[seqfan] Re: a(n)+a(a(n)) is prime

Lars Blomberg lars.blomberg at visit.se
Mon Apr 13 16:04:01 CEST 2015


Hello,

Spinning on the same theme using n, a(n), a(a(n)) the following sequences can be generated:

1. a[n] + a[a[n]] is Prime (Angelini)	
	1,3,2,5,6,7,10,9,8,13,12,11,16,15,14,21,18,19,22,23,20,25,24,29,28,27,26,31,30,37,36,33,34,39,38,35,42,41,40,43
2. n + a[a[n]] is Prime	
	1,3,5,2,4,7,11,9,15,6,10,13,17,8,14,12,16,19,23,21,27,18,22,25,29,20,26,24,28,31,37,33,35,32,34,30,36,39,41,38
3. n + a[n] + a[a[n]] is Prime	
	1,3,6,5,8,2,9,4,13,11,16,14,7,15,12,10,18,24,20,22,23,19,27,17,26,28,21,25,30,38,32,34,35,31,33,37,40,29,41,36
4. Abs(a[n] - a[a[n]]) is NOT Prime = A103889
	2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19,22,21,24,23,26,25,28,27,30,29,32,31,34,33,36,35,38,37,40,39
5. Abs(a[n] - a[a[n]]) is Prime	
	2,4,5,6,3,8,9,10,7,12,13,14,11,16,17,18,15,20,21,22,19,24,25,26,23,28,29,30,27,32,33,34,31,36,37,38,35,40,41,42
6. a[n] and a[a[n]] has opposite property Prime
	2,1,4,3,6,5,8,7,10,11,12,13,14,17,16,19,18,23,20,29,22,31,24,37,26,41,28,43,30,47,32,53,34,59,36,61,38,67,40,71
7. a[n] and a[a[n]] has same property Prime
	2,3,5,6,7,4,11,9,8,12,13,10,17,15,14,18,19,16,23,21,20,24,29,22,26,25,28,27,31,32,37,30,34,33,36,35,41,39,38,42
8. a[n]*a[a[n]] + 1 is Prime	
	1,3,2,5,6,7,10,9,8,13,12,15,24,16,18,21,19,22,30,23,20,28,26,25,42,33,29,34,32,35,36,38,40,37,66,31,48,39,50,43
9. n*a[a[n]] + 1 is Prime	
	1,3,5,2,4,7,10,9,11,6,12,8,14,24,13,17,21,16,20,22,18,23,19,15,26,28,29,33,30,32,34,27,25,36,37,39,42,40,31,41
10. n*a[n]*a[a[n]] + 1 is Prime	
	1,3,2,5,9,7,8,6,4,11,12,10,14,15,13,17,20,19,23,16,22,25,18,26,21,29,28,38,24,31,32,30,34,39,36,35,40,27,33,45
11. n*a[n] + a[a[n]] is Prime	
	1,3,5,2,8,7,11,13,6,12,20,17,23,15,19,14,25,21,22,31,41,39,32,26,36,29,28,53,33,34,57,37,40,43,38,47,45,51,49,61
12. n + a[n] + a[a[n]] is Square	
	2,6,4,9,7,1,13,10,3,18,12,26,5,15,20,17,31,8,21,14,24,23,36,19,27,11,29,30,25,42,16,33,35,37,32,22,50,39,44,41
13. n + a[a[n]] is Square	
	2,3,7,5,12,8,1,10,11,17,16,20,14,23,18,25,6,21,22,4,31,30,35,26,9,40,19,29,36,27,15,33,49,37,13,52,47,24,41,38
14. a[n] + a[a[n]] is Square	
	2,7,4,5,11,8,9,17,16,12,14,13,23,22,18,20,19,31,30,29,24,27,26,25,39,38,37,32,35,34,33,49,48,47,46,40,44,43,42,41
15. Abs(a[a[n] - a[n]]) is Square = A065190
	1,3,2,5,4,7,6,9,8,11,10,13,12,15,14,17,16,19,18,21,20,23,22,25,24,27,26,29,28,31,30,33,32,35,34,37,36,39,38,41
16. n + a[n] + a[a[n]] is Pentagonal
	2,9,4,5,3,7,22,10,1,17,12,28,14,24,16,20,8,19,33,15,23,6,26,13,27,21,40,11,30,58,32,54,18,35,48,37,44,39,68,25
17. n + a[a[n]] is Pentagonal
	2,4,1,3,6,7,16,9,14,11,12,24,8,13,17,5,20,15,21,18,32,23,29,10,26,45,28,43,47,19,33,30,39,35,36,57,31,40,37,54
18. a[n] + a[a[n]] is Pentagonal
	2,3,9,5,7,8,15,14,13,11,24,16,22,21,20,19,18,17,32,31,30,29,25,27,26,44,43,33,41,40,39,38,37,35,57,42,55,54,53,52
19. n + a[n] + a[a[n]] is Triangular
	1,3,5,6,2,11,8,13,10,17,4,14,7,19,16,24,9,20,12,28,22,23,21,15,26,27,25,18,30,32,33,29,41,35,36,34,38,45,40,57
20. n + a[a[n]] is Triangular
	2,5,4,7,8,3,6,1,10,12,9,11,14,15,22,17,20,19,27,28,23,13,24,32,26,30,36,16,25,29,33,21,35,37,45,18,44,39,40,52
21. a[n] + a[a[n]] is Triangular
	2,1,4,6,7,9,8,13,12,11,10,16,15,17,21,20,19,22,26,25,24,23,32,31,30,29,28,27,37,36,35,34,38,44,43,42,41,40,45,51
22. n + a[n] + a[a[n]] is Fibonacci
	1,3,8,5,12,7,21,2,10,15,13,4,31,16,9,25,18,20,22,17,6,48,24,42,14,27,36,29,32,33,11,28,81,35,75,26,38,69,40,65
23. n + a[a[n]] is Fibonacci
	1,3,6,5,9,10,8,14,16,2,12,23,7,13,17,4,19,20,38,37,22,34,43,25,31,27,29,26,28,24,30,33,57,67,18,39,35,15,53,41
24. a[n] + a[a[n]] is Fibonacci
	1,3,2,5,8,7,6,13,10,11,23,14,21,20,16,18,19,37,36,35,34,24,32,31,26,29,28,27,60,33,58,57,56,55,54,53,52,39,50,41
25. a[n]*n == a[a[n]]
	1,3,6,5,20,18,8,56,10,90,12,132,14,182,16,240,19,108,323,100,22,462,24,552,26,650,28,756,30,870,32,992,34,1122,36,1260,38,1406,40,1560
26. 2*a[n]*n == a[a[n]]
	2,4,5,16,30,7,84,9,144,11,220,13,312,15,420,128,18,612,20,760,22,924,24,1104,26,1300,28,1512,31,300,1798,33,2112,35,2380,37,2664,39,2964,41
27. n and a[a[n]] contain NOT the same digits approx= A103889
	2,3,1,5,6,4,8,9,7,11,12,10,14,15,13,17,18,16,20,21,19,23,24,22,26,27,25,29,30,28,32,33,31,35,36,34,38,39,37,41
28. a[n] and a[a[n]] contain NOT the same digits
	2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15,18,17,20,19,22,21,24,23,26,25,28,27,30,29,32,31,34,33,36,35,38,37,40,39
29. a[n] and a[a[n]] contain the same digits
	1,3,33,5,55,7,77,9,99,11,111,13,31,15,51,17,71,19,91,21,112,23,32,25,52,27,72,29,92,34,113,223,333,43,36,63,38,83,40,400
30. (n + a[a[n]])/2 = a[n]
	1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41
31. a[n] divides NOT a[a[n]]
	2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41
32. DigitSum(a[n]) = DigitSum(a[a[n]])
	1,3,12,5,14,7,16,9,18,11,20,21,15,23,24,25,19,27,28,101,30,26,32,33,34,35,36,37,31,102,40,41,42,43,44,45,46,39,48,103
33. a[n] and a[a[n]] has same oddity	
	1,3,5,6,7,4,9,10,11,8,13,14,15,12,17,18,19,16,21,22,23,20,25,26,27,24,29,30,31,28,33,34,35,32,37,38,39,36,41,42
34. a[n] divides a[a[n]]+3
	1,3,6,2,7,9,11,5,15,8,19,13,23,16,27,29,18,33,35,21,39,24,43,45,26,49,51,30,55,57,32,61,63,36,67,69,38,73,75,41
35. a[n] divides a[a[n]]+2
	1,3,4,6,7,10,12,9,16,18,13,22,24,15,28,30,19,34,36,21,40,42,25,46,48,27,52,54,31,58,60,33,64,66,37,70,72,39,76,78
36. a[n] divides a[a[n]]+1
	1,3,2,5,9,7,13,10,17,19,12,23,25,15,29,18,33,35,37,21,41,24,45,47,49,27,53,30,57,59,32,63,65,36,69,71,73,39,77,42

/Lars

-----Ursprungligt meddelande-----
Från: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] För Eric Angelini
Skickat: den 11 april 2015 20:44
Till: Sequence Fanatics Discussion list
Ämne: [seqfan] Re: [LIKELY_SPAM] Re: a(n)+a(a(n)) is prime

Admiration! Lars, many thanks 4 the 2 seq!

Catapulté de mon aPhone

> Le 11 avr. 2015 à 19:36, "Lars Blomberg" <lars.blomberg at visit.se> a écrit :
> 
> Hello,
> 
> I find
> 1, 3, 2, 5, 6, 7, 10, 9, 8, 13, 12, 11, 16, 15, 14, 21, 18, 19, 22, 
> 23, 20, 25, 24, 29, 28, 27, 26, 31, 30, 37, 36, 33, 34, 39, 38, 35, 
> 42, 41, 40, 43, 48, 47, 46, 45, 44, 51, 50, 49, 52, 53, 56, 55, 54, 
> 59, 58, 57, 70, 69, 68, 61, 66, 63, 64, 67, 71, 73, 72, 81, 80, 79, 
> 78, 77, 76, 75, 74, 87, 86, 85, 84, 83, 82, 91, 90, 89, 88, 93, 92, 
> 103, 102, 101, 100, 99, 98, 95, 96, 97, 114, 113, 112, 111, 110, 109, 
> 108, 105, 106, 117, 115, 119, 118, 123, 116, 121, 120, 125, 124, 135, 
> 122, 133, 132, 131, 130, 129, 128, 127, 126, 137, 136, 141, 134, 139, 
> 138, 145, 144, 143, 142, 147, 140, 155, 154, 153, 152, 151, 150, 149, 
> 148, 156, 160, 159, 158, 157, 162, 161, 164, 163, 176, 175, 174, 173, 
> 172, 171, 170, 169, 168, 167, 166, 165, 180, 179, 178, 177, 182, 181, 
> 186, 185, 184, 183, 190, 189, 188, 187, 192, 191, 196, 195, 194, 193, 
> 202, 201, 200, 199, 198, 197, 204, 203, 206, 205, 212, 211, 210, 209, 
> 208, 207, 216, 215, 214, 213, 224, 223, 222, 221, 220, 219, 218, 217, 
> 228, 227, 226, 225, 230, 229
> 
> S[22] seems to be 25 instead of 26. 
> 
> The first few unused values after 10,000 terms are 4, 17, 32, 60, 62, 
> 65, 94, 104, 107, 146, 269, 272, 451, 574, 637, 672, 674, 676, 678, 
> 683
> 
> Regards,
> Lars
> 
> -----Ursprungligt meddelande-----
> Från: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] För Eric Angelini
> Skickat: den 11 april 2015 15:40
> Till: Sequence Discussion list
> Ämne: [seqfan] a(n)+a(a(n)) is prime
> 
> 
> Hello SeqFans,
> S is the lexicographically first seq where a(n)+a(a(n)) is prime.
> 
> [This is not http://oeis.org/A083569
> as the integers 4, 17, etc. will never appear here].
> 
> S=1,3,2,5,6,7,10,9,8,13,12,11,16,15,14,21,18,19,22,23,20,26,24,...
> 
> Example:
> 1 + (the 1st integer of S) is prime (1+1=2)
> 3 + (the 3rd integer of S) is prime (3+2=5)
> 2 + (the 2nd integer of S) is prime (2+3=5)
> 5 + (the 5th integer of S) is prime (5+6=11)
> 6 + (the 6th integer of S) is prime (6+7=13)
> 7 + (the 7th integer of S) is prime (7+10=17)
> 10 + (the 10th integer of S) is prime (10+13=23)
> 9 + (the 9th integer of S) is prime (9+8=17)
> 8 + (the 8th integer of S) is prime (8+9=17) etc.
> S is always extended with the smallest integer not yet in S and not 
> leading to a contradiction.
> 
> Best,
> É.
> 
> 
> _______________________________________________
> 
> Seqfan Mailing list - http://list.seqfan.eu/
> 
> 
> _______________________________________________
> 
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