A332558 -id:A332558 - cp's OEIS frontend

A333542 The primes missing from A333541.

2, 11, 53, 59, 71, 73, 89, 97, 103, 107, 127, 131, 163, 173, 179, 181, 191, 193, 197, 223, 229, 233, 241, 251, 263, 271, 281, 293, 311, 331, 337, 347, 349, 359, 367, 383, 401, 419, 421, 431, 443, 449, 457, 461, 463, 467, 479, 487, 491, 509, 521, 523, 541, 547, 557, 563

Offset: 1

N. J. A. Sloane, Apr 20 2020

nonn

These are the primes that are not record high values in A333537.

			We have A333541(k) = 7 for some k and the term after that A333541(k + 1) = 13. As 11 is a prime between 7 and 13, 11 is in the sequence. - _David A. Corneth_, Apr 21 2020

David A. Corneth, Table of n, a(n) for n = 1..100
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. A332558, A333537, A333538, A333541.

More terms from David A. Corneth, Apr 21 2020

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.

A333545 Indices k such that A217287(k) < A061836(k).

Original entry on oeis.org

1, 2, 5, 6, 9, 11, 12, 13, 14, 15, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 40, 43, 44, 45, 46, 47, 48, 49, 50, 51, 57, 58, 59, 60, 61, 62, 67, 68, 69, 70, 75, 76, 77, 78, 85, 89, 90, 91, 92, 93, 94, 95, 100, 101, 102, 103, 104, 105, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 139

Offset: 1

N. J. A. Sloane, Apr 25 2020

nonn

			Table of values at powers of 2, from _Rémy Sigrist_, Apr 25 2020:
k a(2^k)
-- ------
0 1
1 2
2 6
3 13
4 27
5 51
6 119
7 248
8 535
9 1311
10 2994
11 6838
12 15945
13 36740
14 83716
15 186184
16 427070
17 971213
18 2203026
19 4964477

Rémy Sigrist, Table of n, a(n) for n = 1..50000 (first 2787 terms from N. J. A. Sloane)
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. A332558, A061836, A217287.

cp's OEIS Frontend

A333542 The primes missing from A333541.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A333545 Indices k such that A217287(k) < A061836(k).

Views

Author

Keywords

Examples

Links

Crossrefs