A094097 -id:A094097 - cp's OEIS frontend

A067957 Number of divisor chains of length n: permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

1, 1, 1, 2, 2, 4, 5, 7, 7, 24, 22, 29, 39, 67, 55, 386, 235, 312, 347, 451, 1319, 5320, 3220, 4489, 20237, 36580, 52875, 197103, 216562, 289478, 567396, 659647, 1111153, 3131774, 2200426, 29523302, 34214028, 48161995, 32616148, 242860900, 293579041, 363415618

Offset: 0

Floor van Lamoen, Mar 06 2002

nonn

Apparently this sequence originated in a problem composed by Matthijs Coster in 2002.

Let M = floor(n/2), then the following permutations always work: for n even: M+1, 1, M+2, 2, ..., n-1, M-1, n, M; for n odd: M+1, 1, M+2, 2, ..., M-1, n-1, M, n. - Daniel Asimov, May 04 2004

			Examples of divisor chains of lengths 1 through 9:
  1
  2 1
  3 1 2
  4 2 3 1
  5 1 2 4 3
  6 2 4 3 5 1
  7 1 2 5 3 6 4
  8 2 5 3 6 4 7 1
  8 4 3 5 1 7 2 6 9
The five divisor chains of length 6 are:
  4 1 5 2 6 3
  4 2 6 3 5 1
  5 1 2 4 6 3
  5 1 6 4 2 3
  6 2 4 3 5 1. - Eugene McDonnell, May 21 2004

Matthijs Coster, Sequences
Matthijs Coster, Problem 2001/3-A of the Universitaire Wiskunde Competitie, Nieuw Arch. Wisk. 5/3 (2002), 92-94.

Cf. A093313, A093314, A093315, A094097, A094098, A094099.

a(31)-a(35) from Jud McCranie, May 06 2004
a(0)=1 prepended by Alois P. Heinz, Aug 26 2017
a(36)-a(41) from Zhao Hui Du, May 12 2024

A094099 Number of divisor chains of length 2n+1 which are both cyclic and anchored.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 4, 47, 6, 6, 462, 372, 1589, 20896, 4118, 12815, 130528, 1942045, 163775, 18340336, 5065878, 3607762

Offset: 0

Christopher Landauer, May 04 2004

A divisor chain of length n is an arrangement of 1..n such that each term is a divisor of the sum of the preceding terms.

Cf. A067957, A094097, A094098, A094099.

a(n) = A094097(2n+1). - Martin Fuller, Jul 18 2025

a(14)-a(21) from A094097 by Martin Fuller, Jul 18 2025

A094098 Number of divisor chains of length n in which the first term is a divisor of n(n+1)/2 ("cyclic" divisor chains).

Original entry on oeis.org

1, 0, 2, 0, 2, 0, 3, 0, 5, 0, 6, 0, 6, 0, 147, 1, 22, 2, 27, 165, 519, 0, 516, 2021, 1912, 506, 45658, 514, 7308, 1535, 30746, 68918, 145920, 1370

Offset: 1

Christopher Landauer, May 04 2004

A divisor chain of length n is an arrangement of 1..n such that each term is a divisor of the sum of the preceding terms.

Cf. A067957, A094097, A094099.

a(29)-a(34) from John W. Layman, May 07 2004

A093323 Triangle read by rows giving number of divisor chains of length n beginning with k (1 <= k <= n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 3, 1, 1, 0, 0, 0, 0, 1, 0, 1, 5, 0, 0, 0, 0, 1, 3, 4, 12, 4, 0, 0, 0, 0, 0, 4, 5, 7, 3, 3, 0, 0, 0, 0, 0, 4, 7, 9, 3, 4, 2, 0, 0, 0, 1, 0, 0, 2, 5, 4, 8, 11, 8, 0, 0, 0, 1, 0, 0, 2, 7, 11, 12, 19, 11, 4, 0, 0, 0, 0, 0, 0

Offset: 1

Eugene McDonnell (eemcd(AT)mac.com), May 11 2004

A divisor chain of length n is an arrangement of 1..n such that each term is a divisor of the sum of the preceding terms.

			Triangle begins:
1
0 1
0 1 1
0 0 1 1
0 0 1 2 1
0 0 0 2 2 1
0 0 0 2 3 1 1
0 0 0 0 1 0 1 5
0 0 0 0 1 3 4 12 4
0 0 0 0 0 4 5 7 3 3
0 0 0 0 0 4 7 9 3 4 2
0 0 0 1 0 0 2 5 4 8 11 8
0 0 0 1 0 0 2 7 11 12 19 11 4
0 0 0 0 0 0 0 4 12 4 14 7 8 6
0 0 0 1 0 2 3 14 32 42 64 41 77 63 47
0 0 0 1 0 0 0 0 16 34 39 26 20 24 31 44
0 0 0 1 0 0 0 0 16 44 55 27 34 31 42 56 6
0 0 0 0 0 2 3 2 2 21 13 20 19 31 51 70 76 37
0 0 0 0 0 4 3 3 7 21 17 24 25 34 54 91 113 49 6
0 0 0 0 0 2 0 8 17 12 31 41 43 91 60 121 223 144 360 166
0 0 0 0 0 7 0 20 31 26 57 197 314 383 283 706 938 473 969 454 462
0 0 0 0 0 6 0 17 18 21 0 124 131 220 148 445 538 232 443 222 423 232
0 0 0 0 0 6 0 17 22 29 9 138 164 279 188 520 640 309 616 302 521 357 372
0 0 0 6 0 6 0 44 76 219 86 155 314 545 389 1354 1296 819 727 1246 1959 2619 6247 2130
0 0 0 8 0 11 7 60 112 257 102 273 323 1519 579 2388 2828 1600 2193 2535 3532 3955 9554 3155 1589
0 0 0 3 5 14 15 80 53 139 34 556 453 3063 1160 3194 1739 1756 2015 3648 5311 2903 7496 4084 6061 9093

Cf. A094097 (right diagonal).

cp's OEIS Frontend

A067957 Number of divisor chains of length n: permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

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A094099 Number of divisor chains of length 2n+1 which are both cyclic and anchored.

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A094098 Number of divisor chains of length n in which the first term is a divisor of n(n+1)/2 ("cyclic" divisor chains).

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Author

Keywords

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Extensions

A093323 Triangle read by rows giving number of divisor chains of length n beginning with k (1 <= k <= n).

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