A046397
Palindromes which are the product of exactly 7 distinct primes.
Original entry on oeis.org
22444422, 24266242, 26588562, 35888853, 36399363, 43777734, 47199174, 51066015, 53588535, 53888835, 55233255, 59911995, 60066006, 62588526, 62700726, 62888826, 81699618, 87788778, 89433498, 122434221, 202040202
Offset: 1
The first two palindromes with 7 distinct prime factors are 20522502 = 2 * 3^2 * 7 * 11 * 13 * 17 * 67 and 21033012 = 2^2 * 3 * 7 * 11 * 13 * 17 * 103, but these are excluded since one of the prime factors occurs to a higher power.
a(1) = 22444422 = 2 * 3 * 7 * 11 * 13 * 37 * 101, which is squarefree, is therefore the first term of this sequence.
Cf.
A046333 (similar but prime factors counted with multiplicity),
A373467 (similar but counting just the distinct prime divisors).
Cf.
A002113 (palindromes),
A123321 (products of 7 distinct primes),
A176655 (numbers with omega = 7 distinct prime divisors).
-
digrev:= proc(n) local L,i;
L:= convert(n,base,10);
add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local F;
F:= ifactors(n)[2];
nops(F) = 7 and map(t -> t[2],F)=[1$7]
end proc:
Res:= NULL:
count:= 0:
for d from 2 while count < 100 do
if d::even then
m:= d/2;
for n from 10^(m-1) to 10^m-1 while count < 100 do
v:= n*10^m+digrev(n);
if filter(v) then count:= count+1; Res:= Res, v; fi;
od;
else
m:= (d-1)/2;
for n from 10^(m-1) to 10^m-1 while count < 100 do
for y from 0 to 9 while count < 100 do
v:= n*10^(m+1)+y*10^m+digrev(n);
if filter(v) then count:= count+1; Res:= Res, v; fi;
od od
fi
od:
Res; # Robert Israel, Jan 20 2020
-
A046397_upto(N, start=vecprod(primes(7)), num_fact=7)={ my(L=List()); is_A002113(start)&& start--; while(N >= start = nxt_A002113(start), omega(start)==num_fact && issquarefree(start) && listput(L, start)); L} \\ M. F. Hasler, Jun 06 2024
A373465
Palindromes with exactly 5 distinct prime divisors.
Original entry on oeis.org
6006, 8778, 20202, 28182, 40404, 41514, 43134, 50505, 60606, 63336, 66066, 68586, 80808, 83538, 86268, 87978, 111111, 141141, 168861, 171171, 202202, 204402, 209902, 210012, 212212, 219912, 225522, 231132, 232232, 239932, 246642, 249942, 252252, 258852, 262262, 266662, 272272
Offset: 1
a(1) = 6006 = 2 * 3 * 7 * 11 * 13 is a palindrome (A002113) with 5 prime divisors.
a(5) = 40404 = 2^2 * 3 * 7 * 13 * 37 also is a palindrome with 5 prime divisors, although the divisor 2 occurs twice as a factor in the factorization.
Cf.
A046331 (same but counting prime factors with multiplicity),
A046395 (same but squarefree),
A373466 (same with omega = 6),
A373467 (with omega = 7).
-
Select[Range[300000],PalindromeQ[#]&&Length[FactorInteger[#]]==5&] (* James C. McMahon, Jun 08 2024 *)
Select[Range[300000],PalindromeQ[#]&&PrimeNu[#]==5&] (* Harvey P. Dale, Sep 01 2024 *)
-
A373465_upto(N, start=1, num_fact=5)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}
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}
Showing 1-3 of 3 results.
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