A014741 -id:A014741 - cp's OEIS frontend

A125227 A014741(n)/6 for n>2.

1, 3, 7, 9, 21, 27, 49, 57, 63, 81, 147, 171, 189, 219, 243, 301, 343, 399, 441, 513, 567, 657, 729, 889, 903, 1029, 1083, 1197, 1323, 1533, 1539, 1701, 1971, 2107, 2187, 2359, 2401, 2667, 2709, 2793, 3087, 3249, 3591, 3969, 4161, 4401, 4599, 4617, 5103, 5913

Offset: 3

Alexander Adamchuk, Jan 15 2007

nonn

A014741(n) is divisible by 6 for n>2.
All powers of 3 are terms. All powers of 7 are terms. The prime divisors of terms of this sequence (for n up to 10^6) in order of their first appearance are 3, 7, 19, 73, 43, 127, 337, 163, 379, 571, 5419, 487, 2593, 439, 1459, 431, 883.
The sequence is multiplicative in the sense that if two numbers k and m are terms, then k*m is too.

			a(3) = A014741(3)/6 = 6/6 = 1.
a(4) = A014741(4)/6 = 18/6 = 3.

Amiram Eldar, Table of n, a(n) for n = 3..10000

Cf. A014741, A000295, A086787.

Select[Range[3,6000], PowerMod[2,#+1,# ]==2&]/6

a(n) = A014741(n)/6 = A014945(n-1)/3. - Max Alekseyev, Nov 14 2012

A000295 Eulerian numbers (Euler's triangle: column k=2 of A008292, column k=1 of A173018).

Original entry on oeis.org

0, 0, 1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519, 131054, 262125, 524268, 1048555, 2097130, 4194281, 8388584, 16777191, 33554406, 67108837, 134217700, 268435427, 536870882, 1073741793, 2147483616, 4294967263, 8589934558

Offset: 0

N. J. A. Sloane

There are 2 versions of Euler's triangle:
* A008292 Classic version of Euler's triangle used by Comtet (1974).
* A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
Euler's triangle rows and columns indexing conventions:
* A008292 The rows and columns of the Eulerian triangle are both indexed starting from 1. (Classic version: used in the classic books by Riordan and Comtet.)
* A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0. (Graham et al.)
Number of Dyck paths of semilength n having exactly one long ascent (i.e., ascent of length at least two). Example: a(4)=11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having exactly one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004
Number of permutations of {1,2,...,n} with exactly one descent (i.e., permutations (p(1),p(2),...,p(n)) such that #{i: p(i)>p(i+1)}=1). E.g., a(3)=4 because the permutations of {1,2,3} with one descent are 132, 213, 231 and 312.
a(n+1) is the convolution of nonnegative integers (A001477) and powers of two (A000079). - Graeme McRae, Jun 07 2006
Partial sum of main diagonal of A125127. - Jonathan Vos Post, Nov 22 2006
Number of partitions of an n-set having exactly one block of size > 1. Example: a(4)=11 because, if the partitioned set is {1,2,3,4}, then we have 1234, 123|4, 124|3, 134|2, 1|234, 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34. - Emeric Deutsch, Oct 28 2006
k divides a(k+1) for k in A014741. - Alexander Adamchuk, Nov 03 2006
(Number of permutations avoiding patterns 321, 2413, 3412, 21534) minus one. - Jean-Luc Baril, Nov 01 2007, Mar 21 2008
The chromatic invariant of the prism graph P_n for n >= 3. - Jonathan Vos Post, Aug 29 2008
Decimal integer corresponding to the result of XORing the binary representation of 2^n - 1 and the binary representation of n with leading zeros. This sequence and a few others are syntactically similar. For n > 0, let D(n) denote the decimal integer corresponding to the binary number having n consecutive 1's. Then D(n).OP.n represents the n-th term of a sequence when .OP. stands for a binary operator such as '+', '-', '*', 'quotentof', 'mod', 'choose'. We then get the various sequences A136556, A082495, A082482, A066524, A000295, A052944. Another syntactically similar sequence results when we take the n-th term as f(D(n)).OP.f(n). For example if f='factorial' and .OP.='/', we get (A136556)(A000295) ; if f='squaring' and .OP.='-', we get (A000295)(A052944). - K.V.Iyer, Mar 30 2009
Chromatic invariant of the prism graph Y_n.
Number of labelings of a full binary tree of height n-1, such that each path from root to any leaf contains each label from {1,2,...,n-1} exactly once. - Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009
Also number of nontrivial equivalence classes generated by the weak associative law X((YZ)T)=(X(YZ))T on words with n open and n closed parentheses. Also the number of join (resp. meet)-irreducible elements in the pruning-grafting lattice of binary trees with n leaves. - Jean Pallo, Jan 08 2010
Nonzero terms of this sequence can be found from the row sums of the third sub-triangle extracted from Pascal's triangle as indicated below by braces:
                     1;
                  1,    1;
              {1},   2,    1;
           {1,    3},   3,    1;
        {1,    4,    6},   4,    1;
     {1,    5,   10,   10},   5,    1;
  {1,    6,   15,   20,   15},   6,    1;
  ... - L. Edson Jeffery, Dec 28 2011
For integers a, b, denote by a<+>b the least c >= a, such that the Hamming distance D(a,c) = b (note that, generally speaking, a<+>b differs from b<+>a). Then for n >= 3, a(n) = n<+>n. This has a simple explanation: for n >= 3 in binary we have a(n) = (2^n-1)-n = "anti n". - Vladimir Shevelev, Feb 14 2012
a(n) is the number of binary sequences of length n having at least one pair 01. - Branko Curgus, May 23 2012
Nonzero terms are those integers k for which there exists a perfect (Hamming) error-correcting code. - L. Edson Jeffery, Nov 28 2012
a(n) is the number of length n binary words constructed in the following manner:  Select two positions in which to place the first two 0's of the word.  Fill in all (possibly none) of the positions before the second 0 with 1's and then complete the word with an arbitrary string of 0's or 1's. So a(n) = Sum_{k=2..n} (k-1)*2^(n-k). - Geoffrey Critzer, Dec 12 2013
Without first 0: a(n)/2^n equals Sum_{k=0..n} k/2^k. For example: a(5)=57, 57/32 = 0/1 + 1/2 + 2/4 + 3/8 + 4/16 + 5/32. - Bob Selcoe, Feb 25 2014
The first barycentric coordinate of the centroid of the first n rows of Pascal's triangle, assuming the numbers are weights, is A000295(n+1)/A000337(n). See attached figure. - César Eliud Lozada, Nov 14 2014
Starting (0, 1, 4, 11, ...), this is the binomial transform of (0, 1, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015
Also the number of (non-null) connected induced subgraphs in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Aug 27 2017
a(n) is the number of swaps needed in the worst case to transform a binary tree with n full levels into a heap, using (bottom-up) heapify. - Rudy van Vliet, Sep 19 2017
The utility of large networks, particularly social networks, with n participants is given by the terms a(n) of this sequence. This assertion is known as Reed's Law, see the Wikipedia link. - Johannes W. Meijer, Jun 03 2019
a(n-1) is the number of subsets of {1..n} in which the largest element of the set exceeds by at least 2 the next largest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,2,4}, {1,2,5}, {1,3,5}, {2,3,5}, {1,2,3,5}. - Enrique Navarrete, Apr 08 2020
a(n-1) is also the number of subsets of {1..n} in which the second smallest element of the set exceeds by at least 2 the smallest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,4,5}, {1,3,4,5}. - Enrique Navarrete, Apr 09 2020
a(n+1) is the sum of the smallest elements of all subsets of {1..n}. For example, for n=3, a(4)=11; the subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, and the sum of smallest elements is 11. - Enrique Navarrete, Aug 20 2020
Number of subsets of an n-set that have more than one element. - Eric M. Schmidt, Mar 13 2021
Number of individual bets in a "full cover" bet on n-1 horses, dogs, etc. in different races. Each horse, etc. can be bet on or not, giving 2^n bets.  But, by convention, singles (a bet on only one race) are not included, reducing the total number bets by n.  It is also impossible to bet on no horses at all, reducing the number of bets by another 1.  A full cover on 4 horses, dogs, etc. is therefore 6 doubles, 4 trebles and 1 four-horse etc. accumulator.  In British betting, such a bet on 4 horses etc. is a Yankee; on 5, a super-Yankee. - Paul Duckett, Nov 17 2021
From Enrique Navarrete, May 25 2022: (Start)
Number of binary sequences of length n with at least two 1's.
a(n-1) is the number of ways to choose an odd number of elements greater than or equal to 3 out of n elements.
a(n+1) is the number of ways to split [n] = {1,2,...,n} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n} and then select a subset from the first interval (2^i choices, 0 <= i <= n), and one block/cell (i.e., subinterval) from the second interval (n-i choices, 0 <= i <= n).
(End)
Number of possible conjunctions in a system of n planets; for example, there can be 0 conjunctions with one planet, one with two planets, four with three planets (three pairs of planets plus one with all three) and so on. - Wendy Appleby, Jan 02 2023
Largest exponent m such that 2^m divides (2^n-1)!. - Franz Vrabec, Aug 18 2023
It seems that a(n-1) is the number of odd r with 0 < r < 2^n for which there exist u,v,w in the x-independent beginning of the Collatz trajectory of 2^n x + r with u+v = w+1, as detailed in the link "Collatz iteration and Euler numbers?". A better understanding of this might also give a formula for A374527. - Markus Sigg, Aug 02 2024
This sequence has a connection to consecutively halved positional voting (CHPV); see Mendenhall and Switkay. - Hal M. Switkay, Feb 25 2025
a(n) is the number of subsets of size 2 and more of an n-element set. Equivalently, a(n) is the number of (hyper)edges of size 2 and more in a complete hypergraph of n vertices. - Yigit Oktar, Apr 05 2025

			G.f. = x^2 + 4*x^3 + 11*x^4 + 26*x^5 + 57*x^6 + 120*x^7 + 247*x^8 + 502*x^9 + ...

O. Bottema, Problem #562, Nieuw Archief voor Wiskunde, 28 (1980) 115.
L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." Section 6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 51 and 240-246, 1974.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 34.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 17, 18.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Jean-Luc Baril and J. M. Pallo, The pruning-grafting lattice of binary trees, Theoretical Computer Science, 409, 2008, 382-393.
Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See p. 7.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Maximilien Gadouleau, James D. Mitchell, and Yann Peresse, Chains of subsemigroups, arXiv preprint arXiv:1501.06394 [math.GR], 2015. See Table 4.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Shelby Cox, Pratik Misra, and Pardis Semnani, Homaloidal Polynomials and Gaussian Models of Maximum Likelihood Degree One, arXiv:2402.06090 [math.AG], 2024.
Benjamin Hellouin de Menibus and Yvan Le Borgne, Asymptotic behaviour of the one-dimensional "rock-paper-scissors" cyclic cellular automaton, arXiv:1903.12622 [math.PR], 2019.
Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019.
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
J. M. Dusel, Balanced parabolic quotients and branching rules for Demazure crystals, J Algebr Comb (2016) 44: 363. DOI: 10.1007/s10801-016-0673-y.
Pascal Floquet, Serge Domenech and Luc Pibouleau, Combinatorics of Sharp Separation System synthesis : Generating functions and Search Efficiency Criterion, Industrial Engineering and Chemistry Research, 33, pp. 440-443, 1994.
Pascal Floquet, Serge Domenech, Luc Pibouleau and Said Aly, Some Complements in Combinatorics of Sharp Separation System Synthesis, American Institute of Chemical Engineering Journal, 39(6), pp. 975-978, 1993.
E. T. Frankel, A calculus of figurate numbers and finite differences, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]
Joël Gay, Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
R. K. Guy, Letter to N. J. A. Sloane.
Ryota Inagaki, Tanya Khovanova, and Austin Luo, On Chip-Firing on Undirected Binary Trees, Ann. Comb. (2025). See pp. 24-25.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 388.
Wayne A. Johnson, An Euler operator approach to Ehrhart series, arXiv:2303.16991 [math.CO], 2023.
Lucas Kang, Investigation of Rule 73 as Case Study of Class 4 Long-Distance Cellular Automata, arXiv preprint arXiv:1310.3311 [nlin.CG], 2013.
Oliver Kullmann and Xishun Zhao, Bounds for variables with few occurrences in conjunctive normal forms, arXiv preprint arXiv:1408.0629 [math.CO], 2014.
César Eliud Lozada, Centroids of Pascal triangles
Candice A. Marshall, Construction of Pseudo-Involutions in the Riordan Group, Dissertation, Morgan State University, 2017.
Peter Charles Mendenhall and Hal M. Switkay, Consecutively Halved Positional Voting: A Special Case of Geometric Voting, Social Sciences vol. 12 no. 2 (2023), 47-59.
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
J. W. Moon, A problem on arcs without bypasses in tournaments, J. Combinatorial Theory Ser. B 21 (1976), no. 1, 71--75. MR0427129(55 #165).
Agustín Moreno Cañadas, Hernán Giraldo, and Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Nagatomo Nakamura, Pseudo-Normal Random Number Generation via the Eulerian Numbers, Josai Mathematical Monographs, vol 8, p 85-95, 2015.
Emily Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
J. M. Pallo, Weak associativity and restricted rotation, Information Processing Letters, 109, 2009, 514-517.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. Riordan, Review of Frankel (1950) [Annotated scanned copy]
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 20 (1968), 8-16.
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
Markus Sigg, Collatz iteration and Euler numbers?
Eric Weisstein's World of Mathematics, Chromatic Invariant
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Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A008292 (classic version of Euler's triangle used by Comtet (1974)).
Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990)).
Cf. A002662 (partial sums).
Partial sums of A000225.
Row sums of A014473 and of A143291.
Second column of triangles A112493 and A112500.
Sequences A125128 and A130103 are essentially the same.
Column k=1 of A124324.
Cf. A000079, A000108, A000217, A000325, A000975, A001511, A002663.
Cf. A002664, A007814, A008292, A008949, A014741, A016031, A028366.
Cf. A035039, A035040, A035041, A035042, A079583, A107907, A130321.
Cf. A130128, A130330, A131768, A131816.

a000295 n = 2^n - n - 1  -- Reinhard Zumkeller, Nov 25 2013

[2^n-n-1: n in [0..40]]; // Vincenzo Librandi, Jul 29 2015

[EulerianNumber(n, 1): n in [0..40]]; // G. C. Greubel, Oct 02 2024

[ seq(2^n-n-1, n=1..50) ];
A000295 := -z/(2*z-1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation
# Grammar specification:
spec := [S, { B = Set(Z, 1 <= card), C = Sequence(B, 2 <= card), S = Prod(B, C) }, unlabeled]:
struct := n -> combstruct[count](spec, size = n+1);
seq(struct(n), n = 0..33); # Peter Luschny, Jul 22 2014

a[n_] = If[n==0, 0, n*(HypergeometricPFQ[{1, 1-n}, {2}, -1] - 1)];
Table[a[n], {n,0,40}] (* Olivier Gérard, Mar 29 2011 *)
LinearRecurrence[{4, -5, 2}, {0, 0, 1}, 40] (* Vincenzo Librandi, Jul 29 2015 *)
Table[2^n -n-1, {n,0,40}] (* Eric W. Weisstein, Nov 16 2017 *)

a(n)=2^n-n-1 \\ Charles R Greathouse IV, Jun 10 2011

[2^n -(n+1) for n in range(41)] # G. C. Greubel, Oct 02 2024

a(n) = 2^n - n - 1.
G.f.: x^2/((1-2*x)*(1-x)^2).
A107907(a(n+2)) = A000079(n+2). - Reinhard Zumkeller, May 28 2005
E.g.f.: exp(x)*(exp(x)-1-x). - Emeric Deutsch, Oct 28 2006
a(0)=0, a(1)=0, a(n) = 3*a(n-1) - 2*a(n-2) + 1. - Miklos Kristof, Mar 09 2005
a(0)=0, a(n) = 2*a(n-1) + n - 1 for all n in Z.
a(n) = Sum_{k=2..n} binomial(n, k). - Paul Barry, Jun 05 2003
a(n+1) = Sum_{i=1..n} Sum_{j=1..i} C(i, j). - Benoit Cloitre, Sep 07 2003
a(n+1) = 2^n*Sum_{k=0..n} k/2^k. - Benoit Cloitre, Oct 26 2003
a(0)=0, a(1)=0, a(n) = Sum_{i=0..n-1} i+a(i) for i > 1. - Gerald McGarvey, Jun 12 2004
a(n+1) = Sum_{k=0..n} (n-k)*2^k. - Paul Barry, Jul 29 2004
a(n) = Sum_{k=0..n} binomial(n, k+2); a(n+2) = Sum_{k=0..n} binomial(n+2, k+2). - Paul Barry, Aug 23 2004
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k+1)*2^(n-k-2)*(-1/2)^k. - Paul Barry, Oct 25 2004
a(0) = 0; a(n) = Stirling2(n,2) + a(n-1) = A000225(n-1) + a(n-1). - Thomas Wieder, Feb 18 2007
a(n) = A000325(n) - 1. - Jonathan Vos Post, Aug 29 2008
a(0) = 0, a(n) = Sum_{k=0..n-1} 2^k - 1. - Doug Bell, Jan 19 2009
a(n) = A000217(n-1) + A002662(n) for n>0. - Geoffrey Critzer, Feb 11 2009
a(n) = A000225(n) - n. - Zerinvary Lajos, May 29 2009
a(n) = n*(2F1([1,1-n],[2],-1) - 1). - Olivier Gérard, Mar 29 2011
Column k=1 of A173018 starts a'(n) = 0, 1, 4, 11, ... and has the hypergeometric representation n*hypergeom([1, -n+1], [-n], 2). This can be seen as a formal argument to prefer Euler's A173018 over A008292. - Peter Luschny, Sep 19 2014
E.g.f.: exp(x)*(exp(x)-1-x); this is U(0) where U(k) = 1 - x/(2^k - 2^k/(x + 1 - x^2*2^(k+1)/(x*2^(k+1) - (k+1)/U(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n) = A079583(n) - A000225(n+1). - Miquel Cerda, Dec 25 2016
a(0) = 0; a(1) = 0; for n > 1: a(n) = Sum_{i=1..2^(n-1)-1} A001511(i). - David Siegers, Feb 26 2019
a(n) = A007814(A028366(n)). - Franz Vrabec, Aug 18 2023
a(n) = Sum_{k=1..floor((n+1)/2)} binomial(n+1, 2*k+1). - Taras Goy, Jan 02 2025

A014945 Numbers k such that k divides 4^k - 1.

Original entry on oeis.org

1, 3, 9, 21, 27, 63, 81, 147, 171, 189, 243, 441, 513, 567, 657, 729, 903, 1029, 1197, 1323, 1539, 1701, 1971, 2187, 2667, 2709, 3087, 3249, 3591, 3969, 4599, 4617, 5103, 5913, 6321, 6561, 7077, 7203, 8001, 8127, 8379, 9261, 9747, 10773, 11907, 12483

Offset: 1

Olivier Gérard

nonn

This sequence is closed under multiplication. - Charles R Greathouse IV, Nov 03 2016
Conjecture: if k divides 4^k - 1, then (4^k - 1)/k is squarefree. - Thomas Ordowski, Dec 24 2018
Following Greathouse's comment, see A323203 for the primitive terms. - Bernard Schott, Jan 03 2019
All terms except 1 are divisible by 3.  Proof: suppose n>1 is in the sequence, and let p be its smallest prime factor.  Of course p is odd.  Since 4^n-1 is divisible by p, n is divisible by the multiplicative order of 4 mod p, which is less than p.  But since n has no prime factors < p, that multiplicative order can only be 1, which means p=3. - Robert Israel, Jan 24 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..872 from Muniru A Asiru, terms 873..2000 from Alois P. Heinz)

Cf. A014741, A323203.

a:=Filtered([1..13000],n->(4^n-1) mod n=0);; Print(a); # Muniru A Asiru, Dec 28 2018

[n: n in [1..12500] | (4^n-1) mod n eq 0 ]; // Vincenzo Librandi, Dec 29 2018

select(n->modp(4^n-1,n)=0,[$1..13000]); # Muniru A Asiru, Dec 28 2018

Select[Range[12500],Divisible[4^#-1,#]&]  (* Harvey P. Dale, Mar 23 2011 *)

is(n)=Mod(4,n)^n==1 \\ Charles R Greathouse IV, Nov 03 2016

for n in range(1,1000):
    if (4**n-1) % n ==0:
        print(n, end=', ') # Stefano Spezia, Jan 05 2019

a(n) = A014741(n+1)/2.

More terms and better description from Benoit Cloitre, Mar 05 2002

A086787 a(n) = Sum_{i=1..n} ( Sum_{j=1..n} i^j ).

Original entry on oeis.org

1, 8, 56, 494, 5699, 82200, 1419760, 28501116, 651233661, 16676686696, 472883843992, 14705395791306, 497538872883727, 18193397941038736, 714950006521386976, 30046260016074301944, 1344648068888240941017

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Aug 04 2003

nonn

p divides a(p+1) for all prime p except 3. p^2 divides a(p+1) for prime p in A123374.
2 divides a(n) for n = {2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, ...}.
2^2 divides a(n) for n = {2, 3, 6, 7, 8, 10, 11, 14, 15, 16, 18, 19, 22, 23, 24, 26, 27, 30, 31, 32, 34, 35, 38, 39, 40, 42, 43, 46, 47, 48, 50, ...}.
2^3 divides a(n) for n = {2, 3, 6, 7, 10, 11, 14, 15, 16, 18, 19, 22, 23, 26, 27, 30, 31, 32, 34, 35, 38, 39, 42, 43, 46, 47, 48, 50, ...}.
2^4 divides a(n) for n = {7, 14, 15, 18, 23, 30, 31, 32, 34, 39, 46, 47, 50, ...}.
2^5 divides a(n) for n = {15, 30, 31, 34, 47, 62, 63, 64, 66, 79, 94, 95, 98, ...}.
2^6 divides a(n) for n = {31, 62, 63, 66, 95, ...}.
2^7 divides a(n) for n = {63, 126, 127, 130, ...}.
It appears that for k > 2 the least few n such that a(n) is divisible by 2^(k+1) are n = {(2^k-1), 2*(2^k-1), 2*(2^k-1)+1, 2*(2^k-1)+3, 3*(2^k-1)+2, 4*(2^k-1)+2, 4*(2^k-1)+3, 4*(2^k-1)+4, 4*(2^k-1)+6, 5*(2^k-1)+4, 6*(2^k-1)+4, 6*(2^k-1)+5, 6*(2^k-1)+8, ...}. - Alexander Adamchuk, Oct 08 2006
Numbers n that divide a(n) are listed in A014741. - Alexander Adamchuk, Nov 03 2006

			a(2) = 8 = 1 + 1 + 2 + 4 = 1^1 + 1^2 + 2^1 + 2^2.

G. C. Greubel, Table of n, a(n) for n = 1..385

Cf. A000295, A014741, A215084, A349117.

seq(1-Psi(n)-gamma+sum(i^(n+1)/(i-1),i = 2 .. n),n=1..20);

Table[Sum[i^j,{i,1,n},{j,1,n}],{n,1,24}] (* Alexander Adamchuk, Oct 08 2006 *)
Table[ n + Sum[ i*(i^n-1)/(i-1), {i,2,n} ], {n,1,17} ] (* Alexander Adamchuk, Nov 03 2006 *)

a(n)=sum(i=1,n,sum(j=1,n,i^j)) \\ Charles R Greathouse IV, Jul 19 2013

a(n)=round(1-psi(n)-Euler+sum(i=2,n,i^(n+1)/(i-1))) \\ Charles R Greathouse IV, Jul 19 2013

def A086787(n): return sum(i**j for i in range(1,n+1) for j in range(1,n+1)) # Chai Wah Wu, Jan 08 2022

from sympy import digamma, EulerGamma
from fractions import Fraction
def A086787(n): return 1-digamma(n)-EulerGamma + sum(Fraction(i**(n+1),i-1) for i in range(2,n+1)) # Chai Wah Wu, Jan 08 2022

1 - Psi(n) - gamma + Sum_{i=2..n} (i^(n+1)/(i-1)), where Psi(n) is the digamma function and gamma is Euler's constant.
a(n) = Sum[ i^j, {i,1,n}, {j,1,n} ] = n + Sum[ i*(i^n - 1)/(i - 1), {i,2,n} ]. - Alexander Adamchuk, Nov 03 2006
a(n) = Sum_{k=1..n} (B(k+1, n+1) - B(k+1, 1))/(k+1), where B(n, x) are the Bernoulli polynomials. - Daniel Suteu, Jun 25 2018
a(n) ~ c * n^n, where c = 1 / (1 - exp(-1)) = A185393 = 1.58197670686932642438... - Vaclav Kotesovec, Nov 06 2021

Edited by Max Alekseyev, Jan 29 2012

A069051 Primes p such that p-1 divides 2^p-2.

Original entry on oeis.org

2, 3, 7, 19, 43, 127, 163, 379, 487, 883, 1459, 2647, 3079, 3943, 5419, 9199, 11827, 14407, 16759, 18523, 24967, 26407, 37339, 39367, 42463, 71443, 77659, 95923, 99079, 113779, 117307, 143263, 174763, 175447, 184843, 265483, 304039, 308827

Offset: 1

Benoit Cloitre, Apr 03 2002

nonn

These are the prime values of n such that 2^n == 2 (mod n*(n-1)). - V. Raman, Sep 17 2012
These are the prime values p such that n^(2^(p-1)) is congruent to n or -n (mod p) for all n in Z/pZ, the commutative ring associated with each term. This results follows from Fermat's little theorem. - Philip A. Hoskins, Feb 08 2013
A prime p is in this sequence iff p-1 belongs to A014741. For p>2, this is equivalent to (p-1)/2 belonging to A014945. - Max Alekseyev, Aug 31 2016
From Thomas Ordowski, Nov 20 2018: (Start)
Conjecture: if n-1 divides 2^n-2, then (2^n-2)/(n-1) is squarefree.
Numbers n such that b^n == b (mod (n-1)*n) for every integer b are 2, 3, 7, and 43; i.e., only prime numbers of the form A014117(k) + 1. (End)
These are primes p such that p^2 divides b^(2^p-2) - 1 for every b coprime to p. - Thomas Ordowski, Jul 01 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..3314 (First 553 terms from V. Raman)
Mersenne Forum, Prime Conjecture

Cf. A216822, A217468, A211203, A211349, A330382.

a(n)-1 form subsequence of A014741; (a(n)-1)/2 for n>1 forms a subsequence of A014945.

Filtered([1..350000],p->IsPrime(p) and (2^p-2) mod (p-1)=0); # Muniru A Asiru, Dec 03 2018

[p : p in PrimesUpTo(310000) | IsZero((2^p-2) mod (p-1))]; // Vincenzo Librandi, Dec 03 2018

Select[Prime[Range[10000]], Mod[2^# - 2, # - 1] == 0 &] (* T. D. Noe, Sep 19 2012 *)
Join[{2,3},Select[Prime[Range[30000]],PowerMod[2,#,#-1]==2&]] (* Harvey P. Dale, Apr 17 2022 *)

isA069051(p)=Mod(2,p-1)^p==2 && isprime(p); \\ Charles R Greathouse IV, Sep 19 2012

from sympy import prime
for n in range(1,350000):
    if (2**prime(n)-2) % (prime(n)-1)==0:
        print(prime(n)) # Stefano Spezia, Dec 07 2018

a(1) added by Charles R Greathouse IV, Sep 19 2012

A114977 Numbers k such that (j^k + k^j) == 0 (mod k+j), j=2 case.

Original entry on oeis.org

1, 2, 8, 128, 2144, 4808, 12872, 14168, 32377, 33672, 45992, 116192, 185768, 186824, 271208, 426008, 484177, 524288, 601352, 612768, 755792, 996032, 1878368, 2262752, 3094247, 4325960, 4810808, 6331808, 6707352, 10037792, 10908137, 11475128, 12672992, 13705232

Offset: 1

Zak Seidov, Feb 22 2006

nonn

From Robert G. Wilson v, Aug 02 2021: (Start)
Prime terms: 2, then A156048.
The exponents of the powers of two which are terms: 0, 1, then A014741(n)+1.
The vast majority of terms are congruent to 8 (mod 24); no terms are congruent to 4 (mod 6) nor to 3 (mod 10).
(End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1189 (first 645 terms from Hiroaki Yamanouchi).

Cf. A156038 (odd terms), A156048 (odd prime terms).

Cf. A014741, A114978, A114979, A114980, A114981.

fQ[n_] := PowerMod[2, n, n + 2] + 2 == n; Select[ Range[10^7], fQ] (* Robert G. Wilson v, Aug 02 2021 *)

isok(n, k=2) = (k^n+n^k) % (n+k) == 0; \\ Michel Marcus, Oct 10 2013

is(n)=Mod(2, n+2)^n==-4 \\ Charles R Greathouse IV, Oct 19 2013

a(9)-a(22) from Michel Marcus, Oct 10 2013

a(23)-a(34) from Hiroaki Yamanouchi, Sep 26 2015

A192109 Numbers k that divide 2^(k-1) - 2.

Original entry on oeis.org

1, 2, 6, 10, 14, 22, 26, 30, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 170, 178, 182, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526, 538, 542, 554, 562, 566, 586, 614, 622, 626

Offset: 1

Max Alekseyev, Apr 22 2013

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Contains A216090 as subsequence.
Subsequence of A015921, consisting of the terms that are not multiples of 4.
The odd terms form A173572.
Cf. A014741, A000918.

import Data.List (elemIndices)
a192109 n = a192109_list !! (n-1)
a192109_list = map (+ 1) $ elemIndices 0 $ zipWith mod a000918_list [1..]
-- Reinhard Zumkeller, Apr 23 2013

Join[{1,2},Select[Range[700],PowerMod[2,#-1,#]==2&]] (* Harvey P. Dale, May 15 2015 *)

is(n)=Mod(2,n)^(n-1)==2 \\ Charles R Greathouse IV, Nov 04 2016

A296104 Numbers k such that 2^k == 3 (mod k-1).

Original entry on oeis.org

2, 111482, 465794, 79036178, 1781269903308, 250369632905748, 708229497085910, 15673900819204068

Offset: 1

Krzysztof Ziemak and Max Alekseyev, Dec 04 2017

Also, numbers k such that 2^k - 2 is a Fermat pseudoprime, i.e., 2^k - 2 belongs to A015919 and A006935.
a(3) was found by McDaniel (1989).
Some larger terms (maybe not in order): 2338990834231272653582, 341569682872976768698011746141903924998969680638.
Discovered huge even PSP(2) numbers of the form 2*M(n), where n=p*q and M(n)=2^n-1, ensure that the following numbers are also even pseudoprimes of the form 2*M(p)*M(q): 2*M(37)*M(12589), 2*M(131)*M(17854891864360859951), 2*M(179)*M(1398713032993), 2*M(2111)*M(335494787819), 2*M(35267)*M(50508121). - Krzysztof Ziemak, Jan 01 2018

W. L. McDaniel, Some Pseudoprimes and Related Numbers Having Special Forms, Math. Comp. 53:187 (1989), 407-409.

Cf. A000079, A015919, A006935, A014741, A050259, A055685, A296370.

k = 2; lst = {2}; While[k < 1000000001, If[ PowerMod[2, k, k -1] == 3, AppendTo[lst, k]]; k += 10; If[ PowerMod[2, k, k -1] == 3, AppendTo[lst, k]]; k += 2]; lst (* Robert G. Wilson v, Jan 01 2018 *)

is_A296104(n) = Mod(2, n-1)^n == 3; \\ Iain Fox, Dec 07 2017

A296104_list = [n for n in range(2,10**6) if pow(2,n,n-1) == 3 % (n-1)] # Chai Wah Wu, Dec 06 2017

a(n) = A296370(n) + 1.

A330382 Composite numbers k such that k-1 divides 2^k-2.

Original entry on oeis.org

55, 295, 343, 1027, 1135, 1315, 1807, 2059, 2395, 3403, 4375, 5335, 6175, 6499, 7183, 7939, 9235, 10207, 12643, 13123, 14155, 16003, 16255, 19495, 21547, 23815, 27595, 27703, 30619, 35479, 37927, 43219, 45487, 48007, 48763, 50275, 55567, 58483, 64387, 64639, 74899

Offset: 1

Amiram Eldar and Thomas Ordowski, Dec 12 2019

nonn

If k is in this sequence, then 2^k-1 is also a term, so this sequence is infinite.
Also 2^p-1 is in this sequence for such prime p in A069051 that 2^p-1 is composite.
Theorem: if k-1 | 2^k-2, then m-1 | 2^m-2, where m = 2^k-1.
Conjecture: k-1 | 2^k-2 for k = (2^n-1)^3 if and only if n(n-1) | 2^n-2 for n > 2.
It seems that A007013(n)^3 for n > 1 and A007013(n) for n > 4 are in this sequence.
These are the composites k for which M - 1 divides 2^M - 2 where M = 2^k - 1. - Thomas Ordowski, Jul 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A007013, A014741, A054723, A069051, A217468.

A217468 is a subsequence.

Select[Range[75000], CompositeQ[#] && Divisible[PowerMod[2, #, # - 1] - 2, # - 1] &]

forcomposite(k=1,75000,if(!((2^k-2)%(k-1)),print1(k,", "))) \\ Hugo Pfoertner, Dec 12 2019

Composites of A014741(n) + 1. - Thomas Ordowski, Jul 01 2024

A342320 Integers k such that Euler(k, 1) is an integer multiple of Bernoulli(k + 1, 1).

Original entry on oeis.org

0, 1, 5, 17, 41, 53, 125, 161, 293, 341, 377, 485, 881, 1025, 1133, 1313, 1457, 1805, 2057, 2393, 2645, 3077, 3401, 3941, 4373, 5333, 5417, 6173, 6497, 7181, 7937, 9197, 9233, 10205, 11825, 12641, 13121, 14153, 14405, 16001, 16253, 16757, 18521, 19493, 21545

Offset: 0

Peter Luschny, Mar 24 2021

nonn

			Let E(n) = Euler(n, 1) and B(n) = Bernoulli(n, 1).
2*E(0)  = 4*B(1) = 2;
2*E(1)  = 6*B(2) = 1;
2*E(5)  = 42*B(6) = 1;
2*E(17) = 58254*B(18) = 3202291;
2*E(41) = 418861572486*B(42) = 352552873457246307069012458671.

a(n) = A015942(n-1)-1 for n>=2, (a(n)+1)/2 = A014945(n) for n>=1.
a(n) = A014741(n+1) - 1. - Vaclav Kotesovec, Mar 24 2021
Cf. A341759 (subsequence of primes), A198631/A006519 (Euler), A164555/A027642 (Bernoulli).

Join[{0}, Select[Range[1000], BernoulliB[#+1, 1] != 0 && IntegerQ[EulerE[#, 1]/BernoulliB[#+1, 1]] &]] (* Vaclav Kotesovec, Mar 24 2021 *)
Select[Range[100000], IntegerQ[(2*(-1 + 2^#))/#] & ] - 1 (* Vaclav Kotesovec, Mar 24 2021 *)
L342320 := Select[Range[0, 10000], Divisible[2^(# + 2) - 2, # + 1] &];
A342320[n_] := L342320[[n + 1]] (* Peter Luschny, Apr 10 2021 *)

Numbers k such that k + 1 divides 2^(k + 2) - 2. - Vaclav Kotesovec, Mar 24 2021

cp's OEIS Frontend

A125227 A014741(n)/6 for n>2.

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A000295 Eulerian numbers (Euler's triangle: column k=2 of A008292, column k=1 of A173018).

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A014945 Numbers k such that k divides 4^k - 1.

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A086787 a(n) = Sum_{i=1..n} ( Sum_{j=1..n} i^j ).

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A069051 Primes p such that p-1 divides 2^p-2.

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A114977 Numbers k such that (j^k + k^j) == 0 (mod k+j), j=2 case.

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