A034306
Palindromes P such that Fibonacci iterations starting with (1, P) lead to a "nine digits anagram".
Original entry on oeis.org
4004, 630036, 1559551, 4187814, 4870784, 6097906, 6834386, 9530359, 50755705, 51733715, 54988945, 62399326, 62488426, 63299236, 63477436, 64288246, 64377346, 71399317, 71488417, 73199137, 73466437, 74188147, 74366347, 81299218, 81477418, 82199128, 82466428, 84177148, 84266248
Offset: 1
Denote by F(1,P) the Fibonacci type sequence x(n+1) = x(n) + x(n-1) with x(0) = 1, x(1) = P.
Then for P = a(8) = 9530359, F(1,P) = (1, 9530359, 9530360, 19060719, 28591079, 47651798, 76242877, 123894675, ...) where a 9-digits anagram has occurred.
Cf.
A034587 (all starting values leading to 9-digit anagrams),
A034588 (subset of primes),
A034589 (subset of lucky numbers).
A034587
Fibonacci iteration starting with (1, a(n)) leads to a "nine digits anagram".
Original entry on oeis.org
718, 1790, 1993, 2061, 2259, 3888, 3960, 4004, 4396, 5093, 5832, 7031, 7310, 7712, 8039, 8955, 9236, 11598, 11742, 12312, 13295, 15095, 15432, 16044, 16355, 16472, 18109, 18559, 19144, 19950, 19968, 20116, 20180, 20494, 21170, 21376, 21998
Offset: 1
Denote by F(a,b) the Fibonacci-type sequence x(n+1) = x(n) + x(n-1) starting with x(0) = a, x(1) = b.
Then F(1,21998) = (1, 21998, 21999, 43997, 65996, 109993, 175989, 285982, 461971, 747953, 1209924, 1957877, 3167801, 5125678, 8293479, 13419157, 21712636, 35131793, 56844429, 91976222, 148820651, 240796873, 389617524, ...) where a nine-digits anagram has been reached.
The growth is roughly linear in three parts, with a slope of 700 up to a(292967) = 206993812, then an average slope of 1130 before it rises to (9.87e8 - 4.94e8)/2.05e5 ~ 2400 for 546211 <= n <= 750767 (cf. formula & comments): a(100) = 71960, a(200) = 149540, a(500) = 351868, a(1000) = 649921, a(2000) = 1400539, a(5000) = 3209798, a(10^4) = 6595301, a(2e4) = 13351498, a(5e4) = 32441506, a(10^5) = 67090523, a(2e5) = 134759627, a(3e5) = 214973567, a(4e5) = 327136594, a(5e5) = 439256717. - _M. F. Hasler_, Jan 07 2020
- M. F. Hasler, Table of n, a(n) for n = 1..10000, Jan 06 2020. (Full list of 750767 terms is available on request.)
- Patrick De Geest, Nine Digits Digressions
- M. F. Hasler, Graph of A034587, n = 1..750767 (full sequence), Jan 10 2020
- M. F. Hasler, Slope of A034587, averaged over n +- 10 sqrt(n), Jan 10 2020
-
A034587=select( {is_A034587(n,s=1,L=[1..9])=while( 123456789 > n=s+s=n,); n<1e9 && until( 987654321 < n=s+s=n, Set(digits(n))==L&&return(n))}, [1..22222]) \\ Function is_A034587 returns the 9-digit anagram if one is reached; null == false == 0 else.
nxt_A034587(n)={until(is_A034587(n+=1),);n} \\ Returns next larger term
A034587(n)={if(n>546210, A050289(n-387887)-1, #A034587>=n, A034587[n], A034587=concat( A034587, vector(n-#A034587,i, n=nxt_A034587(if(i>1,n,A034587[#A034587])))); n)} \\ Uses the two functions above. Could use Vecsmall(...) in definition of A034587 and vectorsmall in A034587(n) to reduce memory.
\\ M. F. Hasler, Jan 06 2020 and Jan 07 2020
-
def ok(n):
f, g = n, n+1
while g < 10**9:
if g > 123456788 and "".join(sorted(str(g))) == "123456789":
return True
f, g = g, f+g
return False
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Feb 18 2024
A034588
Primes p such that the Fibonacci iterations starting with (1, p) lead to a "nine digits anagram".
Original entry on oeis.org
1993, 8039, 22303, 30013, 31727, 46559, 50207, 63617, 65437, 72617, 83813, 92077, 101869, 102013, 109717, 131479, 136897, 141413, 145283, 156139, 162257, 163771, 204487, 206951, 207301, 209669, 211369, 221587, 221719, 225133, 225349, 233419
Offset: 1
Starting with (1, 233419), Fibonacci iterations x(n+1) = x(n) + x(n-1) yield the sequence (1, 233419, 233420, 466839, 700259, 1167098, 1867357, 3034455, 4901812, 7936267, 12838079, 20774346, 33612425, 54386771, 87999196, 142385967, ...) where a nine-digits anagram is reached.
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