Firdous Ahmad Mala - cp's OEIS frontend

A376064 Number of quasi-orders on an n-set that are not partial orders.

0, 0, 1, 10, 136, 2711, 79504, 3405382, 211055975, 18749246912, 2365988624260, 420564361630293, 104490620009920522, 36030665275081893690, 17132727719926060775277, 11169098098145556139435182, 9930583626219881751366237516, 11985408843042557809380587456695, 19553143146433198202168306753032180

Offset: 0

Firdous Ahmad Mala, Sep 08 2024

nonn

Cf. A000798, A001035.

a[n_]:=Part[ResourceFunction["OEISSequence"]["A000798"],n+1]-Part[ResourceFunction["OEISSequence"]["A001035"],n+1]; Array[a,18,0] (* Stefano Spezia, Sep 08 2024 *)

a(n) = A000798(n) - A001035(n).

A359810 Partial sums of A001035.

Original entry on oeis.org

1, 2, 5, 24, 243, 4474, 134497, 6264356, 437987735, 44949030246, 6656014279029, 1402937691384928, 416267888747238427, 172266996270334297778, 98656591253398541329961, 77665827611694086894379900, 83558195613101851900738636479, 122236099445908424714842019905630, 242061628696647085027612662167990653

Offset: 0

Firdous Ahmad Mala, Jan 13 2023

nonn

Since the number of partial orders is always odd, a(n) alternates in parity.

			a(2) = 1 + 1 + 3 = 5.

Cf. A001035, A354847.

A353179 a(n) is the first nonzero digit in the decimal expansion of 1/prime(n).

Original entry on oeis.org

5, 3, 2, 1, 9, 7, 5, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 8, 7, 7, 7, 7, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Firdous Ahmad Mala, Apr 29 2022

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A052038, A000040.

a:= n-> (p-> floor(10^length(p)/p))(ithprime(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 30 2022

Table[RealDigits[1/Prime[n],10,1][[1]],{n,100}]//Flatten (* Harvey P. Dale, Aug 25 2024 *)

a(n) = my(p=prime(n)); floor(10^(1+logint(p-1, 10))/p) \\ Felix Fröhlich, Apr 29 2022

a(n) = A052038(prime(n)).

More terms from Felix Fröhlich, Apr 29 2022

A348634 Number of transitive relations on an n-set with exactly five ordered pairs.

Original entry on oeis.org

0, 0, 0, 27, 768, 8771, 63468, 340620, 1470784, 5371002, 17153352, 49075521, 128066400, 309124101, 697874996, 1486830618, 3011414784, 5833686340, 10863883728, 19532496375, 34028554944, 57623258007, 95101946940, 153331834040, 241997811264, 374544148830, 569365964440, 851301035325, 1253479866912, 1819599953913, 2606698902276

Offset: 0

Firdous Ahmad Mala, Dec 13 2021

nonn

			No relation containing exactly five ordered pairs on a 2-element set exists. Thus a(2)=0.
Also, there are 27 transitive relations with exactly five ordered pairs on a 3-set. One such relation is {(1,1),(1,2),(1,3),(2,2),(3,2)} on the 3-set {1,2,3}.

Cf. A349919, A349927, A349849, A006905.

def A348634(n): return n*(n - 2)*(n - 1)*(n*(n*(n*(n*(n*(n*(n - 17) + 167) - 965) + 3481) - 7581) + 9060) - 4608)//120 # Chai Wah Wu, Jan 06 2022

a(n) = 27*C(n,3) + 660*C(n,4) + 5201*C(n,5) + 21822*C(n,6) + 54600*C(n,7) + 84000*C(n,8) + 75600*C(n,9) + 30240*C(n,10).
a(n) = (1/120)*(n^10 - 20*n^9 + 220*n^8 - 1500*n^7 + 6710*n^6 - 19954*n^5 + 38765*n^4 - 46950*n^3 + 31944*n^2 - 9216*n).
a(n) = C(n,3)*(n^7 - 17*n^6 + 167*n^5 - 965*n^4 + 3481*n^3 - 7581*n^2 + 9060*n - 4608)/20. - Chai Wah Wu, Jan 06 2022

a(9) corrected by Georg Fischer, Mar 19 2023

A350159 Number of subgroups of the dicyclic group Dic_n.

Original entry on oeis.org

3, 6, 8, 11, 10, 18, 12, 20, 19, 24, 16, 36, 18, 30, 32, 37, 22, 48, 24, 50, 40, 42, 28, 70, 37, 48, 48, 64, 34, 84, 36, 70, 56, 60, 56, 103, 42, 66, 64, 100, 46, 108, 48, 92, 90, 78, 52, 136, 63, 102, 80, 106, 58, 132, 80, 130, 88, 96, 64, 184, 66, 102, 116

Offset: 1

Firdous Ahmad Mala, Dec 17 2021

nonn

			a(2) = A000005(4) + A000203(2) = 3+3 = 6.
Given the fact that Dic_2 is isomorphic to the quaternion group Q_8, the subgroups of Dic_2 are isomorphic to the subgroups of Q_8 which are {1}, {1,-1}, {1,i,-1,-i}, {1,j,-1,-j}, {1,k,-1,-k} and Q_8.

Hayder Baqer Shelash and A. R. Ashrafi, The Number of Subgroups of a Given Type in Certain Finite Groups, Iranian Journal of Mathematical Sciences and Informatics, Vol. 16, No. 2 (2021), pp. 73-87.
Wikipedia, Dicyclic group

Cf. A000005, A000203, A099777, A345628.

a[n_] := DivisorSigma[0, 2*n] + DivisorSigma[1, n]; Array[a, 50] (* Amiram Eldar, Dec 17 2021 *)

a(n) = numdiv(2*n) + sigma(n); \\ Michel Marcus, Dec 18 2021

a(n) = A000005(2n) + A000203(n) = A099777(n) + A000203(n).

A349919 Number of transitive relations on an n-set with exactly two ordered pairs.

Original entry on oeis.org

0, 0, 5, 27, 90, 230, 495, 945, 1652, 2700, 4185, 6215, 8910, 12402, 16835, 22365, 29160, 37400, 47277, 58995, 72770, 88830, 107415, 128777, 153180, 180900, 212225, 247455, 286902, 330890, 379755, 433845, 493520, 559152, 631125, 709835, 795690, 889110, 990527, 1100385, 1219140, 1347260, 1485225, 1633527, 1792670

Offset: 0

Firdous Ahmad Mala, Dec 05 2021

			a(2) = 5. The five relations on a 2-set are {(1,1),(1,2)}, {(1,1),(2,1)}, {(1,1),(2,2)}, {(1,2),(2,2)} and {(2,1),(2,2)}.

Firdous Ahmad Mala, Counting Transitive Relations with Two Ordered Pairs, Journal of Applied Mathematics and Computation, 5(4), 247-251.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A006905, A336535, A349927.

This is a diagonal of the array A285192.

LinearRecurrence[{5,-10,10,-5,1},{0,0,5,27,90},50] (* Harvey P. Dale, Oct 23 2022 *)

a(n) = 5*C(n,2) + 12*C(n,3) + 12*C(n,4).
a(n) = (1/2)*(n^4 - 2*n^3 + 4*n^2 - 3*n).
a(n) = A336535(n) - 1.
From Elmo R. Oliveira, Aug 26 2025: (Start)
G.f.: x^2*(5 + 2*x + 5*x^2)/(1 - x)^5.
E.g.f.: x^2*(5 + 4*x + x^2)*exp(x)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

A349927 Number of transitive relations on an n-set with exactly three ordered pairs.

Original entry on oeis.org

0, 0, 2, 43, 276, 1150, 3710, 10017, 23688, 50556, 99450, 183095, 319132, 531258, 850486, 1316525, 1979280, 2900472, 4155378, 5834691, 8046500, 10918390, 14599662, 19263673, 25110296, 32368500, 41299050, 52197327, 65396268, 81269426, 100234150

Offset: 0

Firdous Ahmad Mala, Dec 05 2021

			a(2) = 2. These two transitive relations are {(1,1),(1,2),(2,2)} and {(1,1),(2,1),(2,2)} on the 2-set {1,2}.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A349919, A349849.

A349927(n) = (1/6)*(n^6 - 6*n^5 + 24*n^4 - 47*n^3 + 38*n^2 - 10*n); \\ Antti Karttunen, Dec 05 2021

a(n) = 2*C(n,2) + 37*C(n,3) + 116*C(n,4) + 180*C(n,5) + 120*C(n,6).

a(n) = (1/6)*(n^6 - 6*n^5 + 24*n^4 - 47*n^3 + 38*n^2 - 10*n).

A349849 Number of transitive relations on an n-set with exactly four ordered pairs.

Original entry on oeis.org

0, 0, 1, 45, 549, 3755, 18120, 69006, 220710, 616554, 1545435, 3544915, 7552611, 15119325, 28699034, 52032540, 90643260, 152465316, 248625765, 394404489, 610396945, 923906655, 1370595996, 1996425530, 2859913794, 4034751150, 5612802975, 7707539151, 10457928495

Offset: 0

Firdous Ahmad Mala, Dec 06 2021

			a(2) = binomial(2,2) = 1. The only transitive relation with four ordered pairs on the 2-set {1,2} is {(1,1),(1,2),(2,1),(2,2)}.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A349919, A349927, A006905.

a(n) = C(n,2) + 42*C(n,3) + 375*C(n,4) + 1450*C(n,5) + 2940*C(n,6) + 3360*C(n,7) + 1680*C(n,8).

a(n) = (1/24)*(n^8 - 12*n^7 + 84*n^6 - 340*n^5 + 814*n^4 - 1130*n^3 + 829*n^2 - 246*n).

A349848 a(n) = Sum_{k=1..n} prime(n)^prime(k).

Original entry on oeis.org

4, 36, 3275, 840742, 285331320285, 304667330108466, 827250200736677741479, 1983900084687573008820254, 20880542756369384174903669400953, 2567686157477937962829648585022637187631942, 17086936018496343189927728440572423322828545911

Offset: 1

Firdous Ahmad Mala, Dec 02 2021

nonn

			For n=3, p=5, a(n) = 5^2 + 5^3 + 5^5 = 25 + 125 + 3125 = 3275.

a(n) = my(p=prime(n)); sum(k=1, n, p^prime(k)); \\ Michel Marcus, Dec 02 2021

from sympy import prime, primerange
def a(n): pn = prime(n); return sum(pn**pk for pk in primerange(1, pn+1))
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Dec 02 2021

More terms from Michel Marcus, Dec 02 2021

A348151 First differences of A006905.

Original entry on oeis.org

1, 11, 158, 3823, 150309, 9260886, 868807341, 121329481093, 24768540218324, 7282559551588341, 3046214096033592769, 1793950037677437221180, 1474265690307795355749075, 1677763495455703210030729685, 2626545934585707736394538773674, 5623547642339635201400382321016283

Offset: 1

Firdous Ahmad Mala, Oct 03 2021

nonn

Number of transitive relations involving a particular element of an n-set.

			a(3) = A006905(3) - A006905(2) = 171 - 13 = 158.

Firdous Ahmad Mala, Some New Integer Sequences of Transitive Relations, J. Appl. Math. Comp. (2023) Vol. 7, No. 1, 108-111.

Cf. A006905.

a(n) = A006905(n) - A006905(n-1).

cp's OEIS Frontend

User: Firdous Ahmad Mala

A376064 Number of quasi-orders on an n-set that are not partial orders.

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A359810 Partial sums of A001035.

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A353179 a(n) is the first nonzero digit in the decimal expansion of 1/prime(n).

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A348634 Number of transitive relations on an n-set with exactly five ordered pairs.

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A350159 Number of subgroups of the dicyclic group Dic_n.

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A349919 Number of transitive relations on an n-set with exactly two ordered pairs.

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A349927 Number of transitive relations on an n-set with exactly three ordered pairs.

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A349849 Number of transitive relations on an n-set with exactly four ordered pairs.

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A349848 a(n) = Sum_{k=1..n} prime(n)^prime(k).

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