A058909 -id:A058909

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A372309

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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

LINKS

Table of n, a(n) for n=2..16.

FORMULA

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

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

EXAMPLE

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.

PROG

(Python)

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

CROSSREFS

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

KEYWORD

nonn,base,more

AUTHOR

Scott R. Shannon, Apr 26 2024

EXTENSIONS

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

STATUS

approved

A138165

Prime numbers that contain each of the digits 0,1,4,6,8,9 exactly once.

+10
2

104869, 108649, 140689, 140869, 148609, 164089, 164809, 168409, 184609, 186049, 401689, 406981, 408169, 408691, 409861, 416089, 418069, 460189, 460891, 460981, 468019, 468109, 469801, 480169, 486091, 489061, 498061, 601849, 604189, 604819

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

There are 66 terms. Each product 2*3*5*7*a(n) is a squarefree number whose prime factorization (ignoring exponents) contains exactly one of each decimal digit, so each product is a term of A058909. (The primes 2,3,5,7 are the only single-digit primes in base 10.)

LINKS

Rick L. Shepherd, Table of n, a(n) for n = 1..66 (full sequence)

MATHEMATICA

Select[Prime[Range[10000, 50000]], SequenceCount[DigitCount[#], {1, _, _, 1, _, 1, _, 1, 1, 1}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 07 2020 *)

CROSSREFS

Cf. A058909.

KEYWORD

base,easy,fini,full,nonn

AUTHOR

Rick L. Shepherd, Mar 03 2008

STATUS

approved

A213737

Odd numbers whose set of prime factors (taken with multiplicity) uses each digit from 0 to 9 exactly once.

+10
1

42279945, 42315045, 42514845, 43092645, 43767645, 45981645, 46149045, 46321845, 52226745, 52654695, 53159595, 56789745, 56841045, 57321645, 58193745, 59869345, 61277145, 61421595, 61860445, 62146545, 62866645, 62936295, 62969845, 63395295, 63411595

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence also contains numbers not ending in 5 (i.e., 78369189).

a(1916) = 240510701 is the first semiprime with this property.

No pandigital number is in the sequence.

LINKS

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Wikipedia, Prime factor

EXAMPLE

42279945 = 3*5*1049*2687 is in the sequence since the set {3, 5, 1049, 2687} can be formed from the digits 0 to 9 and each digit is used only once.

MATHEMATICA

lst = {}; Do[If[Equal[Sort@Flatten@IntegerDigits@FactorInteger[n][[All, {1}]], {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}] && SquareFreeQ[n], AppendTo[lst, n]], {n, 4*10^7 + 1, 7*10^7, 2}]; lst

ed1Q[n_]:=Module[{fi=FactorInteger[n]}, Max[Transpose[fi][[2]]]==1 && Union[ Flatten[IntegerDigits/@Transpose[fi][[1]]]]==Range[0, 9]]; Select[Range[ 4*10^7+ 1, 6.4*10^7, 2 ], ed1Q] (* Harvey P. Dale, Dec 19 2014 *)

CROSSREFS

Subsequence of A058909.

KEYWORD

base,fini,nonn

AUTHOR

Arkadiusz Wesolowski, Jun 19 2012

STATUS

approved

A290385

Base-ten pandigital factorization integers. The normal factorization (primes raised to greater-than-one exponents) of these numbers uses each digit exactly once.

+10
1

15618090, 20824120, 22022490, 22816290, 22908090, 23294190, 23427135, 23507490, 24843120, 26104560, 26152080, 26679990, 27114690, 27687090, 28275690, 29218704, 29363320, 29447898, 29544690, 29582490, 29670378, 29688144, 29910138, 30120144

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The sequence contains 14856143 terms, the largest being 7^986543210.

The corresponding zeroless sequence contains 2295201 terms, from 2992890 = 2*3*5*67*1489 to 7^98654321. - Giovanni Resta, Jul 29 2017

LINKS

Hans Havermann, Table of n, a(n) for n = 1..10000

Hans Havermann, Pandigital factorization integer cascades

EXAMPLE

20824120 is in the sequence because 2^3*5*487*1069 is pandigital.

MATHEMATICA

pop[d_, mn_] := Union @@ Table[ Select[ FromDigits /@ Flatten[ Permutations /@ Subsets[d, {k}], 1], # > mn && PrimeQ[#] &], {k, IntegerLength@ mn, Length[d]}]; ric[w_, d_, p_] := If[d == {}, cnt++; If[Max[Last /@ w] < 30 && Times @@ (Power @@@ w) <= 4*10^7, AppendTo[L, w]], Block[{pp = pop[d, p], v}, Do[v = Complement[d, IntegerDigits@ x]; ric[Append[w, {x, 1}], v, x]; Do[If[e > 1, ric[Append[w, {x, e}], Complement[v, IntegerDigits@e], x]], {e, Union[ FromDigits /@ Flatten[ Permutations /@ Subsets[v, {1, Length@v}], 1]]}], {x, pp}]]]; Monitor[cnt = 0; L = {}; ric[{}, Range[0, 9], 1]; , cnt]; Print["cnt = ", cnt]; Sort[(Times @@ (Power @@@ #)) & /@ L] (* Giovanni Resta, Jul 29 2017 *)

CROSSREFS

Cf. A058909, A288818.

KEYWORD

nonn,base,fini

AUTHOR

Hans Havermann, Jul 28 2017

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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

LINKS

Table of n, a(n) for n=2..12.

FORMULA

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

EXAMPLE

a(2) = 3 = 3 whose prime factors in base 2 is: 11.

a(3) = 5 = 5 whose prime factors in base 3 is: 12.

a(4) = 14 = 2*7 whose prime factors in base 4 is: 2, 13.

a(5) = 133 = 7*19 whose prime factors in base 5 is: 12, 34.

a(6) = 706 = 2*353 whose prime factors in base 6 is: 2, 1345.

a(7) = 2490 = 2*3*5*83 whose prime factors in base 7 is: 2, 3, 5, 146.

a(8) = 24258 = 2*3*13*311 whose prime factors in base 8 is: 2, 3, 15, 467.

a(9) = 217230 = 2*3*5*13*557 whose prime factors in base 9 is: 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 is: 2, 3, 5, 18, 67, 49a.

a(12) = 647850030 = 2*3*5*19*1136579 whose prime factors in base 12 is: 2, 3, 5, 17, 4698ab.

PROG

(Python)

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)

CROSSREFS

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

KEYWORD

nonn,base,more

AUTHOR

Chai Wah Wu, Apr 30 2024

STATUS

approved

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