A187824 -id:A187824 - cp's OEIS frontend

A220890 a(n) = smallest m such that A187824(m) = n, or -1 if A187824 never takes the value n.

-1, -1, -1, 2, 3, 4, 5, 29, 41, 55, 71, 881, 791, 9360, 10009, 1079, 30239, -1, 246960, -1, 636481, 1360800, 3160079, -1, 2162161

Offset: 0

N. J. A. Sloane, Dec 30 2012

sign

a(17) = -1. Proof: If x mod 9 and x mod 12 are both in {-1, 0, 1} then so is x mod 18. So if x is a number which is congruent to -1, 0 or 1 mod k for k=1..17, then also x mod 18 is congruent to -1, 0 or 1. So there is no x such that A187824(x) = 17. QED
From M. F. Hasler, Dec 30 2012 and Dec 31 2012: (Start)
Similarly, a(19) = -1. Indeed, if x == 0, 1 or -1 (mod 15) and (mod 12), then also (mod 60). [Proof: Write x = 15*(4k+d)+e, |e| < 2, then d = 1, 2, 3 all give impossible x (mod 12).] Therefore A187824 cannot have the value 19 (nor 29, nor 59).
Also, a(23) = -1, because x == 0, 1 or -1 (mod 8) and (mod 12) implies the same (mod 24). [To see this, write x = 12*(2k+d)+e, |e| < 2, then d = 1 gives impossible x (mod 8).] Therefore A187824 cannot have the value 23.
From A220891 one may deduce the values for n = 26, 28, 31, 36, 40, 42, 46, 48, 52, 58, 60, 61 to be a(n) = 39412801, 107881201, 3625549201, 170918748000, 2355997644001, 237662810985599, 4614209634434399, 7522575180120001, 362645725505263201, 10684484093105222399, 442709913651892286399, 5205240636387758366399. (End)
Don Reble shows that a(n) > -1 iff n + 1 is either 12, 2p, 3p or p^k > 3, where p is a prime, k >= 1. - M. F. Hasler, Mar 17 2020

Robert Israel, Table of n, a(n) for n = 0..70
Don Reble, Division gets rough: OEIS A187824 and A220890.

Cf. A187824, A056697, A220891.

  N:= 70: # maximum m
V[0]:= -1: V[1]:= -1: V[2]:= -1:
S[3]:= {$0..5}: M[3]:= 6:
# M[m] is the lcm of 1..m
# S[m] is the set of residues mod M[m] for numbers n with A187824(n)>=m
# A[m] is the set of residues mod M[m] for numbers n with A187824(n)=m-1
for m from 4 to N+1 do
   M[m]:= ilcm(M[m-1], m); p:= M[m]/M[m-1];
   if p = 1 then T:= S[m-1]
   else T:= {seq(seq(a+b*M[m-1], a=S[m-1]), b=0..p-1)}
   end if;
   S[m],A[m]:= selectremove(t -> member(mods(t, m), {1, 0, -1}), T);
   if A[m] = {} then V[m-1]:= -1
   else V[m-1]:= min(A[m])
   end if;
end do:
seq(V[j], j=0..N);
# Robert Israel, Dec 31 2012

a(26) = 39412801. Double-checked all lower given values. - M. F. Hasler, Dec 30 2012

A220891 Where record values occur in A187824.

Original entry on oeis.org

2, 3, 4, 5, 29, 41, 55, 71, 791, 1079, 30239, 246960, 636481, 1360800, 2162161, 39412801, 107881201, 3625549201, 170918748000, 2355997644001, 237662810985599, 4614209634434399, 7522575180120001, 362645725505263201, 10684484093105222399, 442709913651892286399, 5205240636387758366399

Offset: 1

N. J. A. Sloane, Dec 30 2012

nonn

Since A187824 is unbounded, this sequence is infinite.

Robert Israel, Table of n, a(n) for n = 1..30

Cf. A187824, A220890, A056697.

N:= 20: # number of record values wanted
R[1]:= 2: R[2]:= 3: r:= 3: count:= 2:
S[3]:= {$0..5}: M[3]:= 6:
# M[m] is the lcm of 1..m
# S[m] is the set of residues mod M[m] for numbers n with A187824(n)>=m
# R[i] is the i'th record value
for m from 4 while count < N do
  M[m]:= ilcm(M[m-1],m); p:= M[m]/M[m-1];
  if p = 1 then T:= S[m-1]
  else T:= {seq(seq(a+b*M[m-1],a=S[m-1]),b=0..p-1)}
  end if;
  S[m]:= select(t -> member(mods(t,m),{1,0,-1}),T);
  r:= min(S[m] minus {0,1});
  if r > R[count] then
    count:= count+1; R[count]:= r
  end if;
end do:
[seq(R[j],j=1..count)];
# Robert Israel, Dec 31 2012

{m=0;for(n=1,9e9,m<A187824(n) || next; print1(n","); m=A187824(n))} \\ For illustrative purpose (values < 10^8) only. - M. F. Hasler, Dec 31 2012

A192129 a(n) is difference of indices of terms >= 9 in A187824.

Original entry on oeis.org

16, 90, 55, 233, 55, 1, 54, 16, 56, 34, 55, 1, 70, 90, 128, 70, 56, 34, 55, 1, 70, 1, 55, 34, 56, 70, 128, 90, 70, 1, 55, 34, 56, 16, 54, 1, 55, 233, 55, 90, 16, 54, 1, 1, 54, 16, 90, 55, 233, 55, 1, 54, 16, 56, 34, 55, 1, 70, 90, 128, 70, 56, 34, 55, 1, 70

Offset: 1

Kival Ngaokrajang, Dec 31 2012

nonn

The repeat cycle contains 45 numbers which have 22 Fibonacci numbers symmetrically distributed as below.
1      F(1), F(2)
54
16
90
55     F(10)
233    F(13)
55     F(10)
1      F(1), F(2)
54
16
56
34     F(9)
55     F(10)
1      F(1), F(2)
70
90
128
70
56
34     F(9)
55     F(10)
1      F(1), F(2)
70
1      F(1), F(2)
55     F(10)
34     F(9)
56
70
128
90
70
1      F(1), F(2)
55     F(10)
34     F(9)
56
16
54
1      F(1), F(2)
55     F(10)
233    F(13)
55     F(10)
90
16
54
1      F(1), F(2)

			For A187824(m(i)) >= 9:
m(1) = 55, m(2) = 71, m(3) = 161, m(4) = 216,
...
a(1) = 71 - 55 = 16, a(2) = 161 - 71 = 90,
a(3) = 216 - 161 = 55,
...

Cf. A187824.

A187824[n_ /; n > 1] := Catch[For[k = 4, True, k++, m = Mod[n, k, -Floor[k/2]]; If[m != m^3, Throw[k - 1]]]]; Differences[Select[Range[2, 6000], A187824[#] >= 9 &] ] (* Jean-François Alcover, Jan 09 2013 *)

A209282 a(n) is difference of indices of terms = 9 in A187824.

Original entry on oeis.org

161, 288, 1, 70, 90, 470, 89, 1, 70, 1, 89, 470, 90, 70, 1, 288, 161, 288, 1, 70, 90, 470, 89, 1, 70, 1, 89, 470, 90, 70, 1, 288, 161, 288, 1, 70, 90, 470, 89, 1, 70, 1, 89, 470, 90, 70, 1, 288, 161, 288, 1, 70, 90, 470, 89, 1, 70, 1, 89, 470, 90, 70, 1, 288

Offset: 1

Kival Ngaokrajang, Jan 16 2013

The repeat cycle contains 17 numbers which have 6 Fibonacci numbers symmetrically distributed as below.
1           F(1), F(2)
89          F(11)
470
90
70
1           F(1), F(2)
288
161
110
161
288
1           F(1), F(2)
70
90
470
89          F(11)
1           F(1), F(2)

			For A187824(m(i)) = 9:
m(1) = 55, m(2) = 216, m(3) = 504,
...
a(1) = 216 - 55 = 161, a(2) = 504 - 216 = 288,
...

Cf. A187824, A192129.

A209283 a(n) is difference of indices of terms >= 6 in A187824.

Original entry on oeis.org

6, 8, 5, 1, 4, 2, 4, 1, 5, 8, 6, 4, 1, 1, 4, 6, 8, 5, 1, 4, 2, 4, 1, 5, 8, 6, 4, 1, 1, 4, 6, 8, 5, 1, 4, 2, 4, 1, 5, 8, 6, 4, 1, 1, 4, 6, 8, 5, 1, 4, 2, 4, 1, 5, 8, 6, 4, 1, 1, 4, 6, 8, 5, 1, 4, 2, 4, 1, 5, 8, 6, 4, 1, 1, 4, 6, 8, 5, 1, 4, 2, 4, 1, 5, 8, 6, 4

Offset: 1

Kival Ngaokrajang, Jan 16 2013

The repeat cycle contains 15 numbers which have 9 Fibonacci number symmetrically distributed as below.
1         F(1), F(2)
4
6
8         F(6)
5         F(5)
1         F(1), F(2)
4
2         F(3)
4
1         F(1), F(2)
5         F(5)
8         F(6)
6
4
1         F(1), F(2)

			For A187824(m(i)) >= 6:
m(1) = 5, m(2) = 11, m(3) = 19,
...
a(1) = 11 - 5 = 6, a(2) = 19 - 11 = 8,
...

Cf. A187824, A192129.

A209285 a(n) is difference of indices of terms = 6 in A187824.

Original entry on oeis.org

6, 8, 5, 1, 6, 28, 1, 1, 4, 14, 10, 6, 1, 5, 8, 6, 6, 10, 13, 1, 4, 2, 5, 23, 1, 5, 6, 8, 6, 10, 6, 8, 6, 5, 1, 23, 5, 2, 4, 1, 13, 10, 6, 6, 8, 5, 1, 6, 10, 14, 4, 1, 1, 28, 6, 1, 5, 8, 6, 10, 6, 8, 5, 1, 6, 28, 1, 1, 4, 14, 10, 6, 1, 5, 8, 6, 6, 10, 13, 1, 4, 2, 5, 23, 1, 5, 6, 8, 6, 10, 6, 8, 6

Offset: 1

Kival Ngaokrajang, Jan 16 2013

The repeat cycle contains 60 numbers which have 30 Fibonacci numbers symmetrically distributed as below.
.6.5.5.1...6....1..1...6..1....1...7.3.1.6..
   1 3 7   5    1  5   5  1    5   1 5 5
           1       6   1       6
Where: "1" = F(1) or F(2), "." = non-Fibonacci numbers,
       consecutive Fibonacci numbers are showed as
       vertical group.

			For A187824(m(i)) = 6:
m(1) = 5, m(2) = 11, m(3) = 19,
...
a(1) = 11 - 5 = 6, a(2) = 19 - 11 = 8,
...

Cf. A187824, A192129.

A187761 Number of maps f: [n] -> [n] with f(x)<=x and f(f(x)) = f(f(f(x))).

Original entry on oeis.org

1, 1, 2, 6, 23, 106, 568, 3459, 23544, 176850, 1451253, 12904312, 123489888, 1264591561, 13790277294, 159466823794, 1948259002647, 25066729706582, 338670605492700, 4792623436607059, 70873649458154500, 1092969062435462254, 17543703470388927229, 292600906102204630092

Offset: 0

Joerg Arndt, Jan 04 2013

This sequence and A187771 and A187824 are winners in the contest held at the 2013 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 14 2013
Number of monotonic-labeled forests on n vertices with rooted trees of height less than 3. A labeled rooted tree is monotonic-labeled if the label of any parent vertex is (strictly) smaller than the label of any offspring vertex. (See comment by Dennis P. Walsh at A000110; with "greater" instead of "smaller".) To see this, consider the maps [1,2,...,n] -> [0,1,...,n-1] with f(x) < x.
As the maps are valid (left) inversion tables for permutations (see example), we obtain a simple bijection between permutations and such forests.
For n>=3 column 3 of A179455; maps such that f^[k](x) = f^[k-1](x) correspond to column k of A179455 (for n>=k).
Explanation of the Maple routine by Alois P. Heinz, Jan 15 2013: (Start)
b(n,x,y) is the number of forests consisting of trees we want to count, where n nodes are still to be inserted and x nodes at level 0 (the roots) and y nodes at level 1 are already present, plus perhaps some nodes at level 2 (whose number is not of interest).
If the next node is inserted at level 0 then n-1 remaining nodes are to be inserted (and level 0 has one more node: x+1).  There is only one possibility to do that.
If the next node is inserted at level 1 then again n-1 nodes are to be inserted (and level 1 has one more node: y+1).  The inserted node can have x different predecessors (at level 0), accounted by the multiplication by x.
If a node is inserted at level 2 then (again) n-1 nodes are to be inserted and level 2 has one more node (which is not counted).  The inserted node can have y predecessors, accounted by the multiplication by y.
b(0,x,y) = 1 counts any fixed forest that has received all its nodes.
b(n,0,0) counts all forests that can be constructed by inserting n nodes into the empty forest.
(End)
Also the row sums of the Bell transform of the Bell numbers. Since the Bell numbers are the row sums of the Bell transform of the Stirling_2 numbers they might also be called second order Bell numbers. (Also note that the Stirling_2 numbers are the row sums of the Bell transform of the simplest sequence of positive numbers 1,1,1,... which in turn are the row sums of the Bell transform of the characteristic function of 0. See the link 'Bell Transform' for more about this hierarchy which might be called the Bell hierarchy.) - Peter Luschny, Jan 23 2016

			There are a(4)=23 such maps f: [0,1,2,3] -> [0,1,2,3], all 4-digit mixed-radix numbers [f(0), f(1), f(2), f(3)] where 0 <= f(k) <= k (rising factorial basis) except for [ 0 0 1 2 ], as f(3)=2 and f(f(3)) = f(2) = 1 != f(f(f(3))) = f(f(2)) = f(1) = 0.
The exception corresponds to the tree 0 -- 1 -- 2 -- 3 (0 is the root), which can be identified with the map [1,2,3,4] -> [0,1,2,3] where f(k)=k-1.

Alois P. Heinz, Table of n, a(n) for n = 0..493
Peter Luschny, The Bell transform
Peter Luschny, Permutation Trees
D. P. Walsh, Counting forests with Stirling and Bell numbers

Cf. A000110 (Number of maps f: [n] -> [n] where f(x)<=x and f(f(x))=f(x) ).
Cf. A179455 (permutation trees of power n and height <= k+1 ).
Cf. A000949 (maps f: [n] -> [n] where f(f(x)) = f(f(f(x))) ).

b:= proc(n, x, y) option remember; `if`(n=0, 1,
       b(n-1, x+1, y) +x*b(n-1, x, y+1) +y*b(n-1, x, y))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 09 2013
# The function BellMatrix is defined in A264428.
B := BellMatrix(n -> combinat:-bell(n), 24):
seq(add(i, i=LinearAlgebra:-Row(B,n)), n=1..24); # Peter Luschny, Jan 23 2016
# alternative Maple program:
b:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*b(j-1, h-1)*b(n-j, h), j=1..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 21 2017

b[n_, x_, y_] := b[n, x, y] = If[n == 0, 1, b[n-1, x+1, y]+x*b[n-1, x, y+1]+y*b[n-1, x, y]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)
Table[Sum[BellY[n, k, BellB[Range[n] - 1]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

/* using e.g.f. A(x) */
x = 'x + O('x^66);
B = exp( exp(x) - 1 );  /* e.g.f. of Bell numbers */
A = serconvol( x * B, -log(1-x) );
/* A = intformal(B) */ /* alternative to last line */
A = exp( A );
Vec( serlaplace( A ) )

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1)])
def a(n): return b(n, 2)
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 21 2017, after second Maple program by Alois P. Heinz

Conjecture (confirmed below) about the e.g.f. A(x): Let B(x) = Sum_{n>=0} b(n) * x^n/n! = exp(exp(x)-1) the e.g.f. of the Bell numbers (A000110). Then A(x) = Sum_{n>=0} a(n) * x^n/n! = exp( Sum_{n>=0} b(n) * x^(n+1)/(n+1)! ), see PARI program.
From Joerg Arndt, Jan 14 2013: (Start)
Conjecture (confirmed below):  Let C(0,x) = 1 and for k>=1 C(k, x) = exp( integral(C(k-1,x)) ) then C(k,x) is the e.g.f. for monotonic-labeled forests on n vertices with rooted trees of height less than k (column k of A179455(n,k) for k>=1, n>=k).
For k=1 (C(1,x)=exp(x)) and k=2 (C(2,x)=exp(exp(x)-1)) this is known to be true, for k=3 this is the conjecture above. (End)
From Joerg Arndt, Jan 15 2013: (Start)
Gareth McCaughan, on the math-fun mailing list (Jan 14 2013), writes
"If F is the e.g.f. for Things Of Size n, then exp(F) is the e.g.f. for Multisets Of Things Whose Sizes Add Up To n. (The factorials turn into multinomial coefficients.)
"Which means the conjecture is right. (The integral turns that into 'multisets of things whose sizes plus 1 add up to n'; a tree is a forest together with a new node on top.)"
(End)

A056697 a(n) is least N > 1 congruent to -1,0, or 1 mod i for all i=1,...,n.

Original entry on oeis.org

2, 2, 2, 3, 4, 5, 29, 41, 55, 71, 791, 791, 1079, 1079, 1079, 30239, 246960, 246960, 636481, 636481, 1360800, 2162161, 2162161, 2162161, 39412801, 39412801, 107881201, 107881201, 3625549201, 3625549201, 3625549201, 170918748000, 170918748000, 170918748000, 170918748000, 170918748000, 2355997644001

Offset: 1

Ted Alper, Aug 10 2000

nonn

			a(9) = 55 because 55 gives remainder -1 when divided by 2,4,7 and 8, gives remainder 0 when divided by (1 and) 5, and gives remainder 1 when divided by 3,6 and 9. All smaller integers greater than 1 give remainders other than -1, 0, or 1 for at least one of 5,6,7,8, or 9.

Don Reble, Table of n, a(n) for n = 1..156
Don Reble, Notes on the computation of A056697.

One version of an inverse function to A187824. For another version see A220890. See also A220891. - N. J. A. Sloane, Dec 30 2012

Since n! - 1 == -1 (mod i) for all i = 1..n, a(n) <= n! - 1 for n > 2. - N. J. A. Sloane, Dec 30 2012

More terms from Don Reble, Dec 30 2012

A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

Original entry on oeis.org

245180, 612408, 639198, 1698862, 1721182, 5154168, 7824284, 15817596, 20441848, 25969788, 27688078, 28404862, 35860609, 67149432, 77378782, 91397838, 96462862, 179302264, 191550135, 289772221, 306901244, 311657084, 392802179, 441839706, 572673855, 652117774, 988918364

Offset: 1

Manuel Valdivia, Jan 04 2013

This sequence and A187824 and A187761 are winners in the contest held at the 2013 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 14 2013
The identity sigma(k) = (sopf(k))^m only occurs for m = 3 (this sequence) in the given range, however it is likely that it also occurs for other powers m in higher numbers.
The smallest k such that sigma(k) = sopf(k)^m, for m=4,5,6 are 1056331752 (A221262), 213556659624 (A221263) and 45770980141656, respectively. - Giovanni Resta, Jan 07 2013
Prime divisors are taken without multiplicity. - Harvey P. Dale, Dec 17 2016

			a(13) = 35860609 = 41 * 71 * 97 * 127, then sigma(35860609) = 37933056 = (41 + 71 + 97 + 127)^3.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Donovan Johnson and Robert Gerbicz, Table of n, a(n) for n = 1..1105 (first 100 terms from _Donovan Johnson_)
W. Sierpinski, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Cf. A000203, A008472, A020477, A070222.

Cf. A221262 (sigma(k)=sopf(k)^4), A221263 (sigma(k)=sopf(k)^5).

d[n_]:= If[Plus@@Divisors[n]-Power[Plus@@Select[Divisors[n], PrimeQ], 3]==0, n]; Select[Range[2,10^9], #==d[#]&]
Select[Range[2, 10^9],DivisorSigma[1,#]==Total[FactorInteger[#][[All, 1]]]^3&] (* Harvey P. Dale, Dec 17 2016 *)

is(n)=my(f=factor(n));sum(i=1,#f~,f[i,1])^3==sigma(n) \\ Charles R Greathouse IV, Jun 29 2013

a(n) = k if sigma(k) = (sopf(k))^3, where sigma(k) = A000203(k) and sopf(k) = A008472(k).

A201971 a(n) is the largest m such that n is congruent to -2, -1, 0, 1 or 2 mod k for all k from 1 to m.

Original entry on oeis.org

5, 6, 7, 8, 9, 10, 5, 6, 6, 7, 7, 8, 5, 9, 6, 6, 7, 7, 5, 8, 8, 6, 6, 9, 5, 7, 7, 8, 6, 6, 5, 9, 7, 7, 7, 6, 5, 8, 8, 8, 7, 7, 5, 6, 9, 8, 8, 8, 5, 6, 6, 9, 9, 9, 5, 8, 6, 6, 7, 10, 5, 9, 9, 6, 6, 7, 5, 10, 10, 10, 6, 6, 5, 7, 7, 8, 11, 6, 5, 10, 7, 7, 7, 8, 5

Offset: 3

Robert G. Wilson v, Jan 09 2013

Cf. A187824.

f[n_] := Block[{k = 4, r}, While[r = Mod[n, k]; r < 3 || k - r < 3, k++]; k - 1]; Array[f, 100, 3]

cp's OEIS Frontend

A220890 a(n) = smallest m such that A187824(m) = n, or -1 if A187824 never takes the value n.

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A220891 Where record values occur in A187824.

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PARI

A192129 a(n) is difference of indices of terms >= 9 in A187824.

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A209282 a(n) is difference of indices of terms = 9 in A187824.

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A209283 a(n) is difference of indices of terms >= 6 in A187824.

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A209285 a(n) is difference of indices of terms = 6 in A187824.

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A187761 Number of maps f: [n] -> [n] with f(x)<=x and f(f(x)) = f(f(f(x))).

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A056697 a(n) is least N > 1 congruent to -1,0, or 1 mod i for all i=1,...,n.

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A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

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