A023183 -id:A023183 - cp's OEIS frontend

A020344 Fibonacci(a(n)) is the least Fibonacci number beginning with n.

0, 1, 3, 4, 19, 5, 15, 25, 6, 16, 21, 45, 26, 7, 12, 17, 41, 22, 46, 27, 51, 8, 56, 13, 37, 18, 42, 66, 23, 47, 71, 28, 52, 119, 9, 33, 57, 14, 148, 38, 62, 19, 86, 43, 67, 134, 24, 225, 48, 72, 139, 29, 230, 53, 254, 10, 278, 34, 302, 58, 259, 15, 283, 39, 240, 63, 197, 20, 154, 288

Offset: 0

David W. Wilson

Fixed points of this sequence are in A038546. - Alois P. Heinz, Jul 08 2022

T. D. Noe, Table of n, a(n) for n = 0..10000
Ron Knott, Every number starts some Fibonacci Number, The Mathematical Magic of the Fibonacci Numbers.

Cf. A000045, A020345, A023183, A023184, A038546.

nn = 100; t = tn = Table[0, {nn}]; found = 0; n = 0; While[found < nn, n++; f = Fibonacci[n]; d = IntegerDigits[f]; i = 1; While[i <= Length[d], k = FromDigits[Take[d, i]]; If[k > nn, Break[]]; If[t[[k]] == 0, t[[k]] = f; tn[[k]] = n; found++]; i++]]; tn = Join[{0}, tn] (* T. D. Noe, Apr 02 2014 *)

def aupton(nn):
    ans, f, g, k = dict(), 0, 1, 0
    while len(ans) < nn+1:
        sf = str(f)
        for i in range(1, len(sf)+1):
            if int(sf[:i]) > nn:
                break
            if sf[:i] not in ans:
                ans[sf[:i]] = k
        f, g, k = g, f+g, k+1
    return [int(ans[str(i)]) for i in range(nn+1)]
print(aupton(70)) # Michael S. Branicky, Jul 08 2022

A000045(a(n)) = A020345(n).

A020345 Smallest Fibonacci number beginning with n.

Original entry on oeis.org

0, 1, 2, 3, 4181, 5, 610, 75025, 8, 987, 10946, 1134903170, 121393, 13, 144, 1597, 165580141, 17711, 1836311903, 196418, 20365011074, 21, 225851433717, 233, 24157817, 2584, 267914296, 27777890035288, 28657, 2971215073, 308061521170129, 317811

Offset: 0

David W. Wilson

The graph of the indices A020344 is much more interesting. - T. D. Noe, Apr 02 2014

a(1382) is the first term with > 1000 digits (1004). - Michael S. Branicky, Jul 08 2022

			a(4) = 4181 is a Fibonacci number starting with 4.

Michael S. Branicky, Table of n, a(n) for n = 0..1389 (terms 1..400 from T. D. Noe)
Ron Knott, Every number starts some Fibonacci Number, The Mathematical Magic of the Fibonacci Numbers.

Cf. A000045, A020344, A023183, A023184.

nn = 31; t = tn = Table[0, {nn}]; found = 0; n = 0; While[found < nn, n++;  f = Fibonacci[n]; d = IntegerDigits[f]; i = 1; While[i <= Length[d], k = FromDigits[Take[d, i]]; If[k > nn, Break[]]; If[t[[k]] == 0, t[[k]] = f; tn[[k]] = n; found++]; i++]]; t = Join[{0}, t] (* T. D. Noe, Apr 02 2014 *)

def aupton(nn):
    ans, f, g, k = dict(), 0, 1, 0
    while len(ans) < nn+1:
        sf = str(f)
        for i in range(1, len(sf)+1):
            if int(sf[:i]) > nn:
                break
            if sf[:i] not in ans:
                ans[sf[:i]] = f
        f, g, k = g, f+g, k+1
    return [int(ans[str(i)]) for i in range(nn+1)]
print(aupton(31)) # Michael S. Branicky, Jul 08 2022

a(n) = A000045(A020344(n)).

A079185 Number of isomorphism classes of commutative closed binary operations (groupoids) on a set of order n, listed by class size.

Original entry on oeis.org

1, 0, 4, 1, 4, 8, 116, 0, 0, 0, 8, 0, 28, 504, 43428

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

A079184(n)+A079185(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 1; 0,4; 1,4,8,116; 0,0,0,8,0,28,504,43428
A079176(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row n is given by A079177(n).

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A001425, A023183, A079184. a(n, A027423(n)) = A030255(n).

A374026 a(n) = the smallest k such that Fibonacci(k) begins and ends with n, where Fibonacci(k) > n, or -1 if there are none.

Original entry on oeis.org

22, 114, 124, 72, 10, 39, 116, 207, 169, 5715, 2428, 5634, 2366, 189, 2620, 4668, 3137, 2673, 5812, 12090, 721, 11583, 20086, 3798, 1975, 999, 10636, 846, 2071, 9105, 1162, 2076, 8287, 11091, 2770, 2928, 12943, 8493, 172, 2220, 5359, 4737, 28126, 11838, 10460, 7479, 10996

Offset: 1

Gonzalo Martínez, Jun 26 2024

Since the smallest number beginning and ending with n is the same n, the condition that Fibonacci(k) > n is imposed. Partial overlap of the start and end is allowed, but not full overlap.

			As Fibonacci(22) = 17711 is the smallest Fibonacci number greater than 1 that begins and ends with 1, a(1) = 22.
As Fibonacci(10) = 55 is the smallest Fibonacci number greater than 5 that begins and ends with 5, a(5) = 10.

Cf. A000045, A272623, A023183, A020344.

isok(s,t) = my(ss=strsplit(s, t)); (#ss >= 3) && (ss[1]=="") && (ss[#ss]=="");
a(n) = my(k=7); while(!isok(Str(fibonacci(k)), Str(n)), k++); k; \\ Michel Marcus, Jun 26 2024

# uses A023183() in A023183
from itertools import count
def a(n):
    if A023183(n) == -1:
        return -1
    f, g, s = 1, 2, str(n)
    pow10 = 10**len(s)
    for k in count(3):
        if g%pow10 == n:
            sfib = str(g)
            if g > n and sfib.startswith(s):
                return k
        f, g = g, f+g
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jul 03 2024

a(n) >= max{A023183(n), A020344(n)} except that a(n) = -1 when A023183(n) = -1. - Michael S. Branicky, Jun 27 2024

More terms from Michel Marcus, Jun 26 2024

A245456 Index of least Fibonacci number congruent to n mod 100.

Original entry on oeis.org

0, 1, 3, 4, 192, 5, 81, 76, 6, 32, 15, 22, 216, 7, 111, 130, 168, 37, 27, 112, 60, 8, 117, 64, 198, 25, 99, 136, 204, 29, 105, 88, 174, 13, 9, 70, 222, 43, 93, 172, 30, 41, 63, 124, 12, 55, 21, 154, 186, 49, 75, 148, 36, 67, 129, 10, 162, 23, 87, 118, 180, 61, 57, 166, 72, 20, 141, 94, 24, 31, 45, 142, 54, 47, 51, 100, 48, 14, 33, 58, 210, 19, 123, 106, 18, 80, 39, 16, 66, 11, 135, 82, 156, 26, 69, 160, 42, 17, 147, 52

Offset: 0

Charles R Greathouse IV, Jul 22 2014

The analogous sequence mod 1000 does not exist, because no Fibonacci number is congruent to, e.g., 4 mod 1000.

			a(4) = 192 because F(192) = 5972304273877744135569338397692020533504 is 4 mod 100.

Cf. A023183.

Join[{0},With[{fibs=Fibonacci[Range[230]]},Flatten[Table[Position[fibs,?(Mod[#,100]==n&),{1},1],{n,100}]]]] (* _Harvey P. Dale, Mar 31 2015 *)

a(n)=for(k=0,222,if(fibonacci(k)%100==n,return(k)))

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A020344 Fibonacci(a(n)) is the least Fibonacci number beginning with n.

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A020345 Smallest Fibonacci number beginning with n.

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A079185 Number of isomorphism classes of commutative closed binary operations (groupoids) on a set of order n, listed by class size.

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A374026 a(n) = the smallest k such that Fibonacci(k) begins and ends with n, where Fibonacci(k) > n, or -1 if there are none.

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A245456 Index of least Fibonacci number congruent to n mod 100.

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PARI