A332580 -id:A332580 - cp's OEIS frontend

A332830 a(n) = minimal positive k such that the concatenation of decimal digits n and n+1 is a divisor of the concatenation of n+2, n+2+1, ..., n+2+k.

3, 4, 3, 24, 13, 7, 33, 7, 749, 125, 1019, 3643, 123, 1319, 1199, 1424, 1481, 664, 659, 734, 6139, 933, 607, 549, 165, 8124, 63, 296, 1339, 13817, 1691, 6979, 3, 704, 2187, 156, 987, 2521, 1459, 1277, 6047, 25565, 3179, 1954, 7127, 1115, 6139, 18749, 1149

Offset: 1

Scott R. Shannon, Feb 25 2020

Like A332580 a heuristic argument, based on the divergent sum of reciprocals which approximates the probability that the concatenation of n and n+1 will divide the concatenation of n+2, n+3, ..., suggests that k should always exist.

			a(1) = 3 as '1'||'2' = 12 and '3'||'4'||'5'||'6' = 3456, which is divisible by 12 (where '||' denotes decimal concatenation).
a(4) = 24 as '4'||'5' = 45 and '6'||'7'||....||'29'||'30' = 6789101112131415161718192021222324252627282930, which is divisible by 45.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A332580, A332867, A007908.

a:= proc(n) local i, t, m; t, m:= parse(cat(n,n+1)), 0;
      for i from n+2 do m:= parse(cat(m,i)) mod t;
      if m=0 then break fi od; i-n-2
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 29 2020

a(n) = {my(k=1, small=eval(concat(Str(n), Str(n+1))), big=n+2); while( big % small, big = eval(concat(Str(big), Str(n+2+k))); k++); k--;} \\ Michel Marcus, Feb 29 2020

A332867 a(n) = minimal positive k such that the concatenation of decimal digits 1,2,...,n is a divisor of the concatenation of n+1,n+2,...,n+k.

Original entry on oeis.org

1, 4, 20, 1834, 14995, 6986, 2888370, 795412, 37366487, 8803255100

Offset: 1

Scott R. Shannon, Feb 27 2020

As with A332580 a heuristic argument, based on the divergent sum of reciprocals which approximates the probability that the concatenation of 1,2,...,n will divide the concatenation of n+1,n+2,...,n+k suggests that k should always exist. However an examination of the prime factors of the concatenation of 1,2,...,n shows that most of these numbers contain one or more very large primes, suggesting the values of k will likely become extremely large as n increases.

The author thanks Joseph Myers for suggestions for finding the larger terms of this sequence.

			a(2) = 4 as '1'||'2' = 12 and '3'||'4'||'5'||'6' = 3456, which is divisible by 12 (where '||' denotes decimal concatenation).
a(3) = 20 as '1'||'2'||'3' = 123 and '4'||'5'||...||'22'||'23' = 4567891011121314151617181920212223, which is divisible by 123.

J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A332580, A332830, A007908.

a:= proc(n) local i, t, m; t, m:= parse(cat($1..n)), 0;
      for i from n+1 do m:= parse(cat(m,i)) mod t;
      if m=0 then break fi od; i-n
    end:
seq(a(n), n=1..6);  # Alois P. Heinz, Feb 29 2020

a(n) = {my(k=1, small="", big = n+1); for (j=1, n, small=concat(small, Str(j))); small = eval(small); while (big % small, k++; big = eval(concat(Str(big), Str(n+k)))); k;} \\ Michel Marcus, Feb 29 2020

def A332867(n):
    m, k = int(''.join(str(d) for d in range(1,n+1))), 1
    i = n+k
    i2, l = i % m, len(str(i))
    t = 10**l
    t2, r = t % m, i % m
    while r != 0:
        k += 1
        i += 1
        i2 = (i2+1) % m
        if i >= t:
            l += 1
            t *= 10
            t2 = (10*t2) % m
        r = (r*t2 + i2) % m
    return k # Chai Wah Wu, May 20 2020

A360376 a(n) = minimal nonnegative k such that prime(n) * prime(n+1) * ... * prime(n+k) + 1 is divisible by prime(n+k+1), or -1 if no such k exists.

Original entry on oeis.org

0, 99, 14, 1, 2, 73, 33, 10, 137, 277856, 1

Offset: 1

Scott R. Shannon, Feb 04 2023

Assuming a(12) exists it is greater than 2.25 million.

			a(1) = 0 as prime(1) + 1 = 2 + 1 = 3, which is divisible by prime(2) = 3.
a(3) = 14 as prime(3) * ... * prime(17) + 1 = 320460058359035439846, which is divisible by prime(18) = 61.
a(10) = 277856 as prime(10) * ... * prime(277866) + 1 = 645399...451368 (a number with 1701172 digits), which is divisible by prime(277867) = 3919259.
a(11) = 1 as prime(11) * prime(12) + 1 = 31 * 37 + 1 = 1148, which is divisible by prime(13) = 41.

Cf. A360406, A360297, A000040, A007504, A332542, A332580.

from sympy import prime, nextprime
def A360376(n):
    p = prime(n)
    s, k = p, 0
    while (s+1)%(p:=nextprime(p)):
        k += 1
        s *= p
    return k # Chai Wah Wu, Feb 07 2023

A332585 Number of digits in the number formed by concatenating the digits of n, n+1, ..., A332584(n), or -1 if A332584(n) = -1.

Original entry on oeis.org

2, 154, 6443, 26258, 2, 86, 25, 4, 4165, 38, 505, 42, 108, 319, 2906, 90, 445, 636086, 711, 54, 245, 22, 12, 126, 32, 154843, 20, 30, 883, 2057, 4970, 577, 76, 70, 139, 749, 40, 89959, 380407, 42715, 805, 8548, 2031

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 17 2020

a(44) is currently unknown.

			For n=2, A332584(2) = 88, and the concatenation 2 || 3 || ... || 82 is
  23456789101112131415161718192021222324252627282930313233343536373839\
  40414243444546474849505152535455565758596061626364656667686970717273747\
  576777879808182, which has 154 digits. So a(2) = 154.

J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A332580, A332584.

Let f(i) = A058183(i). Assuming A332584(n)>0, a(n) = f(A332584(n))-f(n-1) for n>1. - N. J. A. Sloane, Feb 20 2020

A332586 a(n) = minimal value of n+k+1 such that the concatenation of the binary expansions of n,n+1,...,n+k is divisible by n+k+1, or -1 if no such n+k+1 exists.

Original entry on oeis.org

3, 9, 257, 165, 29, 13, 585, 23, 11, 15, 395, 21, 1605, 33, 185, 59, 1897, 229, 77, 41, 91, 1377, 37, 111, 251, 1559, 605, 329, 43, 61, 6451, 345, 30673, 47, 187, 45, 127, 2759, 69, 5871, 43, 1493, 239, 523, 101, 166575, 175, 1123, 3609, 303, 93, 1139465, 4495201

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 25 2020

For n up to 128 the presently unknown values are a(52) and a(53). If these values of k exist, they are at least 1000000.

Michael S. Branicky, Table of n, a(n) for n = 1..137
Michael S. Branicky, Table of n, a(n) for n = 1..332, with -1 if k is presently unknown (the current search limit is 2000000). Note that this does not mean that a(n) = -1.
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
Scott R. Shannon, Table of n, a(n) for n = 1..128, with -1 if k is presently unknown (the current search limit is 1000000). Note that this does not mean that a(n) = -1.
Scott R. Shannon, The quotient after the final division, for n = 1..15

Cf. A332580, A332584, A332563.

Table[k=0;While[Mod[FromDigits[Flatten@IntegerDigits[Range[n,n+ ++k],2],2],n+k+1]!=0];n+k+1,{n,20}] (* Giorgos Kalogeropoulos, Apr 27 2021 *)

a(52) from Michael S. Branicky, Apr 25 2021

a(53) from Michael S. Branicky, Apr 28 2021

A360311 The sum of the primes prime(n) + prime(n+1) + ... + prime(n+k) in A360297.

Original entry on oeis.org

5, 26, 124, 318, 1703, 1133, 2086, 7641, 10912775, 60927, 8764, 184252585101, 144329, 474, 1090

Offset: 1

Scott R. Shannon, Feb 03 2023

See A360297 for further details.

Cf. A360297, A360312, A000040, A007504, A332542, A332580.

from sympy import prime, nextprime
def A360311(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p+q, 1
    while s%(q:=nextprime(q)):
        k += 1
        s += q
    return s # Chai Wah Wu, Feb 06 2023

A360312 The dividing prime prime(n+k+1) in A360297.

Original entry on oeis.org

5, 13, 31, 53, 131, 103, 149, 283, 14081, 883, 313, 2281229, 1429, 79, 109

Offset: 1

Scott R. Shannon, Feb 03 2023

See A360297 for further details.

Cf. A360297, A360311, A000040, A007504, A332542, A332580.

from sympy import prime, nextprime
def A360312(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p+q, 1
    while s%(q:=nextprime(q)):
        k += 1
        s += q
    return q # Chai Wah Wu, Feb 06 2023

A332562 a(n) = number formed by concatenating the decimal digits of 44, 45, 46, ..., 44+n.

Original entry on oeis.org

44, 4445, 444546, 44454647, 4445464748, 444546474849, 44454647484950, 4445464748495051, 444546474849505152, 44454647484950515253, 4445464748495051525354, 444546474849505152535455, 44454647484950515253545556, 4445464748495051525354555657

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

This is an instance of a sequence arising in A332580.
As of Feb 24 2020, it is an open question as to whether there is an N such that a(N) is divisible by 44+N+1. If such an N exists, N > 10^11, as shown by Joseph Myers (see A332580).
We have now shown that N = 2783191412912. See A332580 and the attached paper. - N. J. A. Sloane, Apr 28 2020

Robert Israel, Table of n, a(n) for n = 0..350
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.

Cf. A332580.

[Seqint(Reverse(&cat[Reverse(Intseq(k)): k in [44..n]])): n in [44..60]]; // Vincenzo Librandi, Feb 26 2020

a:= n-> parse(cat($44..44+n)):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 24 2020

Nest[Append[#, 10^IntegerLength[#2]*#1[[-1]] + #2 ] & @@ {#, 44 + Length@ #} &, {44}, 13] (* Michael De Vlieger, Feb 24 2020 *)
Table[FromDigits[Flatten[IntegerDigits/@Range[44,44+n]]],{n,0,20}] (* Harvey P. Dale, Jul 19 2024 *)

A333687 a(n) is the minimal value of k >= 0, such that the concatenation of the decimal digits of n,n+1,...,n+k is divisible by the digit sum of the concatenation, or -1 if no such k is known.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 2, 42, 4, 3, 0, 1, 0, 0, 1, 17, 0, 131, 26, 0, 16, 11, 0, 1, 2, 37, 1, 1, 0, 1, 2, 21, 0, 3, 0, 7, 8, 0, 6, 83, 0, 1, 0, 89, 8, 26, 0, 97, 142783940, 3, 1, 1, 0, 4, 8, 0, 14, 37, 49994, 380, 20, 17, 0, 65, 0, 62, 1, 3, -1, 29, 46, 235, 0, 0, 18, 29, 0, 1, 53

Offset: 1

Scott R. Shannon, Apr 02 2020

As with A332580 a heuristic argument based on the divergent sum of reciprocals which approximates the probability that the digit sum of the concatenation of n+1,n+2,...,n+k will divide the concatenation suggests that k should always exist. However in the first one thousand terms there are currently fourteen terms which are unknown and have a k value of at least 10^9. These are n = 76, 250, 273, 546, 585, 663, 695, 744, 749, 760, 790, 866, 867, 983. The largest known k value in this range is k = 600747353 for n = 693, which has a corresponding digit sum of 23123615211.
See the companion sequence A333830 for the corresponding digit sum for each value of n.
The author acknowledges Joseph Myers whose algorithm to find terms in A332580 was modified and used to find the large k values in this sequence.

			a(1) = 0 as 1 is divisible by its digit sum 1 so no concatenation of additional numbers is required. This is also true for n = 2 to 10.
a(11) = 2 as 11 requires the concatenation of two more numbers, 12 and 13, to form 111213, which is divisible by its digit sum 9.
a(12) = 0 as 12 is divisible by its digit sum 3.
a(16) = 4 as 16 requires the concatenation of four more numbers, 17,18,19 and 20, to form 1617181920, which is divisible by its digit sum 36.

Scott R. Shannon, Table of n, a(n) for n = 1..1000

Cf. A333830, A007953, A332580, A332558, A332563, A332542, A332830, A332867, A005349.

A333830 a(n) is the digit sum of the concatenation of the decimal digits of n,n+1,...,n+k, where k >= 0 and minimal, such that the concatenation is divisible by its digit sum, or -1 if no such sum is known.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 9, 3, 9, 18, 333, 36, 29, 9, 12, 2, 3, 9, 135, 6, 1218, 216, 9, 126, 90, 3, 9, 18, 355, 15, 17, 9, 21, 27, 198, 4, 26, 6, 75, 81, 9, 64, 810, 12, 18, 5, 855, 90, 297, 9, 936, 5050737477, 45, 27, 20, 6, 45, 99, 9, 174, 446, 1000260, 4209

Offset: 1

Scott R. Shannon, Apr 07 2020

A heuristic argument, see the companion sequence A333687, suggests that the digit sum should always exist. Also see A333687 for the corresponding values of k for each digit sum and for details of the currently unknown terms.

The first escape value is a(76) = -1. - Georg Fischer, Jul 16 2020

			a(1) = 1 as the digit sum 1 divides 1 itself. Similarly a(2),...,a(9) equal 2,...,9 respectively.
a(10) = 1 as the digit sum of 10 is 1 which divides 10.
a(11) = 9 as A333687(11) = 2 giving the decimal concatenation 111213 which has a digit sum of 9.
a(16) = 36 as A333687(16) = 4 giving the decimal concatenation 1617181920 which has a digit sum of 36.

Scott R. Shannon, Table of n, a(n) for n = 1..1000

Cf. A333687, A007953, A332580, A332558, A332563, A332542, A332830, A332867, A005349.

cp's OEIS Frontend

A332830 a(n) = minimal positive k such that the concatenation of decimal digits n and n+1 is a divisor of the concatenation of n+2, n+2+1, ..., n+2+k.

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A332867 a(n) = minimal positive k such that the concatenation of decimal digits 1,2,...,n is a divisor of the concatenation of n+1,n+2,...,n+k.

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A360376 a(n) = minimal nonnegative k such that prime(n) * prime(n+1) * ... * prime(n+k) + 1 is divisible by prime(n+k+1), or -1 if no such k exists.

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A332585 Number of digits in the number formed by concatenating the digits of n, n+1, ..., A332584(n), or -1 if A332584(n) = -1.

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A332586 a(n) = minimal value of n+k+1 such that the concatenation of the binary expansions of n,n+1,...,n+k is divisible by n+k+1, or -1 if no such n+k+1 exists.

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A360311 The sum of the primes prime(n) + prime(n+1) + ... + prime(n+k) in A360297.

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A360312 The dividing prime prime(n+k+1) in A360297.

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A332562 a(n) = number formed by concatenating the decimal digits of 44, 45, 46, ..., 44+n.

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A333687 a(n) is the minimal value of k >= 0, such that the concatenation of the decimal digits of n,n+1,...,n+k is divisible by the digit sum of the concatenation, or -1 if no such k is known.