A046395
Palindromes that are the product of 5 distinct primes.
Original entry on oeis.org
6006, 8778, 20202, 28182, 41514, 43134, 50505, 68586, 87978, 111111, 141141, 168861, 202202, 204402, 209902, 246642, 249942, 262262, 266662, 303303, 323323, 393393, 399993, 438834, 454454, 505505, 507705, 515515, 516615, 519915, 534435, 535535, 543345
Offset: 1
505505 = 5 * 7 * 11 * 13 * 101.
Cf.
A046331 (palindromes with 5 prime factors counted with multiplicity),
A373465 (counting only distinct prime divisors).
Corrected at the suggestion of Sean A. Irvine by
Harvey P. Dale, Apr 09 2021
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}
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).
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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 1-3 of 3 results.
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