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
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.
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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
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
444577 is a term as 444577 = 7 * 7 * 43 * 211, and 444577 has nondecreasing digits while its prime factor concatenation "7743211" has nonincreasing digits.
-
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
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
42 is a term as 42 = 2 * 3 * 7, and 42 has nonincreasing digits while its prime factor concatenation "237" has nondecreasing digits.
-
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
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.
-
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
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
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.
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
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.
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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)
A374225
Irregular triangle read by rows: T(n,k), n > 1 and k <= n, is the smallest composite number x whose set of digits and the set of digits in all prime factors of x, when written in base n, contain exactly k digits in common, or -1 if no such number exists.
Original entry on oeis.org
-1, 9, 4, 4, 8, 6, 15, 4, 6, 14, 30, 114, 4, 12, 10, 35, 190, 894, 4, 8, 33, 188, 377, 2355, 13155, 4, 16, 14, 66, 462, 3269, 22971, 127041, 4, 10, 66, 85, 762, 5359, 36526, 279806, 2219826, 4, 12, 39, 102, 1118, 9096, 62959, 572746, 5053742, 44489860, 4, 12, 95, 132
Offset: 2
T(2, 1) = 9 = 3^2 -> 1001_2 = 11_2^2, have the digit 1 in common, and no lesser composite has this property.
T(6, 2) = 33 = 3 * 11 -> 53_6 = 3_6 * 15_6, have this 2 digits 3 and 5 in common, and no lesser composite has this property.
T(11, 6) = 174752 = 2^5 * 43 * 127 -> 10A326_11 = 2_11^5 * 3A_11 * 106_11, have the 6 digits 0, 1, 2, 3, 6 and A in common, and no lesser composite has this property.
The array begins:
n\k:0, 1, 2, 3, 4, 5, 6,
2: -1, 9, 4;
3: 4, 8, 6, 15;
4: 4, 6, 14, 30, 114;
5: 4, 12, 10, 35, 190, 894;
6: 4, 8, 33, 188, 377, 2355, 13155;
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card(base,x)=my(m=factor(x),u=[],v=[],w=[]);my(u=Set(digits(x,base)));for(i=1,#m~,w=Set(digits(m[i,1],base));v=setunion(v,w));#setintersect(u,v)
T(n,k)=my(x);if(k>n,return(0));if(n==2&&k==0,return(-1));forcomposite(m=max(2,n^(k-1)),oo,x=card(n,m);if(x==k,return(m)))
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.
Original entry on oeis.org
-1, -1, -1, -1, -1, 104, 440, 1440, 4830, 15552, 72240, 282240, 1039104, 4244940, 24108000
Offset: 2
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
Showing 1-8 of 8 results.
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