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
A372335
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) alternate being larger than and then smaller than the previous digit.
Original entry on oeis.org
14, 15, 78, 161, 591, 1214, 1317, 1318, 1326, 1407, 1418, 1438, 1506, 1509, 1514, 1527, 1538, 1618, 1626, 1646, 1658, 1703, 1714, 1718, 1734, 1739, 1758, 1814, 1834, 1838, 1839, 1857, 1858, 1934, 1938, 2307, 2427, 2509, 2517, 2534, 2535, 2715, 2757, 2758, 2869, 2958, 3419, 3439, 3514, 3523
Offset: 1
161 is a term as 161 = 7 * 23 which when concatenated give "161723", the digits of which alternate from being larger than and then smaller than the previous digit.
A372336
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) alternate being smaller than and then larger than the previous digit.
Original entry on oeis.org
6, 51, 91, 106, 219, 323, 406, 435, 437, 518, 529, 609, 614, 626, 629, 634, 658, 703, 705, 818, 826, 838, 878, 906, 938, 978, 2051, 2093, 2173, 3053, 3241, 4151, 4171, 4281, 5041, 5063, 5141, 5183, 5241, 6251, 6591, 7021, 7081, 7251, 8051, 8121, 8491, 8571, 8781, 9121, 9231, 9291, 9583
Offset: 1
106 is a term as 106 = 2 * 53 which when concatenated give "106253", the digits of which alternate from being smaller than and then larger than the previous digit.
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.
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;
-
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)))
Showing 1-6 of 6 results.
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