A046399
Smallest squarefree palindrome with exactly n distinct prime factors.
Original entry on oeis.org
1, 2, 6, 66, 858, 6006, 222222, 22444422, 244868442, 6434774346, 438024420834, 50146955964105, 2415957997595142, 495677121121776594, 22181673755737618122, 5521159517777159511255, 477552751050050157255774
Offset: 0
- J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 315, Pour la Science, Paris 2000.
-
r[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Do[k = 1; While[r[k] != k || !SquareFreeQ[k] || Length[Select[Divisors[k], PrimeQ]] != n, k++ ]; Print[k], {n, 0, 30}] (* Ryan Propper, Sep 16 2005 *)
A046332
Palindromes with exactly 6 prime factors (counted with multiplicity).
Original entry on oeis.org
2772, 2992, 6776, 8008, 21112, 21712, 21912, 23632, 23832, 25452, 25752, 25952, 27472, 28782, 29392, 40104, 40304, 40404, 42024, 42924, 44044, 44144, 44744, 44944, 45954, 46764, 46864, 48984, 53235, 54945, 55755, 59895, 60606, 61216
Offset: 1
Cf.
A046396 (similar but terms must be squarefree),
A373466 (similar, but only distinct prime divisors are counted).
-
N:= 6: # to get all terms of up to N digits
digrev:= proc(n) local L,Ln; L:= convert(n,base,10);Ln:= nops(L);
add(L[i]*10^(Ln-i),i=1..Ln);
end proc:
Res:= NULL:
for d from 2 to N do
if d::even then
m:= d/2;
Res:= Res, select(numtheory:-bigomega=6,
[seq](n*10^m + digrev(n), n=10^(m-1)..10^m-1));
else
m:= (d-1)/2;
Res:= Res, select(numtheory:-bigomega=6,
[seq](seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1));
fi
od:
map(op,[Res]); # Robert Israel, Dec 23 2014
-
A046332_upto(N, start=1, num_fact=6)={ my(L=List()); while(N >= start = nxt_A002113(start), bigomega(start)==num_fact && listput(L, start)); L} \\ M. F. Hasler, Jun 06 2024
-
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])
A046332_list = [x for x in palQgen10(4) if sum(list(factorint(x).values())) == 6]
# Chai Wah Wu, Dec 21 2014
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).
-
Select[Range[3000000],PalindromeQ[#]&&Length[FactorInteger[#]]==6&] (* James C. McMahon, Jun 08 2024 *)
-
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}
A373467
Palindromes with exactly 7 (distinct) prime divisors.
Original entry on oeis.org
20522502, 21033012, 22444422, 23555532, 24266242, 25777752, 26588562, 35888853, 36399363, 41555514, 41855814, 42066024, 43477434, 43777734, 44888844, 45999954, 47199174, 51066015, 51666615, 52777725, 53588535, 53888835, 55233255, 59911995, 60066006, 60366306, 61777716, 62588526, 62700726
Offset: 1
Obviously all terms must be palindromic; let us consider the prime factorization:
a(1) = 20522502 = 2 * 3^2 * 7 * 11 * 13 * 17 * 67 has exactly 7 distinct prime divisors, although the factor 3 appears twice in the factorization. (Without the second factor 3 the number would not be palindromic.)
a(2) = 21033012 = 2^2 * 3 * 7 * 11 * 13 * 17 * 103 has exactly 7 distinct prime divisors, although the factor 2 appears twice in the factorization. (Without the second factor 2 the number would not be palindromic.)
a(3) = 22444422 = 2 * 3 * 7 * 11 * 13 * 37 * 101 is the product of 7 distinct primes (cf. A123321), hence the first squarefree term of this sequence.
Cf.
A046333 (same with bigomega = 7: counting prime factors with multiplicity),
A046397 (same but only squarefree terms),
A373465 (same with omega = 5),
A046396 (same with omega = 6).
-
A373467_upto(N, start=vecprod(primes(7)), num_fact=7)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}
Showing 1-4 of 4 results.
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