A067957 -id:A067957 - cp's OEIS frontend

A093315 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_1 = 4 and such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

0, 0, 0, 1, 2, 2, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 6, 8, 3, 14, 12, 18, 13, 14, 6, 26, 13, 198, 152, 220, 118, 1033, 807

Offset: 1

Matthijs Coster, Apr 26 2004

The beginning elements of the permutation are 4, (either 1 or 2), ...

The total number of permutations with this property is given in A067957.

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

Cf. A067957, A093313, A093315.

A094097 Number of divisor chains of length n which begin with n ("anchored" divisor chains).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 4, 3, 2, 8, 4, 6, 47, 44, 6, 37, 6, 166, 462, 232, 372, 2130, 1589, 9093, 20896, 20314, 4118, 32367, 12815, 167796, 130528, 59173, 1942045, 2607312, 163775, 1297794, 18340336, 22304602, 5065878, 21005347, 3607762, 129164605

Offset: 1

N. J. A. Sloane, following a suggestion of R. K. Guy, May 04 2004

nonn

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.

			Examples of divisor chains of lengths 1 through 8:
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

Cf. A067957, A094098, A094099.

Right diagonal of A093323.

a(1)-a(9) from R. K. Guy and Paul Vaderlind
a(10)-a(19) from Ed Clark, Jr. and Chuck Seggelin
a(20)-a(28) from Christopher Landauer, May 04 2004
a(29)-a(37) from Joseph Myers, May 04 2004
a(38) from Jud McCranie, May 07 2004
a(39)-a(44) from Joseph Myers, May 21 2004

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

A291445 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Sum_{i=1..j} s_i^3.

Original entry on oeis.org

1, 2, 6, 12, 30, 78, 186, 414, 912, 2064, 4338, 9798, 20106, 40974, 80196, 158322, 309414, 615558, 1212402, 2417136, 4776654, 9497508, 18726708, 37056150, 72946116, 144230640, 284660874, 564451830, 1118803818, 2224792026, 4420041210, 8791590168, 17456783136

Offset: 1

Seiichi Manyama, Aug 23 2017

nonn

The permutation [1,...,n] satisfies the conditions since Sum_{i=1..n} i^3 = (Sum_{i=1..n})^2. Similarly, [n,...,1] satisfies the conditions since Sum_{i=m..n} i^3 = (Sum_{i=m..n} i)*(n*(n+1)+m*(m-1))/2. Thus a(n) >= 2 for n > 1 and a(n) is nondecreasing. Seems to approximately double for each successive n. - Chai Wah Wu, Aug 24 2017

			5                 divides 5^3,
5 + 4             divides 5^3 + 4^3,
5 + 4 + 3         divides 5^3 + 4^3 + 3^3,
5 + 4 + 3 + 2     divides 5^3 + 4^3 + 3^3 + 2^3,
5 + 4 + 3 + 2 + 1 divides 5^3 + 4^3 + 3^3 + 2^3 + 1^3.
So [5, 4, 3, 2, 1] satisfies all the conditions.
1                     divides 1^3,
1 + 2                 divides 1^3 + 2^3,
1 + 2 + 6             divides 1^3 + 2^3 + 6^3,
1 + 2 + 6 + 5         divides 1^3 + 2^3 + 6^3 + 5^3,
1 + 2 + 6 + 5 + 4     divides 1^3 + 2^3 + 6^3 + 5^3 + 4^3,
1 + 2 + 6 + 5 + 4 + 3 divides 1^3 + 2^3 + 6^3 + 5^3 + 4^3 + 3^3.
So [1, 2, 6, 5, 4, 3] satisfies all the conditions.
-------------------------------------------------------
a(1) = 1: [[1]];
a(2) = 2: [[1, 2], [2, 1]];
a(3) = 6: [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]];
a(4) = 12: [[1, 2, 3, 4], [1, 3, 2, 4], [2, 1, 3, 4], [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 3, 1], [3, 1, 2, 4], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 2, 1], [4, 2, 3, 1], [4, 3, 2, 1]];
a(5) = 30: [[1, 2, 3, 4, 5], [1, 3, 2, 4, 5], [2, 1, 3, 4, 5], [2, 3, 1, 4, 5], [2, 3, 4, 1, 5], [2, 3, 4, 5, 1], [2, 3, 5, 4, 1], [2, 4, 3, 1, 5], [2, 4, 3, 5, 1], [2, 5, 3, 4, 1], [3, 1, 2, 4, 5], [3, 2, 1, 4, 5], [3, 2, 4, 1, 5], [3, 2, 4, 5, 1], [3, 2, 5, 4, 1], [3, 4, 2, 1, 5], [3, 4, 2, 5, 1], [3, 4, 5, 2, 1], [3, 5, 2, 4, 1], [3, 5, 4, 2, 1], [4, 2, 3, 1, 5], [4, 2, 3, 5, 1], [4, 3, 2, 1, 5], [4, 3, 2, 5, 1], [4, 3, 5, 2, 1], [4, 5, 3, 2, 1], [5, 2, 3, 4, 1], [5, 3, 2, 4, 1], [5, 3, 4, 2, 1], [5, 4, 3, 2, 1]].

Chai Wah Wu, Table of n, a(n) for n = 1..100

Cf. A067957, A291355, A291518, A291519.

a(13)-a(33) from Chai Wah Wu, Aug 24 2017

A093313 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_1 = 2 and such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 1, 11, 9, 15, 14, 14, 23

Offset: 1

Matthijs Coster, Apr 26 2004; revised Aug 05 2005

An easy calculation turns out that the beginning elements are always: 2,1,3,6,(then either 4 or 12),...

The total number of permutations with this property is given in A067957.

			There is a unique permutation of the numbers 1..38, starting with 2, namely:
2 1 3 6 12 24 8 28 21 35 14 22 4 20 25 5 23 11 33 27 9 37 10 19 7 29 15 30 16 31 17 32 36 34 38 18 26 13
with corresponding sums
2 3 6 12 24 48 56 84 105 140 154 176 180 200 225 230 253 264 297 324 333 370 380 399 406 435 450 480 496 527 544 576 612 646 684 702 728 741.

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

Cf. A067957, A093314, A093315.

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

A093314 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_1 = 3 and such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 12, 20, 20, 1, 163, 55

Offset: 1

Matthijs Coster, Apr 26 2004

The beginning elements of the permutation are 3,1,(either 2 or 4),...

The total number of permutations with this property is given in A067957.

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

Cf. A067957, A093313, A093315.

A291504 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i is not a prime.

Original entry on oeis.org

1, 1, 0, 1, 3, 8, 48, 206, 1838, 13336, 133764, 1081556, 11046816, 108196128, 1555323224, 16279258144, 289771660328, 3495882548784, 66923393467216, 942785369844048, 15625264115770992, 315553823251866304, 5974132307015712032, 104979988889030774848

Offset: 0

Seiichi Manyama, Aug 25 2017

nonn

			1             is not a prime,
1 + 3         is not a prime,
1 + 3 + 2     is not a prime,
1 + 3 + 2 + 4 is not a prime.
So [1, 3, 2, 4] satisfies all the conditions.
---------------------------------------------
a(1) = 1: [[1]];
a(3) = 1: [[1, 3, 2]];
a(4) = 3: [[1, 3, 2, 4], [1, 3, 4, 2], [4, 2, 3, 1]];
a(5) = 8: [[1, 3, 2, 4, 5], [1, 3, 4, 2, 5], [1, 5, 2, 4, 3], [1, 5, 4, 2, 3], [4, 2, 3, 1, 5], [4, 2, 3, 5, 1], [4, 5, 1, 2, 3], [4, 5, 3, 2, 1]].

Cf. A067957, A291355.

Table[Count[Permutations@ Range@ n, ?(AllTrue[Accumulate@ #, ! PrimeQ@ # &] &)], {n, 0, 10}] (* _Michael De Vlieger, Aug 26 2017 *)

a(0), a(12)-a(23) from Alois P. Heinz, Aug 25 2017

cp's OEIS Frontend

A093315 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_1 = 4 and such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

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A094097 Number of divisor chains of length n which begin with n ("anchored" divisor chains).

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

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A291445 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Sum_{i=1..j} s_i^3.

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A093313 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_1 = 2 and such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

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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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A093314 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_1 = 3 and such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

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A291504 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i is not a prime.

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