A034306 -id:A034306 - cp's OEIS frontend

A034587 Fibonacci iteration starting with (1, a(n)) leads to a "nine digits anagram".

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

Patrick De Geest, Oct 15 1998

By "nine digits anagram" the author means a number whose digits are a permutation of {1, ..., 9}. These are more commonly known as restricted zeroless pandigital numbers and form the first 9! terms of A050289.
The largest term is a(750767) = 987654320.
More generally, the last N = 9! - 158323 = 204557 (> 56% of 9!) terms are given as A050289(k)-1 with indices k = 9!-N+1, ..., 9!. Indeed, a number > (987654321-1)/2 = 493827160 is a term if and only if it equals a "9-digit anagram" minus 1, since all results beyond the first iteration (1 + n = n+1) will be too large. Since 493827165 = A050289(158324) > 493827160, starting with a(546211) = 493827164 the terms are given by A050289(158324 .. 9!) - 1, for a total of 546211 + N - 1 = 750767 terms. (The term 493827164 is preceded by 493827160 (which yields 987654321 but is not in A050289 - 1) and 493827155 = A050289(158323) - 1.) - M. F. Hasler, Jan 07 2020
The ratio between consecutive terms in a Fibonacci sequence x(n+1) = x(n) + x(n-1) tends quickly to the golden ratio Phi = (sqrt(5)+1)/2 = A001622. We can tell whether a starting value N is in this sequence or not from the terms between 123456789 and 987654321 ~ 1e9. From N*Phi^k = 1e9 we get k = log(1e9/N)/log(Phi) ~ 43 - 2*log(N) for the maximum (and 3 less for the minimum) number of required iterations. - M. F. Hasler, Jan 06 2020

			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

Subsequences: A034588 (primes), A034589 (lucky numbers), A034306 (palindromes).

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

a(n) = A050289(m) with n = 387887 + m for 158324 <= m <= 9! or 546211 <= n <= 750767 = total number of terms in this sequence. - M. F. Hasler, Jan 07 2020

Edited and offset changed to 1 by M. F. Hasler, Jan 06 2020

Results confirmed by Giovanni Resta, Jan 07 2020

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

Patrick De Geest, Oct 15 1998

A "nine digits anagram" is a number whose digits are a permutation of {1, ..., 9}, or one of the first 9! terms of A050289.
Largest term is a(46494) = 987653411.
Subset of primes in A034587. There are 767 (resp. 2982, resp. 6045) primes among the first 10^4 (resp. 5*10^4, resp. 10^5) terms of A034587, and (0, 1, 14, 129, 1566) terms among the first (100, 10^3, 10^4, 10^5, 10^6) primes, the last of which is 15480869 = prime(999708).  - M. F. Hasler, Jan 06 2020
The terms larger than 987654320/2 = 493827160 are primes of the form A050289(k)-1 with 158324 <= k <= 9!, cf. A034587. There are exactly 13005 of these which are therefore the last 13005 terms of this sequence, starting with 493851671 = A050289(158332)-1 = prime(26048750). - M. F. Hasler, Jan 09 2020
The graph of this sequence has a distinct slope for values below, between, and above the two limits of 2.07e8 and 4.94e8, as for the graph of A034587 (cf. link). - M. F. Hasler, Jan 11 2020

			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.

M. F. Hasler, Table of n, a(n) for n = 1..46494 (full sequence), Jan 09 2020
Patrick De Geest, Nine Digits Digressions
M. F. Hasler, Graph of A034587, n = 1..750767 (full sequence), Jan 10 2020

Cf. A034587 (full sequence), A034589 (lucky numbers), A034306 (palindromes).

select( is_A034587, primes(22222)) \\ or, if a vector A034587 is available:
select(isprime, A034587) \\ e.g., using b034587.txt. - M. F. Hasler, Jan 06 2020

Intersection of A000040 and A034587.

Edited and offset changed to 1 by M. F. Hasler, Jan 06 2020

A034589 Lucky numbers N (A000959) such that Fibonacci iterations starting with (1, N) lead to a "nine digits anagram".

Original entry on oeis.org

8955, 24405, 30013, 59325, 62025, 71493, 72123, 76885, 85461, 92077, 99165, 106185, 109717, 112251, 119077, 148773, 153007, 155077, 163771, 163803, 196797, 211369, 221137, 223365, 227119, 228271, 228631

Offset: 1

Patrick De Geest, Oct 15 1998

"Nine digits anagram" is a number whose digits are a permutation of {1, ..., 9}, also called restricted zeroless pandigital number. These are listed as the first 9! terms of A050289. - M. F. Hasler, Jan 10 2020

			Denote by F(1, N) the Fibonacci sequence x(k+1) = x(k) + x(k-1) starting with x(0) = 1, x(1) = N.
Then for N = 228631, F(1, N) = (1, 228631, 228632, 457263, 685895, 1143158, 1829053, 2972211, 4801264, 7773475, 12574739, 20348214, 32922953, 53271167, 86194120, 139465287, ...), where a nine-digits anagram has been reached.

M. F. Hasler, Table of n, a(n) for n = 1..275 (using A000959(1..2e5) from b-file), Jan 10 2020
Patrick De Geest, Nine Digits Digressions
Eric Weisstein's World of Mathematics, Lucky Numbers

Cf. A000959 (lucky numbers), A050289 (zeroless pandigital numbers).

Cf. A034587 (all starting values leading to 9-digit anagrams), A034588 (subset of primes), A034306 (subset of palindromes).

A034589_upto(N)=[n|n<-A000959_upto(N),is_A034587(n)] \\ or select(is_A034587, A959=from("b000959.txt")). - M. F. Hasler, Jan 10 2020

Intersection of A000959 (lucky numbers) and A034587. - M. F. Hasler, Jan 10 2020

Name, example & crossrefs edited, offset changed to 1 by M. F. Hasler, Jan 06 2020

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A034587 Fibonacci iteration starting with (1, a(n)) leads to a "nine digits anagram".

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A034588 Primes p such that the Fibonacci iterations starting with (1, p) lead to a "nine digits anagram".

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A034589 Lucky numbers N (A000959) such that Fibonacci iterations starting with (1, N) lead to a "nine digits anagram".

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions