A007908 -id:A007908 - cp's OEIS frontend

A181373 Least m>0 such that prime(n) divides S(m)=A007908(m)=123...m and all numbers obtained by cyclic permutations of its digits; 0 if no such m exists.

0, 2, 0, 100, 106, 120, 196, 102, 542, 400, 181, 21, 216, 372, 10446, 127, 10086, 616, 399, 1703, 196, 2009, 118, 12350, 516, 416, 13244, 884, 15462, 15146, 106, 1006942, 10762, 10814, 11634, 5808, 12408, 576, 30076, 4996, 25290, 1015092, 1108, 26874, 24036, 5994

Offset: 1

M. F. Hasler and Marco Ripà, Jan 27 2011

The first three primes 2, 3 and 5 are particular cases, cf. examples. It happens that all other primes < 47 are in A180346 (and therefore have a(n) < 1000). P=37 is the only one among them with a(n) < 100 (but m=123 is another possibility for this prime).
Conjecture: a(n) > 0 for n <> 1 and n <> 3. - Chai Wah Wu, Oct 06 2023
Least m>0 such that prime(n) divides both A007908(m) and 10^A058183(m)-1; or 0 if no such m exists. - Chai Wah Wu, Oct 07 2023

			For prime(1)=2, no such m can exist (consider e.g. the initial 1 is permuted to the end), therefore a(1)=0.
For prime(2)=3, we have S(2)=12 and the permutation 21 both divisible by 3, thus a(2)=2. (There are many m for which the divisibility property is satisfied; it is equivalent to 1+...+m=0 (mod 3), or equivalently the sum of all these digits is divisible by 3. Therefore, the permutations do not need to be checked.)
For prime(3)=5, similar to prime(1)=2, no such m can exist.
For prime(4)=7, it turns out the m=100 is the least possibility, i.e., 123...99100 and the permutations 234...991001, 345...9910012, ... 100123...99, (00)123...991, (0)123...9910 are all divisible by 7.

Chai Wah Wu, Table of n, a(n) for n = 1..4042

Cf. A000040, A007908, A058183, A180346 (see references there), A181373.

A181373(p,LIM=999,MIN=1)={ p=prime(p); p!=2 & p!=5 & for(n=MIN,LIM, my(S=eval(concat(vector(n,i,Str(i)))),L=#Str(S)-1); S%p & next; for(k=1,L, (S=[1,10^L]*divrem(S,10)) % p & next(2)); return(n)) } /* highly unoptimized code, for illustration purpose */

from sympy import prime
def A181373(n):
    s, p, l = '', prime(n), 0
    for m in range(1,10**6):
        u = str(m)
        s += u
        l += len(u)
        t = s
        if not int(t) % p:
            for i in range(l-1):
                t = t[1:]+t[0]
                if int(t) % p:
                    break
            else:
                return m
    else:
        return 'search limit reached.' # Chai Wah Wu, Nov 12 2015

from itertools import count
from sympy import prime
def A181373(n):
    if n == 1 or n == 3: return 0
    p, c, q, a, b = prime(n), 0, 1, 10, 10
    for m in count(1):
        if m >= b:
            a = 10*a%p
            b *= 10
        c = (c*a + m) % p
        q = q*a % p
        if not (c or (q-1)%p):
            return m # Chai Wah Wu, Oct 07 2023

A007908( A181373(n) ) = 0 (mod A000040(n)).

a(15)-a(31) from Chai Wah Wu, Nov 12 2015

a(32)-a(46) from Chai Wah Wu, Oct 06 2023

A078205 a(n) = A078204(n) / A007908(n).

Original entry on oeis.org

1, 18, 261, 3502, 44003, 530004, 6200005, 71000006, 800000008, 8900000074, 8999000074522, 98099900812365, 10628099988011209, 11446281093786559142, 122644629115519160114, 13082644736437172941406, 139008265613936288720365

Offset: 1

Amarnath Murthy, Nov 22 2002

			a(3) = 261 = 32103/123.

Cf. A007908, A078204.

More terms from Sascha Kurz, Jan 04 2003

A179069 Array read by antidiagonals: row b lists the base-b analog of the base-10 sequence 1, 12, 123, ..., 123456789, 12345678910, ... (A007908).

Original entry on oeis.org

1, 1, 3, 1, 6, 6, 1, 5, 27, 10, 1, 6, 48, 220, 15, 1, 7, 27, 436, 1765, 21, 1, 8, 38, 436, 3939, 14126, 28, 1, 9, 51, 194, 6981, 35367, 113015, 36, 1, 10, 66, 310, 4855, 111702, 318310, 1808248

Offset: 1

Jonathan Vos Post, Jun 27 2010

The numbers in the row b of the array are constructed in base b, but are converted to base 10 for display here.

R. K. Guy writes [UPINT, A3, pp. 9-10]: Selfridge asked if the sequence (in decimal notation) 1, 12, 123, 1234, ... [A007908] ... contains infinitely many primes.... The question can be asked for other scales of notation. There are (trivially) an infinite number of primes in the n=2 column, as that converges to k+2. In the n=3 column, the first prime is A[3,8] = 83 (base 10) = 123 (base 8). In the n=7 column, the first prime is A[8,7] = 342391 (base 10) = 1234567 (base 8). This can be continued to bases higher than 10, where A, B, C, ... are conventionally used as numerals. For example, A[12,5] = 12345 (base 12) = 24677 (base 10) is prime, as is A[12,17] = 656998737209054448298001 (base 10). A[13,3] = 227 (base 10) = 123 (base 13) is prime. Similarly, to pick the 9th row but go further than the table shown here, A[9,14] = 1709671414851143033 (base 10) is prime. Existing OEIS sequences stop at A048447, the concatenation of first n numbers in base 16.

			The array begins:
====================================================================
....|n=1.|.n=2.|.n=3.|.n=4.|..n=5.|..n=6.|...n=7.|.....n=8.|.in OEIS
b=1.|.1..|...3.|...6.|..10.|...15.|...21.|....28.|......36.|.A000217
b=2.|.1..|...6.|..27.|.220.|.1765.|.14126|.113015|.1808248.|.A047778
b=3.|.1..|...5.|..48.|.436.|.3929.|.35367|.318310|.2864798.|.A048435
b=4.|.1..|...6.|..27.|.436.|.6981.|111702|1787239|28595832.|.A048436
b=5.|.1..|...7.|..38.|.194.|.4855.|121381|3034532|75863308.|.A048437
b=6.|.1..|...8.|..51.|.310.|.1865.|.67146|2417263|87021476.|.A048438
b=7.|.1..|...9.|..66.|.466.|.3267.|.22875|1120882|54923226.|.A048439
b=8.|.1..|..10.|..83.|.668.|.5349.|.42798|.342391|21913032.|.A048440
...
b=10|.1..|..12.|.123.|1234.|12345.|123456|1234567|12345678.|.A007908
=====================================================================

Richard K. Guy, Unsolved Problems In Number Theory, 2nd Edn., Springer Verlag, 1994.

Cf. A000217, A007908, A033307, A047778, A048435, A048436, A048437, A048438, A048439, A048440, A048441, A048442, A048443, A048444, A048445, A048446, A048447.

A[b,n] = n-th integer concatenated from consecutive integers in base b.

Should be revised to start with base 2, rather than the ill-defined "base 1". - N. J. A. Sloane, Jul 05 2010

A180346 Primes that divide every circular permutation of the digits of at least one number of the form 123...(n-1)(n) (see A007908), where n is 3 digits long (that is, for some n in the range 99

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 61, 67, 73, 83, 97, 101, 107, 127, 163, 211, 271, 277, 1009, 18973

Offset: 1

Marco Ripà, Jan 22 2011

Every a(i) divides at least 192 permutations of the digits of an element belonging to [A007908]. Skipping the trivial case a(1)=3, the most recurring elements are a(2)=7 and a(10)=37. The occurrences in our 1386450 terms set are the following [A181373]:
a(2) | 7  ⇒  n=100+14*v       (v=0,1,2,...,64)
a(3) | 11 ⇒  n=106+22*v       (v=0,1,2,...,40)
a(4) | 13 ⇒  n=120+26*v       (v=0,1,2,...,33)
a(5) | 17 ⇒  n=196+272*v      (v=0,1,2)
a(6) | 19 ⇒  n=102+114*v      (v=0,1,2,3,4,5,6,7)
a(7) | 23 ⇒  n=542
a(8) | 29 ⇒  n=400
a(9) | 31 ⇒  n=181+155*v      (v=0,1,2,3,4,5)
a(10)| 37 ⇒  n=123+d(v),
(where d(v)=0,12,25,12,25,12,25...  for v=0,1,2,3,...,47)
a(11) | 41 ⇒  n=216+205*v     (v=0,1,2,3)
a(12) | 43 ⇒  n=372+301*v     (v=0,1,2)
a(13) | 53 ⇒  n=127+689*v     (v=0,1)
a(14) | 61 ⇒  n=616
a(15) | 67 ⇒  n=399
a(16) | 73 ⇒  n=196+584*v     (v=0,1)
a(17) | 83 ⇒  n=118
a(18) | 97 ⇒  n=516
a(19) | 101 ⇒  n=416+404*v    (v=0,1)
a(20) | 107 ⇒  n=884
a(21) | 127 ⇒  n=106
a(22) | 163 ⇒  n=576
a(23) | 211 ⇒  n=306
a(24) | 271 ⇒  n=936
a(25) | 277 ⇒  n=174
a(26) | 1009 ⇒ n=960
a(27) | 18973 ⇒ n=903
N.B.
Every coefficient of "v" is a multiple of i. This is a general property of [A007908], valid for an arbitrary fixed digits interval of the parameter "n" (10^k-1a(28) >= prime(10^6) if it exists. - Chai Wah Wu, Nov 12 2015
Primes p such that p divides both A007908(m) and 10^A058183(m)-1 for some 99Chai Wah Wu, Oct 07 2023
a(28) > prime(2.3316*10^9) if it exists. Conjecture: 18973 is the last term. - Chai Wah Wu, Oct 09 2023

Vassilev-Missana and K. Atanassov, “Some Smarandache problems”, Hexis, 2004.

Marco Ripà, Strutture modulari associate al problema della primalità di alcune sequenze concatenate, Rudimatematici, Bookshelf, October 2010. In Italian.
Marco Ripà, On prime factors in old and new sequences of integers, vixra, 2011.
Marco Ripa, Patterns related to the Smarandache circular sequence primality problem, Notes Numb. Th. Discr. Math., vol. 18(1) (2012), pp. 29-48.
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.

Cf. A007908, A058183, A181373.

isA180346(p)={ isprime(p) & p!=2 & p!=5 & for(n=100,999, my(S=eval(concat(vector(n,i,Str(i)))),L=#Str(S)-1); S%p & next; for(k=1,L, (S=[1,10^L]*divrem(S,10))%p & next(2));return(n)) }  /* returns the least corresponding n or 0 if not in this sequence */ \\ M. F. Hasler, Jan 23 2011

from itertools import islice
from sympy import nextprime
def A180346_gen(startvalue=1): # generator of terms >= startvalue
    p = max(startvalue-1,0)
    while (p:=nextprime(p)):
        c, q, a, b = 0, 1, 10, 10
        for m in range(1,1000):
            if m >= b:
                a = 10*a%p
                b *= 10
            c = (c*a + m) % p
            q = q*a % p
            if m>99 and not (c or (q-1)%p):
                yield p
                break
A180346_list = list(islice(A180346_gen(),20)) # Chai Wah Wu, Oct 07 2023

For n<10 the only a(i) is 3. If 9

A276200 Largest prime < the concatenation of the numbers from 1 to n (A007908).

Original entry on oeis.org

11, 113, 1231, 12343, 123449, 1234547, 12345653, 123456761, 12345678899, 1234567891003, 123456789101099, 12345678910111207, 1234567891011121309, 123456789101112131383, 12345678910111213141337, 1234567891011121314151561, 123456789101112131415161717, 12345678910111213141516171723

Offset: 2

Ilya Gutkovskiy, Aug 24 2016

			a(5) = 12343, because this is the largest prime less than 12345.

Robert Israel, Table of n, a(n) for n = 2..368
Eric Weisstein's World of Mathematics, Previous Prime

Cf. A000040, A000720, A007908, A074365, A151799.

tcat:= (a,b) -> a*10^(1+ilog10(b))+b:
t:= 1: R:= NULL:
for i from 2 to 20 do
  t:= tcat(t,i);
  R:= R,prevprime(t);
od:
R; # Robert Israel, Oct 29 2024

Table[NextPrime[FromDigits[Flatten[IntegerDigits[Range[n]]]], -1], {n, 2, 19}]

a(n) = A151799(A007908(n)).

a(n) = A000040(A000720(A007908(n)-1)).

A293577 Decimal expansion of number with continued fraction expansion 0, 1, 12, 123, 1234, 12345, 123456, ... (A007908).

Original entry on oeis.org

9, 2, 3, 1, 2, 4, 9, 9, 9, 6, 8, 3, 4, 5, 0, 2, 4, 1, 1, 7, 4, 0, 1, 2, 3, 3, 0, 6, 0, 9, 8, 4, 2, 1, 9, 1, 6, 6, 3, 6, 7, 4, 8, 8, 6, 2, 9, 1, 6, 9, 0, 3, 9, 8, 9, 4, 1, 4, 7, 4, 4, 4, 1, 1, 1, 3, 5, 6, 7, 3, 9, 1, 1, 6, 5, 1, 4, 7, 4, 5, 2, 7, 3, 5, 4, 1, 2, 5, 4, 0, 4, 3, 2, 5, 1, 5, 0, 1, 9, 1

Offset: 0

Ilya Gutkovskiy, Oct 12 2017

			0.92312499968345024117401233060984219166367488... = 1/(1 + 1/(12 + 1/(123 + 1/(1234 + 1/(12345 + 1/(123456 + 1/...)))))).

G. C. Greubel, Table of n, a(n) for n = 0..5000 [a(5000) corrected by _Georg Fischer_, Apr 02 2025]

Cf. A007908, A030167, A052119, A264934.

Take[RealDigits[N[FromContinuedFraction[Table[FromDigits[Flatten[IntegerDigits[Range[n]]]], {n, 0, 20}]], 101]][[1]], 100]  (* modified by Ilya Gutkovskiy, Nov 08 2017 *)

a(99) corrected by G. C. Greubel, Nov 07 2017

A362966 Numbers k such that A007908(k) == 1 (mod k).

Original entry on oeis.org

1, 121487, 293957, 13449179, 549999887

Offset: 1

Max Alekseyev, Jun 06 2023

a(6) > 10^11. - Jason Yuen, Oct 12 2024

Cf. A007908, A029455, A029479, A110740.

# See A029455 for concat_mod
def isok(k): return concat_mod(10, k, k)==1%k # Jason Yuen, Oct 12 2024

A366393 a(n) is the largest number that can be obtained by deleting n digits from A007908(n).

Original entry on oeis.org

9, 91, 912, 9213, 92314, 931415, 9341516, 94151617, 945161718, 9516171819, 95617181920, 961718192021, 9671819202122, 97181920212223, 978192021222324, 9819202122232425, 98920212223242526, 992021222324252627, 9922122232425262728, 99222223242526272829, 992222324252627282930

Offset: 10

Stefano Spezia, Oct 08 2023

Inspired by the Example 1.1 at pp. 1-2 in Andreescu and Feng (see References).

Titu Andreescu and Zuming Feng, A Path to Combinatorics for Undergraduates: Counting Strategies, Birkhäuser, Boston, 2004.

Cf. A007908, A011557, A055642, A366394 (smallest).

a(n) mod 10^A055642(n) = n, for n > 11.

A366394 a(n) is the smallest number that can be obtained by deleting n digits from A007908(n).

Original entry on oeis.org

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111110, 11111111021, 111111102122, 1111110212223, 11111021222324, 111102122232425, 1110212223242526, 11021222324252627, 102122232425262728, 212223242526272829, 1222324252627282930, 12232425262728293031

Offset: 10

Stefano Spezia, Oct 08 2023

The leading zeros are omitted.

Cf. A007908, A366393 (largest).

A019524 Duplicate terms of A007908.

Original entry on oeis.org

11, 1212, 123123, 12341234, 1234512345, 123456123456, 12345671234567, 1234567812345678, 123456789123456789, 1234567891012345678910, 12345678910111234567891011, 123456789101112123456789101112

Offset: 1

R. Muller

nonn

F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sciences, Vol. 16E, No. 2 (1997), pp. 237-240.

Eric Weisstein's World of Mathematics, Smarandache Sequences
F. Smarandache, Collected Papers, Vol. II
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

a(n) = A007908(n)*(1 + 10^A058183(n)) = (n + a(n - 1)*10^L(n)/(1 + 10^(n*L(n - 1) - (10^L(n - 1) - 1)/9)))*(1 + 10^((n + 1)*L(n) - (10^L(n) - 1)/9)) where L(n) = floor(log_10(10n)). - Henry Bottomley, Nov 17 2000 (may need to be adapted for change in offset)

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cp's OEIS Frontend

A181373 Least m>0 such that prime(n) divides S(m)=A007908(m)=123...m and all numbers obtained by cyclic permutations of its digits; 0 if no such m exists.

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A078205 a(n) = A078204(n) / A007908(n).

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A179069 Array read by antidiagonals: row b lists the base-b analog of the base-10 sequence 1, 12, 123, ..., 123456789, 12345678910, ... (A007908).

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A180346 Primes that divide every circular permutation of the digits of at least one number of the form 123...(n-1)(n) (see A007908), where n is 3 digits long (that is, for some n in the range 99

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A276200 Largest prime < the concatenation of the numbers from 1 to n (A007908).

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A293577 Decimal expansion of number with continued fraction expansion 0, 1, 12, 123, 1234, 12345, 123456, ... (A007908).

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A362966 Numbers k such that A007908(k) == 1 (mod k).

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A366393 a(n) is the largest number that can be obtained by deleting n digits from A007908(n).

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