Diana L. Mecum - cp's OEIS frontend

User: Diana L. Mecum

Diana L. Mecum has authored 3 sequences.

A141439 Positions at which disjoint pattern counts differ from overlapping pattern counts related to sequence A117127.

21, 22, 23, 45, 47, 55, 73, 85, 86, 90, 91, 95, 109, 110, 111, 119, 146, 170, 173, 181, 182, 187, 191, 219, 221, 223, 239, 247, 273, 307, 341, 342, 346, 362, 365, 375, 383, 409, 438, 443, 447, 477, 478, 479, 495, 503, 546, 585, 614, 660, 682, 685, 693, 725

Offset: 1

Diana L. Mecum, Aug 06 2008

			When "n" = 21, the base two concatenated range of integers from 1 to 21 is "1101110010111011110001001101010111100110111101111100001000110010100111010010101". 21 in base two is "10101".
Note that there are two instances of "10101" in the concatenated number when disjoint patterns are considered. There are three pattern matches when overlapping patterns are considered.

Diana Mecum, Table of n, a(n) for n = 1..131

Cf. A117127.

f[n_, k_] := StringCount[ ToString@ FromDigits@ Flatten@ IntegerDigits[ Range@n, 2], ToString@ FromDigits@ IntegerDigits[n, 2], Overlaps -> {True, False}[[k]]]; Select[ Range@ 725, f[ #, 1] != f[ #, 2] &] (* Robert G. Wilson v, Aug 30 2008 *)

A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.

Original entry on oeis.org

2, 3, 2, 3, 4, 2, 3, 5, 3, 5, 2, 6, 3, 4, 7, 2, 3, 7, 5, 8, 2, 3, 5, 9, 3, 4, 9, 2, 6, 10, 3, 11, 2, 3, 5, 7, 11, 4, 5, 7, 12, 2, 3, 13, 3, 8, 13, 2, 6, 14, 3, 4, 5, 9, 15, 2, 3, 5, 9, 15, 7, 16, 2, 3, 7, 10, 17, 3, 4, 17, 2, 5, 6, 11, 18, 3, 5, 8, 11, 19, 2, 3, 19, 4, 12, 20, 2, 3, 7

Offset: 3

Diana L. Mecum, Jul 17 2007

A066272 gives the number of terms in each row.
See A066272 for definition of anti-divisor.
2n-1 and 2n+1 are twin primes (that is, n is in A040040) iff n has no odd anti-divisors. For example, because n=15 has no odd anti-divisors, 29 and 31 are twin primes. - Jon Perry, Sep 12 2012
Row n is all the numbers which are: (a) 2n divided by its odd divisors (except 1), and (b) the divisors of 2n-1 and 2n+1 (except 1, 2n+1 and 2n-1). For example, n=18: odd divisors of 36 are {3,9} and 36/{3,9} = {4,12}; divisors of 35 are {5,7} and divisors of 37 are null (37 is prime).  Therefore row 18 is 4,5,7 and 12.  See A066542 for further explanation. - Bob Selcoe, Feb 24 2014

			Anti-divisors of 3 through 20:
3: 2
4: 3
5: 2, 3
6: 4
7: 2, 3, 5
8: 3, 5
9: 2, 6
10: 3, 4, 7
11: 2, 3, 7
12: 5, 8
13: 2, 3, 5, 9
14: 3, 4, 9
15: 2, 6, 10
16: 3, 11
17: 2, 3, 5, 7, 11
18: 4, 5, 7, 12
19: 2, 3, 13
20: 3, 8, 13

T. D. Noe, Rows n=3..1000 of triangle, flattened
Diana Mecum, Rows 3 through 500

f[n_] := Complement[ Sort@ Join[ Select[ Union@ Flatten@ Divisors[{2 n - 1, 2 n + 1}], OddQ@ # && # < n &], Select[ Divisors[2 n], EvenQ@ # && # < n &]], Divisors@ n]; Flatten@ Table[ f@n, {n, 3, 32}] (* Robert G. Wilson v, Jul 17 2007 *)
Table[Select[Range[2, n - 1], Abs[Mod[n, #] - #/2] < 1 &], {n, 3, 31}] // Flatten (* Michael De Vlieger, Jun 14 2016, after Harvey P. Dale at A066272 *)

A101933 Concatenate the first n positive integers written in binary (with each integer written so the most significant 1 is on the left and the concatenated string is from left to right, A030302). a(n) is the number of times n written in binary appears disjointedly in the concatenated string.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 1, 2, 1, 4, 3, 4, 5, 4, 1, 2, 2, 3, 2, 2, 1, 6, 3, 4, 4, 6, 6, 5, 7, 5, 1, 2, 2, 3, 1, 4, 3, 4, 2, 4, 2, 5, 2, 3, 4, 8, 3, 4, 5, 5, 4, 6, 4, 8, 6, 6, 6, 7, 9, 6, 9, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 5, 2, 6, 4, 5, 2, 4, 3, 6, 3, 3, 3, 7, 2, 5, 3, 5, 3, 7, 5, 10, 3, 4, 5, 5, 4, 7, 5, 7, 4, 6

Offset: 1

Diana L. Mecum and Robert G. Wilson v, Aug 31 2008

First occurrence of n: 1, 3, 6, 11, 14, 23, 30, 47, 60, 95, 126, 124, 223, 248, 252, 447, 863, 504, 895, 1008, 1020, 1016, 1919, 2044, 2032, 2040, 3839, 7159, 7871, 4080, 7679, 15855, 8160, 8184, 8176, 15839, ??, 16376, 16352, 16368, ..., to a limit of 25000.

			a(21) = 2 because 21_10 = 10101_2 and that appears in
1101110010111011110001001101010111100110111101111100001000110010100111010010101_2
three times but only two times which do not overlap.

Diana Mecum & Robert G. Wilson v, Table of n, a(n) for n = 1..10000.

Cf. A007088, A030190, A030302, A058935, A047778. Differs from A117127 at positions in A141439.

a[n_] := StringCount[ ToString@ FromDigits@ Flatten@ IntegerDigits[ Range@n, 2], ToString@ FromDigits@ IntegerDigits[n, 2], Overlaps -> False]; Array[a, 105]

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User: Diana L. Mecum

A141439 Positions at which disjoint pattern counts differ from overlapping pattern counts related to sequence A117127.

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A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.

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A101933 Concatenate the first n positive integers written in binary (with each integer written so the most significant 1 is on the left and the concatenated string is from left to right, A030302). a(n) is the number of times n written in binary appears disjointedly in the concatenated string.

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