Palindromic Primes Assignment 3

1. Smallest palindromes with exactly n palindromic prime factors
(counted with multiplicity).

Report extensions also to Sloane's "Online Encyclopedia".

Refer to sequence A046385

n	palindrome	palindromic prime factors
0	1	-
1	2	2
2	4	2 * 2
3	8	2 * 2 * 2
4	88	2 * 2 * 2 * 11
5	252	2 * 2 * 3 * 3 * 7
6	2772	2 * 2 * 3 * 3 * 7 * 11
7	82728	2 * 2 * 2 * 3 * 3 * 3 * 383
8	2112	2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
9	4224	2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
10	8448	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
11	236989632	2 * 2 * 2 * 2 * 2 * 2 * 3 * 11 * 11 * 101 * 101
12	48384	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 7
13	2977792	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 727
14	?	?
15	?	?

2. Smallest palindromes with exactly n prime factors
(counted with multiplicity).

Refer to sequence A076886 [ November 25, 2002 ]

by Shyam Sunder Gupta (extended by Robert G. Wilson v).

n	palindrome	prime factors
0	1	-
1	2	2
2	4	2 * 2
3	8	2 * 2 * 2
4	88	2 * 2 * 2 * 11
5	252	2 * 2 * 3 * 3 * 7
6	2772	2 * 2 * 3 * 3 * 7 * 11
7	27872	2 * 2 * 2 * 2 * 2 * 13 * 67
8	2112	2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
9	4224	2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
10	8448	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 11
11	44544	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 29
12	48384	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 7
13	2977792	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 727
14	27011072	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 11 * 11 * 109
15	405504	2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 11
16	4091904	2¹² * 3³ * 37
17	441606144	2¹³ * 3 * 7 * 17 * 151
18	405909504	2¹² * 3² * 7 * 11² * 13
19	886898688	2¹⁶ * 3 * 13 * 347
20	677707776	2¹⁶ * 3³ * 383
21	4285005824	2¹⁹ * 11 * 743
22	276486684672	2¹⁸ * 3 * 11 * 31 * 1031
23	21128282112	2¹⁹ * 3 * 7 * 19 * 101
24	633498894336	2¹⁷ * 3 * 7⁴ * 11 * 61
25	2701312131072	2²² * 3 * 19 * 11299
26	8691508051968	2²² * 3 * 7 * 101 * 977

3. Smallest palindromes with exactly n distinct prime factors.

Report extensions also to Sloane's "Online Encyclopedia".

Refer to sequence A046399

n	palindrome	distinct prime factors
0	1	-
1	2	2
2	6	2 * 3
3	66	2 * 3 * 11
4	858	2 * 3 * 11 * 13
5	6006	2 * 3 * 7 * 11 * 13
6	222222	2 * 3 * 7 * 11 * 13 * 37
7	22444422	2 * 3 * 7 * 11 * 13 * 37 * 101
8	244868442	2 * 3 * 7 * 13 * 17 * 23 * 31 * 37
9	6434774346	2 * 3 * 7 * 11 * 13 * 17 * 19 * 31 * 107
10	?	?
11	?	?
12	?	?
13	?	?

4. Smallest palindromes with exactly n distinct palindromic prime factors.

n	palindrome	distinct palindromic prime factors
0	1	-
1	2	2
2	6	2 * 3
3	66	2 * 3 * 11
4	6666	2 * 3 * 11 * 101
5	?	?
6	?	?
7	?	?

5. Palindromes divided by the palindromic sum of their prime factors
(counted with multiplicity) is a palindrome.

Report extensions also to Sloane's "Online Encyclopedia".

Refer to sequence A046362

palindrome	prime factors	palindromic sum	palindromic quotient
4	2 * 2	4	1
1345431	3 * 17 * 23 * 31 * 37 (all distinct)	111	12121
12333321	3 * 3 * 7 * 11 * 13 * 37 * 37	111	111111
1425665241	3 * 3 * 7 * 7 * 11 * 13 * 13 * 37 * 47	141	10111101
?	?	?	?
?	?	?	?

6. Palindromes divisible by the palindromic sum of their palindromic prime factors
(counted with multiplicity).

palindrome	palindromic prime factors	palindromic sum	quotient
4	2 * 2	4	1
5445	3 * 3 * 5 * 11 * 11	33	165
?	?	?	?
?	?	?	?

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