author:guy keyword:nice - cp's OEIS frontend

A005353 Number of 2 X 2 matrices with entries mod n and nonzero determinant.

0, 6, 48, 168, 480, 966, 2016, 3360, 5616, 8550, 13200, 17832, 26208, 34566, 45840, 59520, 78336, 95526, 123120, 147240, 181776, 219846, 267168, 307488, 372000, 433446, 505440, 580776, 682080, 762150, 892800, 999936, 1138368, 1284486

Offset: 1

N. J. A. Sloane, R. K. Guy

T. Brenner, personal communication.
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=1..1000
Richard K. Guy, Letter to N. J. A. Sloane, Sep 1989.

Cf. A020478, A059306, A062801.

Table[cnt=0; Do[m={{a, b}, {c, d}}; If[Det[m, Modulus->p] > 0, cnt++ ], {a, 0, p-1}, {b, 0, p-1}, {c, 0, p-1}, {d, 0, p-1}]; cnt, {p, 37}] (* T. D. Noe, Jan 12 2006 *)
f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[1] = 0; a[n_] := n^4 - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2023 *)

a(n) = {my(f = factor(n), p, e); n^4 - prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  p^(2*e - 1)*(p^(e + 1) + p^e - 1));} \\ Amiram Eldar, Oct 31 2023

a(n) = n^4 - A020478(n).

For prime n, a(n) = (n^2-1)(n-1)n. - T. D. Noe, Jan 12 2006

More terms from T. D. Noe, Jan 12 2006

A005575 a(n) = A259095(2n,n).

Original entry on oeis.org

0, 0, 1, 2, 5, 11, 20, 37, 63, 110, 174, 283, 435, 671, 1001, 1492, 2160, 3127, 4442, 6269, 8739, 12109, 16597, 22618, 30576, 41077, 54834, 72788, 96056, 126131, 164829, 214327, 277534, 357810, 459507, 587779, 749220, 951473, 1204501, 1519691, 1911618, 2397247, 2997985, 3738482, 4649981, 5768457, 7138640, 8812704, 10854735, 13339286

Offset: 1

N. J. A. Sloane, R. K. Guy

Computed by R. K. Guy in 1988.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..900
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A259095, A005576, A005577.

b:= proc(n, i, d) option remember; `if`(i*(i+1)/2n, 0, d*b(n-i, i-1, 1))))
    end:
a:= n-> b(n, n-1, 1):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 08 2016

b[n_, i_, d_] := b[n, i, d] = If[i*(i+1)/2 < n, 0, If[n == 0, 1, b[n, i-1, d+1] + If[i > n, 0, d*b[n-i, i-1, 1]]]];
a[n_] := b[n, n-1, 1];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 28 2016, after Alois P. Heinz *)

Edited by N. J. A. Sloane, Jun 20 2015

Terms a(25) and beyond from Joerg Arndt, Apr 09 2016

A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.

Original entry on oeis.org

1, 2, 3, 3, 5, 8, 8, 11, 16, 24, 24, 32, 43, 59, 83, 83, 107, 139, 182, 241, 324, 324, 407, 514, 653, 835, 1076, 1400, 1400, 1724, 2131, 2645, 3298, 4133, 5209, 6609, 6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758

Offset: 0

N. J. A. Sloane, R. K. Guy

			Triangle starts:
[0] [   1]
[1] [   2,    3]
[2] [   3,    5,    8]
[3] [   8,   11,   16,    24]
[4] [  24,   32,   43,    59,    83]
[5] [  83,  107,  139,   182,   241,   324]
[6] [ 324,  407,  514,   653,   835,  1076,  1400]
[7] [1400, 1724, 2131,  2645,  3298,  4133,  5209,  6609]
[8] [6609, 8009, 9733, 11864, 14509, 17807, 21940, 27149, 33758]

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Borders give A038561.

Cf. A011971.

a046937 n k = a046937_tabl !! n !! k
a046937_row n = a046937_tabl !! n
a046937_tabl = [1] : iterate (\row -> scanl (+) (last row) row) [2,3]
-- Reinhard Zumkeller, Jan 13 2013

# Compare the analogue algorithm for the Catalan triangle in A350584.
A046937Triangle := proc(len) local A, P, T, n; A := [2]; P := [1]; T := [[1]];
for n from 1 to len-1 do P := ListTools:-PartialSums([A[-1], op(P)]);
A := P; T := [op(T), P] od; T end:
A046937Triangle(9): ListTools:-Flatten(%); # Peter Luschny, Mar 27 2022

a[0, 0] = 1; a[1, 0] = 2; a[n_, 0] := a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 06 2013 *)

A051488 Numbers k such that phi(k) < phi(k - phi(k)).

Original entry on oeis.org

30, 60, 66, 120, 132, 138, 174, 210, 240, 246, 264, 276, 318, 330, 348, 420, 480, 492, 498, 510, 528, 534, 552, 630, 636, 660, 678, 690, 696, 786, 840, 870, 910, 960, 984, 996, 1020, 1038, 1056, 1068, 1074, 1104, 1122, 1146, 1260, 1272, 1320, 1330, 1356

Offset: 1

R. K. Guy

If p is a Sophie Germain prime greater than 3 and n is a natural number then 2^n*3*p is in the sequence. That is because if m = 2^n*3*p then phi(m) = 2^n*(p-1) and phi(m - phi(m)) = phi(2^n*3*p - 2^n*(p-1)) = phi(2^n*(2p+1)) = 2^n*p so phi(m) < phi(m-phi(m)) and m is in the sequence. - Farideh Firoozbakht, Jun 19 2005

Erdős (1980) proposed the problem to prove that this sequence is infinite and has an asymptotic density 0. Grytczuk et al. (2001) proved that this sequence is infinite with an upper asymptotic density < 0.45637. - Amiram Eldar, May 22 2021

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 209.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Paul Erdős, Problem P. 294, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505.
Aleksander Grytczuk, Florian Luca and Marek Wojtowicz, A conjecture of Erdős concerning inequalities for the Euler totient function, Publ. Math. Debrecen, Vol. 59, No. 1-2, (2001), pp. 9-16.

Cf. A051487, A005384.

Cf. A000010, A051953.

a051488 n = a051488_list !! (n-1)
a051488_list = [x | x <- [2..], let t = a000010 x, t < a000010 (x - t)]
-- Reinhard Zumkeller, Apr 12 2014

Select[Range[1360], EulerPhi[ # ] < EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)

More terms from James Sellers

A005168 n-th derivative of x^x at 1, divided by n.

Original entry on oeis.org

1, 1, 1, 2, 2, 9, -6, 118, -568, 4716, -38160, 358126, -3662088, 41073096, -500013528, 6573808200, -92840971200, 1402148010528, -22554146644416, 385014881294496, -6952611764874240, 132427188835260480, -2653529921603890560, 55802195178451990896

Offset: 1

N. J. A. Sloane, R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..400 (first 100 terms from T. D. Noe)
R. K. Guy, Letter to N. J. A. Sloane, 1986
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. R. Patterson and G. Suri, The derivatives of x^x, date unknown. Preprint. [Annotated scanned copy]

Cf. A005727.

Column k=2 of A295027 (for n>1), A295028.

a:= n-> (n-1)! *coeftayl(x^x, x=1, n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 18 2012

Rest[(NestList[ Factor[ D[ #1, x]] &, x^x, 23] /. (x -> 1))/Range[0, 23]] (* Robert G. Wilson v, Aug 10 2010 *)

from sympy import var, diff
x = var('x')
y = x**x
l = [[y:=diff(y),y.subs(x,1)/(n+1)][1] for n in range(10)]
print(l) # Nicholas Stefan Georgescu, Mar 02 2023

One more term from Robert G. Wilson v, Aug 10 2010

A005576 The limiting sequence [A259095(r(r+1)/2-s,r), s=0,1,2,...,r-1] for very large r.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 9, 13, 17, 25, 32, 43, 56, 73, 95, 122, 155, 196, 248, 309, 388, 480, 595, 731, 899, 1096, 1338, 1624, 1967, 2373, 2860, 3431, 4111, 4911, 5853, 6963, 8263, 9785, 11565, 13646, 16064, 18884, 22155, 25953, 30349, 35441, 41311, 48098, 55906, 64900, 75231, 87103, 100702, 116296, 134130, 154522

Offset: 0

N. J. A. Sloane, R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 143 terms from Joerg Arndt)
Joerg Arndt, C++ program to compute this sequence, 2016
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
E. M. Wright, Stacks (III), Quart. J. Math. Oxford, 23 (1972), 153-158.

Cf. A259095, A005575, A005577.

b:= proc(n, i, d) option remember; `if`(i*(i+1)/2n, 0, d*b(n-i, i-1, 1))))
    end:
a:= n-> b(n*(n-1)/2, n, 1):
seq(a(n), n=0..55);  # Alois P. Heinz, Jul 08 2016

b[n_, i_, d_] := b[n, i, d] = If[i*(i + 1)/2 < n, 0, If[n == 0, 1, b[n, i - 1, d + 1] + If[i > n, 0, d*b[n - i, i - 1, 1]]]];
a[n_] := b[n*(n - 1)/2, n, 1];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jul 28 2016, after Alois P. Heinz *)

Edited by N. J. A. Sloane, Jun 20 2015
Terms a(0)..a(11) computed by R. K. Guy
Terms a(12)=56 and beyond from Joerg Arndt, Apr 10 2016

A005577 Maxima of the rows of the triangle A259095.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 11, 15, 20, 27, 35, 44, 56, 73, 91, 115, 148, 186, 227, 283, 358, 435, 538, 671, 813, 1001, 1233, 1492, 1815, 2223, 2673, 3247, 3933, 4713, 5683, 6850, 8170, 9785, 11725, 13948, 16587, 19783, 23468, 27710, 32942, 38956, 45852, 54133, 63879, 75000, 87909, 103471, 121273, 141629

Offset: 1

N. J. A. Sloane, R. K. Guy

Computed by R. K. Guy in 1988.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Joerg Arndt)
F. C. Auluck, On some new types of partitions associated with generalized ferrers graphs, Math. Proc. Camb. Phil. Soc. 47 (1951) 679-686.
R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
E. M. Wright, Stacks (III), The Quarterly J. of Math. (Oxford Journals), 23 (2) (1972) 153-158. MR0299575

Cf. A259095, A005575, A005576.

b:= proc(n, i, d) option remember; `if`(i*(i+1)/2n, 0, d*b(n-i, i-1, 1))))
    end:
a:= n-> max(seq(b(n-r, r-1, 1), r=1..n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 08 2016

b[n_, i_, d_] := b[n, i, d] = If[i*(i+1)/2 < n, 0, If[n == 0, 1, b[n, i-1, d+1] + If[i > n, 0, d*b[n-i, i-1, 1]]]];
a[n_] := Max[Table[b[n-r, r-1, 1], {r, 1, n}]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 28 2016, after Alois P. Heinz *)

Edited by N. J. A. Sloane, Jun 20 2015

A007045 Second (lower) diagonal of partition triangle A047812.

Original entry on oeis.org

0, 1, 5, 20, 51, 112, 221, 411, 720, 1221, 2003, 3206, 5021, 7728, 11698, 17472, 25766, 37580, 54254, 77617, 110087, 154942, 216488, 300456, 414365, 568113, 774571, 1050572, 1417868, 1904641, 2547152, 3392042, 4498948, 5944158, 7824703, 10263932, 13418043, 17484554

Offset: 2

N. J. A. Sloane and R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 2..500 (terms n=2..53 from Vincenzo Librandi, n=54..103 from Bruno Berselli)
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A007042, A007044, A047812, A051643.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i b((n-3)*(n+1), n$2):
seq(a(n), n=2..40);  # Alois P. Heinz, May 31 2020

s[n_] := s[n] = Series[Product[(1 - q^(2*n - k))/(1 - q^(k + 1)), {k, 0, n - 1}], {q, 0, n^2}]; t[n_, k_] := SeriesCoefficient[s[n], k*(n + 1)]; A007045 = Join[{0}, Table[t[n + 3, n], {n, 0, 25}] ] (* Jean-François Alcover, Apr 25 2012 *)

T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=3, 33, print1(T(n, n-3), ", ")) \\ Petros Hadjicostas, May 31 2020

A061776 Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives number of triangles in n-th generation.

Original entry on oeis.org

1, 3, 6, 12, 18, 30, 42, 66, 90, 138, 186, 282, 378, 570, 762, 1146, 1530, 2298, 3066, 4602, 6138, 9210, 12282, 18426, 24570, 36858, 49146, 73722, 98298, 147450, 196602, 294906, 393210, 589818, 786426, 1179642, 1572858, 2359290, 3145722, 4718586, 6291450

Offset: 0

N. J. A. Sloane, R. K. Guy, Jun 23 2001

Number of 3-colorings of the (n,2)-Turán graph. - Alois P. Heinz, Jun 07 2024

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

A061777 gives total population of triangles at n-th generation.

Cf. A266972.

A061776 := proc(n) if n mod 2 = 0 then 6*(2^(n/2)-1); else 3*(2^((n-1)/2)-1)+3*(2^((n+1)/2)-1); fi; end; # for n >= 1

a[0]=1; a[n_/;EvenQ[n]]:=6*(2^(n/2)-1); a[n_/;OddQ[n]] := 3*(2^((n-1)/2)-1) + 3*(2^((n+1)/2)-1); a /@ Range[0, 37] (* Jean-François Alcover, Apr 22 2011, after Maple program *)
CoefficientList[Series[(1 + 2 x) (1 + x^2) / ((1 - x) (1 - 2 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
LinearRecurrence[{1,2,-2},{1,3,6,12},40] (* Harvey P. Dale, Mar 27 2019 *)

a(n)=([0,1,0; 0,0,1; -2,2,1]^n*[1;3;6])[1,1] \\ Charles R Greathouse IV, Feb 19 2017

Explicit formula given in Maple line.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3) for n>3. G.f.: (1+2*x)*(1+x^2)/((1-x)*(1-2*x^2)). - Colin Barker, May 08 2012
a(n) = 3*A027383(n-1) for n>0, a(0)=1. - Bruno Berselli, May 08 2012

A064164 EHS numbers: k such that there is a prime p satisfying k! + 1 == 0 (mod p) and p !== 1 (mod k).

Original entry on oeis.org

8, 9, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

R. K. Guy, Sep 20 2001

The complement of this sequence (A064295) is a superset of A002981, that is, terms of A002981 do not appear in this sequence.

Hardy & Subbarao prove that this sequence is infinite, see their Theorem 2.12. - Charles R Greathouse IV, Sep 10 2015

Amiram Eldar, Table of n, a(n) for n = 1..120
G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109:6 (2002), pp. 554-559.
H. Mishima, Factors of N!+1

The smallest associated primes p are given in A064229.

Cf. A002981, A064295.

Do[k = 1; While[p = Prime[k]; k < 10^8 && Not[ Nor[ Mod[n! + 1, p] != 0, Mod[p, n] == 1]], k++ ]; If[k != 10^8, Print[n, " ", p]], {n, 2, 88}]

is(n)=my(f=factor(n!+1)[,1]); for(i=1,#f, if(f[i]%n != 1, return(n>1))); 0 \\ Charles R Greathouse IV, Sep 10 2015

Corrected and extended by Don Reble, Sep 23 2001

cp's OEIS Frontend

A005353 Number of 2 X 2 matrices with entries mod n and nonzero determinant.

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A005575 a(n) = A259095(2n,n).

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A046937 Triangle read by rows. Same rule as Aitken triangle (A011971) except T(0,0) = 1, T(1,0) = 2.

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A051488 Numbers k such that phi(k) < phi(k - phi(k)).

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A005168 n-th derivative of x^x at 1, divided by n.

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A005576 The limiting sequence [A259095(r(r+1)/2-s,r), s=0,1,2,...,r-1] for very large r.

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A005577 Maxima of the rows of the triangle A259095.

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