A372309 -id:A372309 - cp's OEIS frontend

A372384 The smallest composite number k such that the digits of k and its prime factors, both written in base n, contain the same set of distinct digits.

4, 8, 30, 25, 57, 16, 27, 192, 132, 121, 185, 169, 465, 32, 306, 289, 489, 361, 451, 2250, 552, 529, 125, 1586, 81, 1652, 985, 841, 1057, 64, 1285, 86166, 2555, 1332, 1387, 1369, 4752, 3240, 2005, 1681, 2649, 1849, 2047, 5456, 2256, 2209, 343, 5050, 2761, 5876, 2862, 2809, 3097, 15512

Offset: 2

Scott R. Shannon, Apr 29 2024

			a(4) = 30 as 30 = 2 * 3 * 5 = 132_4 = 2_4 * 3_4 * 11_4, and both 132_4 and its prime factors contain the same distinct digits 1, 2, and 3.
a(10) = 132 as 132 = 2 * 3 * 11, and both 132 and its prime factors contain the same distinct digits 1, 2, and 3. See also A035141.
a(14) = 465 as 465 = 3 * 5 * 31 = 253_14 = 3_14 * 5_14 * 23_14, and both 253_14 and its prime factors contain the same distinct digits 2, 3, and 5.

Scott R. Shannon, Table of n, a(n) for n = 2..404

Cf. A027746, A035141, A035140, A372309, A371695, A000668.

a(n) = 2*n + 2 if n = 2^k - 1 with k >= 2, otherwise a(n) = n^2 if n is prime.

A376078 The smallest number whose prime factor concatenation when written in all bases k, where k = 2...n, contains all digits 0,1,...,(k-1).

Original entry on oeis.org

2, 6, 38, 190, 5662, 39255, 1206042, 22481939, 392228153, 6329975006

Offset: 2

Scott R. Shannon, Sep 09 2024

Up to a(11) 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(7) = 39255 as 39255 = 3*5*2617 = 11_2*101_2*101000111001_2 = "11101101000111001" which contains all digits 0...1 = 10_3*12_3*10120221_3 = "101210120221" which contains all digits 0...2, 3_4*11_4*220321_4 = "311220321" which contains all digits 0...3 = 3_5*10_5*40432_5 = "31040432" which contains all digits 0...4 = 3_6*5_6*20041_6 = "3520041" which contains all digits 0...5 = 3_7*5_7*10426_7 = "3510426" which contains all digits 0...6.

Cf. A372309, A373645, A055085, A027748.

A373645 The smallest number whose prime factor concatenation, as well as the number itself, when written in base n, contains all digits 0,1,...,(n-1).

Original entry on oeis.org

2, 11, 114, 894, 13155, 127041, 2219826, 44489860, 1023485967, 26436195405, 755182183459, 23609378957430, 802775563829902, 29480898988179429, 1162849454580682365

Offset: 2

Scott R. Shannon, Jun 12 2024

For base 2 and base 3 the number is prime; are there other bases where this is also true?

			a(5) = 894 = 12034_5 which contains all the digits 0..4, and 894 = 2 * 3 * 149 = 2_5 * 3_5 * 1044_5, and the factors contain all digits 0..4.
a(10) = 1023485967 which contains all digits 0..9, and 1023485967 = 3 * 3 * 7 * 16245809, and the factors contain all digits 0..9.
a(15) = 29480898988179429 = 102345C86EA7BD9_15 which contains all the digits 0..E, and 29480898988179429 = 3 * 7 * 17 * 139 * 594097474723 = 3_15 * 7_15 * 12_15 * 94_15 * 106C1A8B5ED_15, and the factors contain all digits 0..E.

Cf. A372309, A358003, A372384, A372280, A372295, A372249, A027746.

Cf. A374225.

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

A378950 Numbers that are a proper substring of the concatenation (with repetition) in decreasing order of their prime factors.

Original entry on oeis.org

95, 132, 995, 9995, 73332, 85713, 93115, 131131, 197591, 632812, 999995, 4285713, 8691315, 58730137, 99999995, 131373333, 507107133, 4870313015

Offset: 1

Scott R. Shannon, Dec 11 2024

All numbers of the form 5*A055558(k), k>=1, are terms.

			95 is a term as 95 = 19 * 5 = "195" when concatenated, which contains "95" as a substring.
632812 is a term as 632812 = 563 * 281 * 2 * 2 = "56328122" when concatenated, which contains "632812" as a substring.
4870313015 is a term as 4870313015 = 748703 * 1301 * 5 = "74870313015" when concatenated, which contains "4870313015" as a substring.

Cf. A027746, A378893, A378894, A376078, A372309.

cp's OEIS Frontend

A372384 The smallest composite number k such that the digits of k and its prime factors, both written in base n, contain the same set of distinct digits.

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A376078 The smallest number whose prime factor concatenation when written in all bases k, where k = 2...n, contains all digits 0,1,...,(k-1).

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A373645 The smallest number whose prime factor concatenation, as well as the number itself, when written in base n, contains all digits 0,1,...,(n-1).

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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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A378950 Numbers that are a proper substring of the concatenation (with repetition) in decreasing order of their prime factors.

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