A332542 -id:A332542 - cp's OEIS frontend

A332543 a(n) = n + A332542(n) + 1.

6, 12, 20, 10, 14, 56, 24, 15, 22, 33, 39, 26, 21, 48, 272, 34, 38, 76, 28, 33, 46, 69, 40, 50, 39, 36, 116, 58, 62, 992, 96, 51, 70, 45, 111, 74, 57, 65, 205, 82, 86, 172, 55, 69, 94, 141, 112, 70, 75, 68, 212, 106, 66, 77, 133, 87, 118, 177, 183, 122

Offset: 3

Scott R. Shannon, Feb 18 2020

nonn

Note that A332542 is well-defined for all n. - N. J. A. Sloane, Feb 20 2020

			See A332542.

Seiichi Manyama, Table of n, a(n) for n = 3..10000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000, April 2020

Cf. A332542, A332544, A082183.

Maple
```
See A332542.
```

A332544 a(n) = (k+1)n + k(k+1)/2, where k = A332542(n).

Original entry on oeis.org

12, 60, 180, 30, 70, 1512, 240, 60, 176, 462, 663, 234, 105, 1008, 36720, 408, 532, 2660, 168, 297, 782, 2070, 480, 900, 390, 252, 6264, 1218, 1426, 491040, 4032, 714, 1820, 360, 5439, 1998, 855, 1300, 20090, 2460, 2752, 13760, 495, 1311, 3290, 8742

Offset: 3

Scott R. Shannon, Feb 18 2020

nonn

Note that A332542 is well-defined for all n. - N. J. A. Sloane, Feb 20 2020

Seiichi Manyama, Table of n, a(n) for n = 3..10000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000, April 2020

Cf. A332542, A332543.

A332558 a(n) is the smallest k such that n(n+1)(n+2)...(n+k) is divisible by n+k+1.

Original entry on oeis.org

4, 3, 2, 3, 4, 5, 4, 3, 5, 4, 6, 5, 6, 5, 4, 7, 6, 5, 4, 3, 6, 7, 6, 5, 4, 8, 7, 6, 6, 5, 8, 7, 6, 5, 4, 8, 7, 6, 5, 7, 6, 5, 10, 9, 8, 9, 8, 7, 6, 9, 8, 7, 6, 5, 4, 6, 12, 11, 10, 9, 8, 7, 6, 7, 6, 5, 12, 11, 10, 9, 8, 7, 6, 5, 8, 7, 6, 11, 10, 9, 8, 7, 6, 5

Offset: 1

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

This is a multiplicative analog of A332542.

a(n) always exists because one can take k to be 2^m - 1 for m large.

Robert Israel, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program
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. A061836 (k+1), A332559 (n+k+1), A332560 (the final product), A332561 (the quotient).
For records, see A333532 and A333533 (and A333537), which give the records in the essentially identical sequence A061836.
Additive version: A332542, A332543, A332544, A081123.
"Concatenate in base 10" version: A332580, A332584, A332585.

f:= proc(n) local k,p;
  p:= n;
  for k from 1 do
    p:= p*(n+k);
    if (p/(n+k+1))::integer then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 25 2020

a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 04 2020, after Maple *)

a(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return(k))); } \\ Jinyuan Wang, Feb 25 2020

PARI
```
\\ See Corneth link
```

def a(n):
    k, p = 1, n*(n+1)
    while p%(n+k+1): k += 1; p *= (n+k)
    return k
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jun 06 2021

a(n) = A061836(n) - 1 for n >= 1.

a(n + 1) >= a(n) - 1. a(n + 1) = a(n) - 1 mostly. - David A. Corneth, Apr 14 2020

A360297 a(n) = minimal positive k such that the sum of the primes prime(n) + prime(n+1) + ... + prime(n+k) is divisible by prime(n+k+1), or -1 if no such k exists.

Original entry on oeis.org

1, 3, 7, 11, 26, 20, 27, 52, 1650, 142, 53, 168234, 212, 7, 13

Offset: 1

Scott R. Shannon, Feb 02 2023

In the first 100 terms there are twenty values for which a(n) is currently unknown; for all of these values a(n) is at least 10^9. These unknown terms are for n = 16, 22, 24, 34, 41, 42, 45, 48, 50, 54, 55, 62, 68, 70, 72, 75, 80, 87, 88, 98. In this same range the largest known value is a(76) = 749597506, where prime(76) = 383 leads to a sum of primes of 6173644601523754801 that is divisible by 16865554301.
See A360311 for the sum of the k+1 primes. See A360312 for prime(n+k+1).
a(16) > 10^10. - Michael S. Branicky, Feb 19 2025
a(16) > 10^12. - Michael S. Branicky, Apr 22 2025

			a(1) = 1 as prime(1) + prime(2) = 2 + 3 = 5, which is divisible by prime(3) = 5.
a(4) = 11 as prime(4) + ... + prime(15) = 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 = 318, which is divisible by prime(16) = 53.

Scott R. Shannon, Terms for n = 1..100. The terms currently unknown are set to -1.

Cf. A360311, A360312, A000040, A007504, A332542, A332580.

from sympy import prime, nextprime
def A360297(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p+q, 1
    while s%(q:=nextprime(q)):
        k += 1
        s += q
    return k # Chai Wah Wu, Feb 03 2023

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

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

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.

A360406 a(n) = minimal positive 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

1, 1, 9, 14, 31, 826, 1, 34

Offset: 1

Scott R. Shannon, Feb 06 2023

Assuming a(9) exists it is greater than 1.75 million.

a(11) = 692, a(12) = 8, a(13) = 792. - Robert Israel, Feb 22 2023

			a(1) = 1 as prime(1) * prime(2) - 1 = 2 * 3 - 1 = 5, which is divisible by prime(3) = 5.
a(2) = 1 as prime(2) * prime(3) - 1 = 3 * 5 - 1 = 14, which is divisible by prime(4) = 7.
a(3) = 9 as prime(3) * ... * prime(12) - 1 = 1236789689134, which is divisible by prime(13) = 41.

Cf. A360376, A360297, A000040, A007504, A332542, A332580.

f:= proc(n) local P,k,p;
P:= ithprime(n); p:= nextprime(P);
for k from 0 to 10^6 do
  if P-1 mod p = 0 then return k fi;
  p:= nextprime(p);
 od;
FAIL
end proc:
map(f, [$1..8]); # Robert Israel, Feb 22 2023

from sympy import prime, nextprime
def A360406(n):
    p = prime(n)
    q = nextprime(p)
    s, k = p*q, 1
    while (s-1)%(q:=nextprime(q)):
        k += 1
        s *= q
    return k # Chai Wah Wu, Feb 06 2023

cp's OEIS Frontend

A332543 a(n) = n + A332542(n) + 1.

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Maple

A332544 a(n) = (k+1)*n + k*(k+1)/2, where k = A332542(n).

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A332558 a(n) is the smallest k such that n*(n+1)*(n+2)*...*(n+k) is divisible by n+k+1.

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Formula

A360297 a(n) = minimal positive k such that the sum of the primes prime(n) + prime(n+1) + ... + prime(n+k) is divisible by prime(n+k+1), or -1 if no such k exists.

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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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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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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.

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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.

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A332544 a(n) = (k+1)n + k(k+1)/2, where k = A332542(n).

A332558 a(n) is the smallest k such that n(n+1)(n+2)...(n+k) is divisible by n+k+1.