Andrew Weimholt - cp's OEIS frontend

A231881 The digits of a(n) and a(n+1) together can be reordered to form a square; lexicographically earliest sequence of distinct positive integers with this property.

1, 6, 3, 16, 9, 4, 14, 8, 29, 5, 2, 11, 25, 12, 15, 21, 37, 69, 13, 27, 19, 26, 10, 24, 30, 42, 39, 52, 20, 34, 18, 28, 81, 43, 36, 31, 48, 7, 56, 47, 61, 33, 46, 17, 23, 40, 32, 49, 60, 57, 22, 45, 63, 54, 67, 41, 38, 44, 62, 55, 126, 58, 108, 76, 50, 92, 35, 64

Offset: 0

N. J. A. Sloane, Nov 17 2013, based on a posting to the Sequence Fans Mailing List by Andrew Weimholt, Nov 12 2013

A231880 and A231881 eventually merge: A231881(2539) = 2541; A231880(2540) = 2536; A231881(2540,2541,..) = A231880(2541,2542,..) = 2544,2551,.. Hans Havermann, Nov 17 2013

Hans Havermann, Table of n, a(n) for n = 0..10000

A variant of A228407. Cf. A231880.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, idm = IntegerDigits@ a[n - 1], t = a@# & /@ Range[0, n - 1]}, Label[start]; While[ MemberQ[t, k], k++]; While[ Select[ Permutations[ Join[idm, IntegerDigits[ k]]], #[[1]] != 0 && IntegerQ[ Sqrt[ FromDigits[ #]]] &] == {}, k++; Goto[start]]; k]; Array[a, 100, 0] (* Robert G. Wilson v, Nov 17 2013 *)

Corrected and extended by Hans Havermann, Nov 17 2013

A231880 The digits of a(n) and a(n+1) together can be reordered to form a square; lexicographically earliest sequence of distinct nonnegative integers with this property.

Original entry on oeis.org

0, 10, 24, 3, 6, 1, 8, 14, 4, 9, 16, 12, 15, 21, 25, 2, 5, 22, 45, 18, 28, 81, 34, 20, 29, 7, 48, 13, 27, 19, 26, 11, 52, 30, 42, 39, 61, 33, 46, 17, 23, 40, 32, 49, 60, 57, 64, 35, 92, 43, 36, 31, 63, 54, 67, 41, 38, 44, 62, 47, 56, 70, 65, 74, 124, 69, 37

Offset: 0

N. J. A. Sloane, Nov 17 2013, based on a posting to the Sequence Fans Mailing List by Andrew Weimholt, Nov 12 2013

A231880 and A231881 eventually merge: A231880(2540) = 2536; A231881(2539) = 2541; A231880(2541,2542,..) = A231881(2540,2541,..) = 2544,2551; ... Hans Havermann, Nov 17 2013

Hans Havermann, Table of n, a(n) for n = 0..10000

A variant of A228407. Cf. A231881.

a[0] = 0; a[n_] := a[n] = Block[{k = 1, idm = IntegerDigits@ a[n - 1], t = a@# & /@ Range[n - 1]}, Label[ start]; While[ MemberQ[t, k], k++]; While[ Select[ Permutations[ Join[idm, IntegerDigits[ k]]], #[[1]] != 0 && IntegerQ[ Sqrt[ FromDigits[ #]]] &] == {}, k++; Goto[ start]]; k]; Array[a, 100, 0] (* Robert G. Wilson v, Nov 17 2013 *)

Corrected and extended by Hans Havermann, Nov 17 2013

More terms added (from b-file) by Jon E. Schoenfield, Dec 22 2013

A222192 a(n) = number of inequivalent ways to choose a subset of the n*2^(n-1) edges of the n-cube so that the resulting figure is connected and fully n-dimensional.

Original entry on oeis.org

1, 3, 78, 7338218

Offset: 1

Andrew Weimholt, Feb 12 2013

"Inequivalent" means that figures differing by a rotation and/or reflection are not regarded as different.
"Fully n-dimensional" means not lying in a proper subspace.
This is a variation on A222186, that was based on a work by the artist Sol LeWitt.

			For n=2 the three figures are: the four edges of a square, or omit one edge, or omit two adjacent edges.

Andrew Weimholt, 3D solutions in numerical representation
Andrew Weimholt, Notes on reading the 3D solutions

Cf. A222186.

a(4) computed by Andrew Weimholt, Feb 13 2013

A185811 a(n) is the k value that corresponds to A185729(n).

Original entry on oeis.org

2, 2, 2, 8, 7, 11, 11, 11, 11, 15, 16, 20, 17, 23, 19, 24, 23, 24, 23, 24, 27, 26, 31, 32, 32, 31, 32, 32, 35, 35, 35, 36, 40, 42, 42, 44, 41, 47, 48, 48, 47, 48, 48, 51, 52, 51, 53, 56, 50, 56, 55, 53, 57, 59, 60, 64, 61, 64, 64, 68, 68, 68, 68, 64

Offset: 1

Andrew Weimholt, Feb 05 2011

nonn

			a(4) = 8 because A185729(4) = 51 which is the sum of its first 8 non-divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A185729, A064510.

A185729 Numbers that are the sum of their first k non-divisors for some k.

Original entry on oeis.org

5, 8, 12, 51, 56, 85, 87, 105, 132, 164, 224, 249, 280, 321, 324, 343, 357, 363, 366, 405, 427, 428, 553, 583, 591, 618, 638, 654, 689, 699, 820, 918, 978, 1028, 1045, 1077, 1144, 1207, 1261, 1267, 1268, 1342, 1377, 1417, 1477, 1505, 1517, 1691, 1692, 1707, 1758, 1794, 1805, 1883, 1927, 2197, 2322, 2334, 2384, 2461, 2481, 2523, 2525, 2568

Offset: 1

Andrew Weimholt, Feb 05 2011

nonn

			51 is in the sequence because 51 = 2 + 4 + 5 + 6 + 7 + 8 + 9 + 10
(the sum of its first 8 non-divisors)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A185811 (k-values associated with this sequence)

Cf. A064510.

Select[Range[2600],MemberQ[Accumulate[Complement[Range[#],Divisors[#]]],#]&]  (* Harvey P. Dale, Feb 16 2011 *)

is(n)=my(s,t=1); while(sCharles R Greathouse IV, Nov 23 2015

A181887 a(0) = 0, and for n > 0, a(n) = A002956(n) - A000041(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 9, 33, 43, 89, 124, 292, 290, 726, 839, 1318, 1904, 3616, 3653, 7446, 7620, 12175, 16474, 27907, 26490, 47651, 56922, 80410, 93525, 160402, 146944, 273510, 286942, 395776, 495852, 659747, 690842

Offset: 0

Andrew Weimholt, Feb 01 2011

nonn

A002956 can be thought of as a modular arithmetic version of the partition numbers (A000041). The number of "modulo n" partitions of n is the number of multisets of integers ranging from 1 to n, such that the sum of members of the multiset is congruent to 0 mod n, and no submultiset exists whose members sum to 0 mod n. Therefore, a(n) is the number of "modulo n" partitions which are not ordinary partitions of n.

			The multisets counted by A002956(5) but not by A000041(5) are
..{1,3,3,3}
..{2,2,2,2,2}
..{2,2,2,4}
..{2,4,4}
..{3,3,3,3,3}
..{3,4,4,4}
..{3,3,4}
..{4,4,4,4,4}
So a(5) = 8.

Finklea, Moore, Ponomarenko and Turner, Invariant Polynomials and Minimal Zero Sequences

Cf. A000041, A002956, A082641

A180427 Lexicographically earliest permutation of the positive integers such that the inverse permutation is also the absolute value of the first differences.

Original entry on oeis.org

1, 2, 4, 10, 13, 3, 19, 38, 16, 5, 9, 73, 48, 43, 23, 6, 15, 42, 7, 14, 45, 8, 49, 64, 12, 72, 17, 50, 97, 154, 20, 95, 27, 98, 18, 83, 21, 99, 91, 173, 22, 107, 89, 103, 190, 169, 28, 117, 104, 127, 155, 24, 118, 219, 26, 135, 29, 142, 258, 25, 147, 36, 181, 11, 35, 159

Offset: 1

Andrew Weimholt, Sep 04 2010

			Let a(n) be this sequence and b(n)=|a(n)-a(n+1)| be the inverse permutation of this sequence.
After a(1)=1, a(2)=2, a(3)=4, the next term, a(4), cannot be a repeat of 1,2, or 4 since by definition a(n) must be a permutation of the positive integers.
It cannot be 3,5, or 6, as that would force b(3)=1 or 2 (a repeat of b(1)=1, or b(2)=2).
We cannot have a(4)=7, because b(3)=3 implies a(3)=3, which contradicts a(3)=4.
We cannot have a(4)=8, because b(3)=4 implies a(4)=3.
We cannot have a(4)=9, because b(3)=5 implies a(5)=3, and b(4)=|a(5)-a(4)|=6 which contradicts b(4)=3 as implied by a(3)=4.
Therefore a(4)=10 is the smallest value of a(4) which will not generate a contradiction.

Cf. A180428 - Inverse Permutation of this sequence. Also the first differences (absolute value) of this sequence.

A180428 Inverse permutation of A180427. Also the absolute values of the first differences of A180427.

Original entry on oeis.org

1, 2, 6, 3, 10, 16, 19, 22, 11, 4, 64, 25, 5, 20, 17, 9, 27, 35, 7, 31, 37, 41, 15, 52, 60, 55, 33, 47, 57, 134, 75, 68, 71, 80, 65, 62, 78, 8, 82, 151, 85, 18, 14, 87, 21, 141, 89, 13, 23, 28, 131, 94, 101, 193, 109, 106, 113, 116, 233, 122, 111, 145, 170, 24, 124

Offset: 1

Andrew Weimholt, Sep 04 2010

nonn

Cf. A180427 Inverse Permutation of this sequence.

a(n) = | A180427(n+1) - A180427(n) |.

A169901 Earliest sequence such that (x+y) | a(xy) for all x>=1, y>=1.

Original entry on oeis.org

2, 3, 4, 20, 6, 35, 8, 18, 30, 77, 12, 728, 14, 45, 16, 680, 18, 1881, 20, 252, 110, 299, 24, 3850, 130, 135, 84, 5104, 30, 75361, 32, 396, 238, 665, 36, 28860, 38, 273, 80, 82082, 42, 218569, 44, 360, 2898, 1175, 48, 193648, 350, 2295, 260, 25228, 54, 33495

Offset: 1

Andrew Weimholt, Jul 05 2010

Alois P. Heinz, Table of n, a(n) for n = 1..20000

a:= n-> ilcm(seq(d+n/d, d=numtheory[divisors](n))):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 09 2016

eq[n_] := And @@ Table[y = n/x; Mod[a[x*y], x + y] == 0, {x, Divisors[n]}]; r[n_] := Reduce[ eq[n], a[n], Integers] /. C[1] -> 1; Table[a[n] /. ToRules[r[n]], {n, 1, 51}] (* Jean-François Alcover, Aug 03 2012 *)

A169900 Earliest sequence such that xy | a(x+y) for all x>=1, y>=1.

Original entry on oeis.org

1, 1, 2, 12, 12, 360, 60, 1680, 2520, 25200, 2520, 332640, 27720, 5045040, 5405400, 2882880, 720720, 220540320, 12252240, 4655851200, 4888643760, 5121436320, 232792560, 128501493120, 26771144400, 696049754400

Offset: 1

Andrew Weimholt, Jul 05 2010

From Robert Israel, Dec 29 2017: (Start)
If n = p^d for prime p, then a(n) = p^(2*d-2)*Product_q q^floor(log_q(n)), where the product is over all primes q < n other than p.
Otherwise, a(n) = n^2*Product_p p^floor(log_p(n/p^(nu(n,p)))),
 where the product is over all primes p < n and nu(n,p) is the p-adic order of n. (End)

			After a(1)=a(2)=1, we must have a(3) >= 2 from 2 | a(1+2), and a(3)=2 works.

Robert Israel, Table of n, a(n) for n = 1..2293

seq(ilcm(seq(x*(n-x),x=1..n/2)),n=1..50); # Robert Israel, Dec 28 2017

a[n_]:=If[n<=2,1,LCM@@Table[x(n-x),{x,Floor[n/2]}]];Table[a[n],{n,30}] (* Zak Seidov, Jul 11 2010 *)

cp's OEIS Frontend

User: Andrew Weimholt

A231881 The digits of a(n) and a(n+1) together can be reordered to form a square; lexicographically earliest sequence of distinct positive integers with this property.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A231880 The digits of a(n) and a(n+1) together can be reordered to form a square; lexicographically earliest sequence of distinct nonnegative integers with this property.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A222192 a(n) = number of inequivalent ways to choose a subset of the n*2^(n-1) edges of the n-cube so that the resulting figure is connected and fully n-dimensional.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A185811 a(n) is the k value that corresponds to A185729(n).

Views

Author

Keywords

Examples

Links

Crossrefs

A185729 Numbers that are the sum of their first k non-divisors for some k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A181887 a(0) = 0, and for n > 0, a(n) = A002956(n) - A000041(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A180427 Lexicographically earliest permutation of the positive integers such that the inverse permutation is also the absolute value of the first differences.

Views

Author

Keywords

Examples

Crossrefs

A180428 Inverse permutation of A180427. Also the absolute values of the first differences of A180427.

Views

Author

Keywords

Crossrefs

Formula

A169901 Earliest sequence such that (x+y) | a(xy) for all x>=1, y>=1.

Views

Author

Keywords

Links

Programs

Maple

Mathematica

A169900 Earliest sequence such that xy | a(x+y) for all x>=1, y>=1.

Views

Author

Keywords