Yasutoshi Kohmoto - cp's OEIS frontend

A357895 Number of partitions of the complete graph on n vertices into strokes.

1, 2, 12, 472, 104800

Offset: 1

Yasutoshi Kohmoto and Max Alekseyev, Oct 18 2022

Partition of graph G=(V,E) into strokes is a collection of directed edge-disjoint trails (viewed as sets of directed edges, cf. A357857) in G such that (i) no two trails can be concatenated into a single one; (ii) the corresponding undirected edges form a partition of E.

Cf. A089243, A131518, A131520, A354228, A131709, A357857.

A354228 Number of partitions of the multigraph G_n (defined below) into "strokes".

Original entry on oeis.org

1, 6, 58, 578, 5766, 57810, 580310, 5829538, 58575686, 588641522, 5915670070, 59451845314, 597489270438, 6004768803090, 60348023150742, 606498938168290, 6095328830488582, 61258206225329970, 615646518692614390, 6187263150038580994

Offset: 1

Yasutoshi Kohmoto and Max Alekseyev, Jul 18 2022

nonn

Here G_n = {V_n, E_n}, V_n = {v_1, v_2, ..., v_n}, E_n = {e_1, e_2, ..., e_{n-1}, f_1, f_2, ..., f_{n-1}}. For all i, e_i = f_i = v_iv_{i+1} although e_i and f_i are considered distinct.

Given an undirected multigraph G = (V,E) with labeled edges, its partition into strokes is a collection of directed edge-disjoint paths (viewed as sets of directed edges) on V such that (i) union of any two paths is not a path; (ii) union of corresponding undirected paths is E.

Index entries for linear recurrences with constant coefficients, signature (13,-22,-88,112).

Previously A131519 was thought to be this sequence.

Cf. A131518, A131520.

CoefficientList[Series[x (1-2x)^2(1-3x-14x^2)/(1-13x+22x^2+88x^3-112x^4),{x,0,20}],x] (* or *) LinearRecurrence[{13,-22,-88,112},{0,1,6,58,578,5766},30] (* Harvey P. Dale, Oct 31 2024 *)

For n >= 6, a(n) = 13*a(n-1) - 22*a(n-2) - 88*a(n-3) + 112*a(n-4).

O.g.f.: x * (1-2*x)^2 * (1 - 3*x - 14*x^2) / (1 - 13*x + 22*x^2 + 88*x^3 - 112*x^4).

A355273 Primes p for which p + q is a multiple of 4, where q is the previous prime if p == 2 (mod 3) or the next prime otherwise.

Original entry on oeis.org

3, 5, 29, 31, 53, 59, 61, 73, 89, 137, 139, 149, 151, 157, 173, 179, 181, 191, 239, 241, 251, 257, 263, 269, 271, 283, 293, 331, 337, 347, 359, 367, 373, 389, 409, 419, 421, 431, 433, 449, 509, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 631

Offset: 1

M. F. Hasler and Yasutoshi Kohmoto, Jun 26 2022

nonn

Naively one might expect p + precprime / nextprime congruent to 0 or to 2 (mod 4) with equal probability. It turns out that, following the given rule, the case 2 is much more frequent than the case 0, especially for small primes. (Observation by Y. Kohmoto.)

See the comment from 2017 in A068228 for an explanation.

Cf. A151799 (previous prime), A151800 (next prime).

Cf. A068228.

select( is(p)=if(p%3==2, precprime(p-1)+p, nextprime(p+1)+p)%4==0, primes(149))

from sympy import nextprime
from itertools import islice
def agen():
    p, q = 2, [3, 1]
    while True:
        if (p + q[int(p%3 == 2)])%4 == 0: yield p
        p, q = q[0], [nextprime(q[0]), p]
print(list(islice(agen(), 54))) # Michael S. Branicky, Jun 26 2022

A355272 Primes p for which p + q is not a multiple of 4, where q is the previous prime if p == 1 (mod 3) or else the next prime.

Original entry on oeis.org

2, 89, 97, 211, 223, 359, 367, 389, 397, 401, 409, 449, 457, 467, 479, 487, 491, 499, 509, 631, 673, 683, 691, 701, 709, 719, 727, 743, 751, 761, 769, 797, 887, 907, 911, 919, 929, 937, 983, 991, 1009, 1109, 1117, 1163, 1171, 1193, 1201, 1213, 1249, 1307, 1373

Offset: 1

M. F. Hasler and Yasutoshi Kohmoto, Jun 26 2022

nonn

Naively one might expect p + precprime / nextprime congruent to 0 or to 2 (mod 4) with equal probability. It turns out that, following the given rule, the case 0 is much more frequent than the case 2, especially for small primes. (Observation by Y. Kohmoto.)

See the comment from 2017 in A068228 for an explanation.

Cf. A151799 (previous prime), A151800 (next prime).

Cf. A068228.

select( is(p)=if(p%3==1, precprime(p-1)+p, nextprime(p+1)+p)%4, primes(199))

from sympy import nextprime
from itertools import islice
def agen():
    p, q = 2, [3, 1]
    while True:
        if (p + q[int(p%3 == 1)])%4 != 0: yield p
        p, q = q[0], [nextprime(q[0]), p]
print(list(islice(agen(), 51))) # Michael S. Branicky, Jun 26 2022

A281791 Ways to tile a 5 X (2n+1) floor with tatami mats, including one monomer.

Original entry on oeis.org

3, 18, 10, 8, 18, 24, 32, 52, 68, 100, 142, 196, 280, 388, 542, 756, 1046, 1452, 2006, 2768, 3816, 5248, 7212, 9896, 13562, 18568, 25392, 34692, 47354, 64580, 88002, 119824, 163034, 221672, 301200, 409004, 555060, 752844, 1020550, 1382732, 1872520, 2534596, 3429206, 4637556, 6269070

Offset: 0

Yasutoshi Kohmoto, Jan 30 2017

Apart from a single 1 X 1 monomer, the area is tiled with 2 X 1 mats. No four mats are permitted to meet at a point.

			For n=0, the 5X1 floor allows the monomer to be placed at one of the two ends or in the middle: a(n=0)=3.

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-2,-1).

Cf. A271786 [3X(2n+1) floor]. 2nd column of A272474.

s1(n)=my(s); forstep(k=(n%4!=1),(n-1)\6,2, s+=((n+3)/4-k/2)*((n-1)/4-k/2)!/(k!*((n-1)/4-3*k/2)!)); 2*s
s3(n)=my(s); forstep(k=(n%4==1),(n-3)\6,2, s+=((n-3)/4-k/2)!/(k!*((n-3)/4-3*k/2)!)); 2*s
s5(n)=my(s); forstep(k=(n%4!=1),(n-5)\6,2, s+=((n+7)/4-k/2)*((n-5)/4-k/2)!/(k!*((n-5)/4-3*k/2)!)); 2*s
a(n)=s1(n) + s3(n) + s5(n) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = S_1(2n+1) + S_5(2n+1) + S_3(2n+1) for n>1 where
S_1(n) = 2* Sum_{k= 0<=k<=[(n-1)/6]} ((n+3)/4-1/2*k) *((n-1)/4-1/2*k)!/(k!*((n-1)/4-3/2*k)!). The sum is over even k if n==1 (mod 4), else over odd k.
S_5(n) = 2* Sum_{0<=k<=[(n-5)/6]} ((n+7)/4-1/2*k) *((n-5)/4-1/2*k)!/(k!*((n-5)/4-3/2*k)!). The sum is over even k if n==1 (mod 4) else over odd k.
S_3(n) = 2* Sum_{0<=k<=[(n-3)/6]} 2*((n-3)/4-1/2*k)!/(k!*((n-3)/4-3/2*k)!). The sum is over odd k if n==1 (mod 4), else over even k.
Where [m] is floor(m).
G.f. x +14*x^3 +2*x*(1 +2*x^2 +3*x^4 -2*x^6 -4*x^8 -2*x^10)/ (1-x^4-x^6)^2. (Includes zeros for even floor widths).- R. J. Mathar, Apr 10 2017
a(n) = 2*(A228577(n-1)+A228577(n+1))+4*(A182097(n-2)+A182097(n-1)), n>1. - R. J. Mathar, Apr 10 2017

A145680 a(n) = smallest number m such that UnitarySigma(m) = nm/(n-1).

Original entry on oeis.org

6, 2, 3, 4, 5, 216, 7, 8, 9, 5292000, 11, 10584000, 13, 4991499040640000, 165375, 16, 17, 235270656, 19, 101867327360000, 8107610881081625211441398582431641600000, 19235716742537891017605454376709022095843377283072000000, 23, 552063590295800832, 25

Offset: 2

Yasutoshi Kohmoto Mar 09 2009

nonn

If n=p^i+1 then a(n)=p^i.
Unitary sigma = A034448 = sum of divisors d with gcd(d,m/d)=1.
Note that a(27) > 10^70. - Jack Brennen, Feb 04 2014

			UnitarySigma(216) = 2^2*3^2*7 = (7/6)*216.

Cf. A034448.

Related sequences include A145681 (and A144672, A144673, A144674) for 3/2-URPMS, and A144949-A144951 for 5/4-URPMS. - M. F. Hasler, Jan 29 2014

Corrected a(13) and added a(22) found by Jack Brennen. M. F. Hasler, Feb 04 2014

a(23)-a(26) from Jack Brennen, Feb 04 2014

A146892 For definition see comments lines.

Original entry on oeis.org

1, 6, 6, 72, 72, 72, 6, 72, 72, 5184, 6, 5184, 72, 5184, 31104, 5184, 5184, 5184, 2592, 5184, 432, 373248, 36, 373248, 31104, 26873856, 26873856, 26873856, 373248, 31104, 36, 31104, 2239488, 2239488, 1934917632, 26873856, 31104, 2239488

Offset: 0

Yasutoshi Kohmoto, Apr 17 2009

Let USigma denote the unitary sigma function, A034448.
As in A146891, let PF_p(n) denote the largest power of the prime p dividing n. PF_2 is A006519, and PF_3 is A038500. Furthermore define PF_1(n)=1.
Extension to multi-prime-indices is done by multiplying the corresponding functions: PF_{p,q,..}(n) = PF_p(n)*PF_q(n)*... An example of this is PF_{2,3} = A065331.
[How to compute c(m)]
Case of Base Primes = {2}{3}
c(0)=2^m, b(0)=2^m
c(n)=c(n-1)/PF_2[USigma[b(n-1)]]*PF_3[USigma[b(n-1)]]
b(n)=USigma[b(n-1)]/ PF_2,3[USigma[b(n-1)]]
IF b(k)=1 THEN END
a(m)=c(k)
Sequence gives a(m)
Factorization of term becomes 2^r*3^s.

Cf. A146891.

A146892 := proc(n) local b,a,k ;
   b := [2^n] ;
   while op(-1,b) <> 1 do
       b := [op(b), A065330(A034448(op(-1,b))) ] ;
   od:
   a := 2^n ;
   for k from 2 to nops(b) do
       a := a/ A006519(A034448(op(k-1,b))) *A038500(A034448(op(k-1,b))) ;
   od:
   a ;
end: # R. J. Mathar, Jun 24 2009

More terms from R. J. Mathar, Jun 24 2009

A152199 Trajectory of 7 under the map m -> A082010(m).

Original entry on oeis.org

7, 12, 6, 3, 5, 9, 15, 25, 41, 66, 33, 53, 85, 137, 220, 110, 55, 89, 143, 229, 367, 588, 294, 147, 236, 118, 59, 95, 153, 245, 393, 629, 1007, 1612, 806, 403, 645, 1033, 1653, 2645, 4233, 6773, 10837, 17340, 8670, 4335, 6937, 11100, 5550, 2775, 4441, 7106

Offset: 1

N. J. A. Sloane, Oct 06 2009, suggested by a posting to the Sequence Fans Mailing List by Yasutoshi Kohmoto, Sep 30 2009

nonn

First differences: 5, -6, -3, 2, 4, 6, 10, 16, 25, -33, 20, 32, 52, 83, -110, -55, 34, 54, 86, 138, 221, -294, -147, 89, -118, -59, 36, 58, 92, 148, 236, 378, 605, -806, -403, 242, 388, 620, 992, 1588, 2540, 4064, 6503, -8670, -4335, 2602, 4163, -5550, -2775, 1666, 2665, -3553, 2132, 3412, 5459, -7278, -3639, 2184, 3494, 5591, -7454, -3727, 2237, -2982, -1491, 895, -1193, 716, 1146, 1834, 2934, 4694, 7511, -10014, -5007, 3005, -4006, -2003, 1202, 1924, 3078, 4925, -6566, -3283, 1970, 3152, 5044, 8070, 12912, 20659, -27545, 16528, 26444, 42311, -56414, -28207, 16925, -22566, -11283, 6770. [Zak Seidov, Oct 07 2009]

Robert Israel, Table of n, a(n) for n = 1..10000

A082010 := proc(n) if type(n,'even') then n/2; else 1+floor(8*n/5) ; fi; end: A152199 := proc(n) option remember; if n = 1 then 7; else A082010(procname(n-1)) ; fi; end: seq(A152199(n),n=1..100) ; # R. J. Mathar, Oct 07 2009

NestList[If[EvenQ[ # ], #/2, Floor[8*#/5] + 1] &, 7, 100] (* Zak Seidov, Oct 07 2009 *)

More terms from R. J. Mathar and Zak Seidov, Oct 07 2009

A146891 Terminal point of a repeated reduction of usigma starting at 2^n.

Original entry on oeis.org

1, 6, 20, 72, 72, 72, 20, 72, 72, 17280, 4800, 17280, 72, 17280, 1152000, 5184, 5184, 5184, 96000, 5184, 345600, 1244160, 320000, 1244160, 82944000, 89579520, 71663616000, 298598400, 1244160, 82944000, 23040000, 82944000, 19906560000

Offset: 0

Yasutoshi Kohmoto, Apr 17 2009

nonn

Let PF_p(n) be the highest power of p dividing n. Examples are PF_2(n) = A006519(n), PF_3(n) = A038500(n) and PF_5(n) = 5^A112765(n) for the cases p = 2, 3, and 5.
Multi-indexed PF_(p1,p2,...)(n) are defined as the products PF_(p1)(n)*PF_(p2)(n)*...
For each n, we define an auxiliary sequence b(k) starting at b(0) = 2^n by b(k+1) = A034448(b(k))/PF_(2,3,5)(A034448(b(k)), that is, repeated removal of all powers of 2, 3 and 5 from the unitary sigma value. b(k) terminates at some k with b(k)=1. In addition there is an auxiliary parallel sequence c(k) defined by c(0)=2^n and recursively c(k+1) = c(k)*PF_(3,5)(A034448(b(k)))/A006519(A034448(b(k))), reducing 2^n by the powers of 2 which are divided out of the sequence b.
The sequence is defined by a(n) = c(k), the auxiliary sequence c at the point where b terminates.
All values of the sequence a(n) are 5-smooth, i.e., members of A051037.

			n=5
b(n) : 2^5 -> 11 -> 1
c(n) : 2^5 -> 2^5*3 -> 2^3*3^2
So a(5) = c(2) = 2^3*3^2 = 72.

Cf. A146892, A151659.

PF := proc(n,p) local nshf,a ; a := 1; nshf := n ; while (nshf mod p ) = 0 do nshf := nshf/p ; a := a*p ; od: a ; end:
A146891 := proc(n) local b,a,k,t ;
    b := [2^n] ;
    while op(-1,b) <> 1 do
        t := A034448(op(-1,b)) ;
        b := [op(b), t/A006519(t)/ A038500(t)/PF(t,5) ] ;
    od:
    a := 2^n ;
    for k from 2 to nops(b) do
        t := A034448(op(k-1,b)) ;
        a := a/ A006519(t) *A038500(t)*PF(t,5) ;
     od:
     a ;
end:
# R. J. Mathar, Jun 24 2009

PF[n_, p_] := p^IntegerExponent[n, p];
usigma[n_] := If[n == 1, 1, Times @@ (1+Power @@@ FactorInteger[n])];
A146891[n_] := Module[{b, a, k, t},
   b = {2^n};
   While[b[[-1]] != 1,
      t = usigma[b[[-1]]];
      b = Append[b, t/PF[t, 2]/PF[t, 3]/PF[t, 5]]];
   a = 2^n;
   For[k = 2, k <= Length[b], k++,
      t = usigma[b[[k-1]]];
      a = a/PF[t, 2]*PF[t, 3]*PF[t, 5]];
   a];
Table[A146891[n], {n, 0, 32}] (* Jean-François Alcover, Apr 09 2024, after R. J. Mathar *)

More terms from R. J. Mathar, Jun 24 2009
Edited by R. J. Mathar, Jul 02 2009
Description of relation between a(n) and c(k) corrected by R. J. Mathar, Jul 07 2009

A145681 Numbers m such that A034448(m) = 3m/2, where A034448 = unitary sigma = sum of divisors d with gcd(d,m/d)=1.

Original entry on oeis.org

2, 20, 24, 360, 816, 1056, 12240, 15840, 29120, 181632, 2724480, 9192960, 13790400, 15288000, 93358848, 199180800, 1130734080, 1400382720, 25115166720, 544161945600, 3089165506560, 11519246340096, 172788695101440, 274358081249280, 5944425093734400, 33746043993661440, 7425442423511040000, 25976914334122752000000, 48787315395486187520000, 56897897650906642513920

Offset: 1

Yasutoshi Kohmoto Mar 09 2009

nonn

See also A144672, A144673 and A144674. These three and the present one all differ only by one or two terms around a(11)-a(12). More explanations would be welcome. - M. F. Hasler, Jan 29 2014

Y. Kohmoto, Unitary RMPN, SeqFan list, Jan. 2014

Cf. A034448.

Corrected and extended by Jack Brennen, Jan 30 2014

cp's OEIS Frontend

User: Yasutoshi Kohmoto

A357895 Number of partitions of the complete graph on n vertices into strokes.

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A354228 Number of partitions of the multigraph G_n (defined below) into "strokes".

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Mathematica

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A355273 Primes p for which p + q is a multiple of 4, where q is the previous prime if p == 2 (mod 3) or the next prime otherwise.

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PARI

Python

A355272 Primes p for which p + q is not a multiple of 4, where q is the previous prime if p == 1 (mod 3) or else the next prime.

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Author

Keywords

Comments

Crossrefs

Programs

PARI

Python

A281791 Ways to tile a 5 X (2n+1) floor with tatami mats, including one monomer.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A145680 a(n) = smallest number m such that UnitarySigma(m) = nm/(n-1).

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Keywords

Comments

Examples

Crossrefs

Extensions

A146892 For definition see comments lines.

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Maple

Extensions

A152199 Trajectory of 7 under the map m -> A082010(m).

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Programs

Maple

Mathematica

Extensions

A146891 Terminal point of a repeated reduction of usigma starting at 2^n.

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Comments

Examples