9 * 108_9, which contain all digits 0..8.
a(10) = 15618090 = 2 * 3 * 5 * 487 * 1069, which contain all digits 0..9. See A058909.
a(11) = 155077890 = 2 * 3 * 5 * 11 * 571 * 823 = 2_11 * 3_11 * 5_11 * 10_11 * 47a_11 * 689_11, which contain all digits 0..a.
a(12) = 5148702870 = 2 * 3 * 5 * 151 * 1136579 = 2_12 * 3_12 * 5_12 * 107_12 * 4698ab_12, which contain all digits 0..b.

Dominic McCarty, Numbers Whose Prime Factorizations Have Every Digit (OEIS A372309), YouTube video (2024).
Dominic McCarty, Java program for A372309
Dominic McCarty, Bounds on a(n) for n <= 36

Cf. A372249, A371993, A027746, A371958, A058909, A185122, A058760.

from math import factorial
from itertools import count
from sympy import factorint
from sympy.ntheory import digits
def a(n):
    for k in count(factorial(n)):
        s = set()
        for p in factorint(k): s.update(digits(p, n)[1:])
        if len(s) == n: return k
print([a(n) for n in range(2, 10)]) # Michael S. Branicky, Apr 28 2024

a(n) >= n!. - Michael S. Branicky, Apr 28 2024

a(n) <= A185122(n). - Michael S. Branicky, Apr 28 2024

a(13)-a(16) from Martin Ehrenstein, May 03 2024

a(17) from Dominic McCarty, Jan 07 2025

A371993 The smallest number k that has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain all, and only, the digits 0..n, where n<=9.

Original entry on oeis.org

100, 20, 120, 102, 120, 240, 1260, 1680, 8598

Offset: 1

Scott R. Shannon, Apr 15 2024

			The factorizations are:
a(1) = 100 = [10, 10]
a(2) = 20 = [2, 10]
a(3) = 120 = [2, 2, 3, 10]
a(4) = 102 = [3, 34]
a(5) = 120 = [2, 3, 4, 5]
a(6) = 240 = [3, 5, 16]
a(7) = 1260 = [3, 3, 4, 5, 7]
a(8) = 1680 = [2, 2, 3, 4, 5, 7]
a(9) = 8596 = [2, 14, 307]

Cf. A371958, A370970, A370972.

A378893 Numbers that are a proper substring of the concatenation (with repetition) in increasing order of their prime factors.

Original entry on oeis.org

333, 22564, 113113, 210526, 252310, 1143241, 3331233, 3710027, 31373137, 217893044, 433100023, 2263178956

Offset: 1

Scott R. Shannon, Dec 10 2024

			333 is a term as 333 = 3 * 3 * 37 = "3337" when concatenated, which contains "333" as a substring.
113113 is a term as 113113 = 7 * 11 * 13 * 113 = "71113113" when concatenated, which contains "113113" as a substring.
433100023 is a term as 433100023 = 433 * 1000231 = "4331000231" when concatenated, which contains "433100023" as a substring.

Cf. A027746, A378894, A376078, A372309, A371958, A371696, A372046, A371641.

A378894 Numbers, when written in binary, that are a proper substring of the concatenation (with repetition) in increasing order of their prime factors, when written in binary.

Original entry on oeis.org

10, 57, 63, 355, 737, 921, 1526, 13803, 22008, 43364, 44016, 48895, 65151, 88032, 130545, 235929, 255987, 563207, 702460, 1456355, 2799617, 3020897, 3137557, 3774873, 4163463, 5697350, 5995862, 14176747, 42172441, 55933611, 87559273, 93206755, 108530173, 126474397, 180677710, 193337441, 249550095, 259779663, 533713761, 536378647, 715881440, 940339099, 1000732491

Offset: 1

Scott R. Shannon, Dec 10 2024

			10 is a term as 10 = 1010_2 = 2 * 5 = 10_2 * 101_2 = "10101" when concatenated, which contains "1010" as a substring.
63 is a term as 63 = 111111_2 = 3 * 3 * 7 = 11_2 * 11_2 * 111_2 = "1111111" when concatenated, which contains "111111" as a substring.
1526 is a term as 1526 = 10111110110_2 = 2 * 7 * 109 = 10_2 * 111_2 * 1101101_2 = "101111101101" when concatenated, which contains "10111110110" as a substring.

Cf. A027746, A378893, A376078, A372309, A371958, A371696, A372046, A371641.

A370612 The smallest number whose prime factor concatenation, when written in base n, does not contain 0 and contains all digits 1,...,(n-1) at least once.

Original entry on oeis.org

3, 5, 14, 133, 706, 2490, 24258, 217230, 2992890, 24674730, 647850030, 4208072190, 82417704810

Offset: 2

Chai Wah Wu, Apr 30 2024

All terms are squarefree. Many thanks to Michael Branicky for pointing out errors in the terms in the original submission.

			a(2) = 3 = 3 whose prime factor in base 2 is: 11.
a(3) = 5 = 5 whose prime factor in base 3 is: 12.
a(4) = 14 = 2*7 whose prime factors in base 4 are: 2, 13.
a(5) = 133 = 7*19 whose prime factors in base 5 are: 12, 34.
a(6) = 706 = 2*353 whose prime factors in base 6 are: 2, 1345.
a(7) = 2490 = 2*3*5*83 whose prime factors in base 7 are: 2, 3, 5, 146.
a(8) = 24258 = 2*3*13*311 whose prime factors in base 8 are: 2, 3, 15, 467.
a(9) = 217230 = 2*3*5*13*557 whose prime factors in base 9 are: 2, 3, 5, 14, 678.
a(10) = 2992890 = 2*3*5*67*1489.
a(11) = 24674730 = 2*3*5*19*73*593 whose prime factors in base 11 are: 2, 3, 5, 18, 67, 49a.
a(12) = 647850030 = 2*3*5*19*1136579 whose prime factors in base 12 are: 2, 3, 5, 17, 4698ab.
a(13) = 4208072190 = 2*3*5*7*61*89*3691 whose prime factors in base 13 are: 2, 3, 5, 7, 49, 6b, 18ac.
a(14) = 82417704810 = 2*3*5*7*23*937*18211 whose prime factors in base 14 are: 2, 3, 5, 7, 19, 4ad, 68cb.

Cf. A371194, A372309, A372249, A371993, A027746, A371958, A058909, A185122.

from math import factorial
from itertools import count
from sympy import primefactors
from sympy.ntheory import digits
def A370612(n): return next(k for k in count(max(factorial(n-1),2)) if 0 not in (s:=set.union(*(set(digits(p,n)[1:]) for p in primefactors(k)))) and len(s) == n-1)

(n-1)! <= a(n) <= A371194(n).

a(13)-(14) from Dominic McCarty, Jan 07 2025

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

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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A372309 The smallest number whose prime factor concatenation, when written in base n, contains all digits 0,1,...,(n-1).

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A371993 The smallest number k that has a factorization k = f1*f2*...*fr where the digits of {k, f1, f2, ..., fr} together contain all, and only, the digits 0..n, where n<=9.

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A378893 Numbers that are a proper substring of the concatenation (with repetition) in increasing order of their prime factors.

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A378894 Numbers, when written in binary, that are a proper substring of the concatenation (with repetition) in increasing order of their prime factors, when written in binary.

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A370612 The smallest number whose prime factor concatenation, when written in base n, does not contain 0 and contains all digits 1,...,(n-1) at least once.

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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.

A371993 The smallest number k that has a factorization k = f1f2...*fr where the digits of {k, f1, f2, ..., fr} together contain all, and only, the digits 0..n, where n<=9.

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.