A046396
Palindromes which are the product of 6 distinct primes.
Original entry on oeis.org
222222, 282282, 474474, 555555, 606606, 646646, 969969, 2040402, 2065602, 2206022, 2417142, 2646462, 2673762, 2875782, 3262623, 3309033, 4179714, 4192914, 4356534, 4585854, 4912194, 5021205, 5169615, 5174715, 5578755
Offset: 1
Cf.
A046332 (similar, but for 6 prime factors counted with multiplicity).
Cf.
A074969 (numbers having 6 distinct prime divisors).
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Select[Range[6*10^6],#==IntegerReverse[#]&&PrimeNu[#]==PrimeOmega[#]==6&] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Mar 17 2016 *)
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A046332_upto(N, start=1, num_fact=6)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && issquarefree(start) && listput(L, start)); L} \\ M. F. Hasler, Jun 06 2024
A373466
Palindromes with exactly 6 distinct prime divisors.
Original entry on oeis.org
222222, 282282, 414414, 444444, 474474, 555555, 606606, 636636, 646646, 666666, 696696, 828828, 888888, 969969, 2040402, 2065602, 2141412, 2206022, 2343432, 2417142, 2444442, 2572752, 2646462, 2673762, 2747472, 2848482, 2875782, 2949492, 2976792
Offset: 1
a(1) = 222222 = 2 * 3 * 7 * 11 * 13 * 37 has exactly 6 distinct prime divisors.
a(3) = 414414 = 2 * 3^2 * 7 * 11 * 13 * 23 has 6 distinct prime divisors, even though the factor 3 occurs twice in the factorization.
Cf.
A046332 (same with bigomega = 6: prime factors counted with multiplicity),
A046396 (similar, but squarefree terms only),
A373465 (same with omega = 5),
A373467 (same with bigomega = 7).
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Select[Range[3000000],PalindromeQ[#]&&Length[FactorInteger[#]]==6&] (* James C. McMahon, Jun 08 2024 *)
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A373466_upto(N, start=1, num_fact=6)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}
A046380
Palindromes with exactly 6 palindromic prime factors (counted with multiplicity).
Original entry on oeis.org
2772, 6776, 25452, 59895, 88788, 549945, 1931391, 8117118, 8447448, 51033015, 52711725, 58344385, 103838301, 535707535, 620434026, 1663223661, 8262112628, 15271417251, 25227972252, 27747974772, 27974547972, 92628082629, 97079897079, 6421339331246, 8401825281048
Offset: 1
-
from sympy import factorint
def palQgen10(l): # generator of palindromes in base 10 of length <= 2*l
if l > 0:
yield 0
for x in range(1,l+1):
for y in range(10**(x-1),10**x):
s = str(y)
yield int(s+s[-2::-1])
for y in range(10**(x-1),10**x):
s = str(y)
yield int(s+s[::-1])
A046380_list = []
for x in palQgen10(6):
a = factorint(x)
if sum(list(a.values())) == 6:
for p in a:
s = str(p)
if s != s[::-1]:
break
else:
A046380_list.append(x) # Chai Wah Wu, Dec 26 2014
A348050
Palindromes setting a new record of their number of prime divisors A001222.
Original entry on oeis.org
1, 2, 4, 8, 88, 252, 2112, 4224, 8448, 44544, 48384, 405504, 4091904, 405909504, 677707776, 4285005824, 21128282112, 29142024192, 4815463645184, 445488555884544, 27874867776847872, 40539458585493504, 63556806860865536, 840261068860162048, 4870324782874230784
Offset: 1
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m=0;lst=Union@Flatten[Table[{FromDigits@Join[s=IntegerDigits@n,Reverse@s],FromDigits@Join[w=IntegerDigits@n,Rest@Reverse@w]},{n,10^5}]];Do[t=PrimeOmega@lst[[n]];If[t>m,Print@lst[[n]];m=t],{n,Length@lst}] (* Giorgos Kalogeropoulos, Oct 25 2021 *)
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from sympy import factorint
from itertools import product
def palsthru(maxdigits):
midrange = [[""], [str(i) for i in range(10)]]
for digits in range(1, maxdigits+1):
for p in product("0123456789", repeat=digits//2):
left = "".join(p)
if len(left) and left[0] == '0': continue
for middle in midrange[digits%2]:
yield int(left+middle+left[::-1])
def afind(maxdigits):
record = -1
for p in palsthru(maxdigits):
f = factorint(p, multiple=True)
if p > 0 and len(f) > record:
record = len(f)
print(p, end=", ")
afind(10) # Michael S. Branicky, Oct 25 2021
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