A008588 -id:A008588 - cp's OEIS frontend

A087409 Multiples of 6 with digits grouped in pairs and leading zeros omitted.

61, 21, 82, 43, 3, 64, 24, 85, 46, 6, 67, 27, 88, 49, 9, 61, 2, 10, 81, 14, 12, 1, 26, 13, 21, 38, 14, 41, 50, 15, 61, 62, 16, 81, 74, 18, 1, 86, 19, 21, 98, 20, 42, 10, 21, 62, 22, 22, 82, 34, 24, 2, 46, 25, 22, 58, 26, 42, 70, 27, 62, 82, 28, 82, 94, 30, 3, 6, 31, 23, 18, 32, 43

Offset: 1

N. J. A. Sloane, Oct 19 2003

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video

Cf. A008588, A059805, A087406, A060936, A087407, A087408, A087075, A087410, A087411.

FromDigits /@ Partition[ Flatten[ IntegerDigits[ Table[ 6n, {n, 1, 60}]]], 2] (* Robert G. Wilson v *)

More terms from Ray Chandler, Oct 20 2003

A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

Offset: 1

Reinhard Zumkeller, May 31 2004

T(n,k) = A003991(n-1,k) for 1 <= k < n;
T(n,k) = T(n,n-1-k) for k < n;
T(n,1) = n-1; T(n,n) = 0;          T(n,2) = A005843(n-2) for n > 1;
T(n,3) = A008585(n-3) for n>2;     T(n,4) = A008586(n-4) for n > 3;
T(n,5) = A008587(n-5) for n>4;     T(n,6) = A008588(n-6) for n > 5;
T(n,7) = A008589(n-7) for n>6;     T(n,8) = A008590(n-8) for n > 7;
T(n,9) = A008591(n-9) for n>8;     T(n,10) = A008592(n-10) for n > 9;
T(n,11) = A008593(n-11) for n>10;  T(n,12) = A008594(n-12) for n > 11;
T(n,13) = A008595(n-13) for n>12;  T(n,14) = A008596(n-14) for n > 13;
T(n,15) = A008597(n-15) for n>14;  T(n,16) = A008598(n-16) for n > 15;
T(n,17) = A008599(n-17) for n>16;  T(n,18) = A008600(n-18) for n > 17;
T(n,19) = A008601(n-19) for n>18;  T(n,20) = A008602(n-20) for n > 19;
Row sums give A000292; triangle sums give A000332;
All numbers m > 0 occur A000005(m) times;
A002378(n) = T(A005408(n),n+1) = n*(n+1).
k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) =  =  = i  = -i . Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016
T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

			From _M. F. Hasler_, Feb 02 2013: (Start)
Triangle begins:
  0;
  1, 0;
  2, 2, 0;
  3, 4, 3, 0;
  4, 6, 6, 4, 0;
  5, 8, 9, 8, 5, 0;
  (...)
If an additional 0 was added at the beginning, this would become:
  0;
  0, 1;
  0, 2, 2;
  0, 3, 4; 3;
  0, 4, 6, 6, 4;
  0, 5, 8, 9, 8, 5;
  ... (End)

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

J_3: A114327; J_1^2, J_2^2: A141387, A268759.
Cf. A000005, A002378, A003991, A005408.
Cf. A000292 (row sums), A000332 (triangle sums).
T(n,k) for values of k:
  A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

/* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

{for(n=1, 13, for(k=1, n, print1(k*(n - k)," ");); print(););} \\ Indranil Ghosh, Mar 12 2017

A138750 a(n) = ceiling(n/2) if n == 2 (mod 3), a(n) = 2n otherwise.

Original entry on oeis.org

0, 2, 1, 6, 8, 3, 12, 14, 4, 18, 20, 6, 24, 26, 7, 30, 32, 9, 36, 38, 10, 42, 44, 12, 48, 50, 13, 54, 56, 15, 60, 62, 16, 66, 68, 18, 72, 74, 19, 78, 80, 21, 84, 86, 22, 90, 92, 24, 96, 98, 25, 102, 104, 27, 108, 110, 28, 114, 116, 30, 120, 122, 31, 126, 128, 33, 132, 134, 34

Offset: 0

M. F. Hasler, Mar 28 2008

This map is inspired by A124123, which hides in fact a variation of the Collatz problem, defined on the set of primes and working mod 3 instead of mod 2. See A138751 for more information.
The use of ceiling() is here equivalent to round().
The main reason for defining this function is to write A124123 as complement of A007918(A138750(A000040)), and to express the recursion function occurring there in terms of this map.
It might have been more natural to define this map as a(n) = 2n if n == 1 (mod 3), a(n) = ceiling(n/2) otherwise, which is equivalent for all primes > 3 (which are either == 1 or == 2 (mod 3)) and would have "better" properties regarding the analysis of orbits of all integers under this map.
However, for the prime n=3 it does make a difference, and in order to reproduce the map occurring in A124123, we had to adopt the present convention.

			a(0) = 2*0 = 0, a(1) = 2*1 = 2, a(3) = 2*3 = 6, a(4) = 2*4 = 8, ... since these indices are not congruent to 2 (mod 3).
a(2) = ceiling(2/2) = 1, a(5) = ceiling(5/2) = 3, a(8) = ceiling(8/2) = 4, a(11) = ceiling(11/2) = 6, ... since these indices are congruent to 2 (mod 3).

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A001281, A124123, A138751, A138752, A138753, A008588 (trisection), A016933 (trisection), A032766 (trisection)

Table[If[Mod[n,3]==2,Ceiling[n/2],2n],{n,0,70}] (* or *) LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{0,2,1,6,8,3,12,14,4},70] (* Harvey P. Dale, Nov 20 2013 *)

A138750(n) = if( n%3==2, ceil(n/2), 2*n )

G.f.: x*(2 + x + 6*x^2 + 6*x^3 + 2*x^4 + 6*x^5 + 4*x^6) / ( (1+x)*(x^2-x+1)*(x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Oct 16 2013
a(n) = a(n-3) + a(n-6) - a(n-9); a(0)=0, a(1)=2, a(2)=1, a(3)=6, a(4)=8, a(5)=3, a(6)=12, a(7)=14, a(8)=4. - Harvey P. Dale, Nov 20 2013
Sum_{n>=1} (-1)^n/a(n) = log(3)/2 - log(2)/3 = log(27/4)/6. - Amiram Eldar, Jul 26 2024

A157176 a(n+1) = a(n - n mod 2) + a(n - n mod 3), a(0) = 1.

Original entry on oeis.org

1, 2, 2, 3, 5, 8, 8, 16, 16, 24, 40, 64, 64, 128, 128, 192, 320, 512, 512, 1024, 1024, 1536, 2560, 4096, 4096, 8192, 8192, 12288, 20480, 32768, 32768, 65536, 65536, 98304, 163840, 262144, 262144, 524288, 524288, 786432, 1310720, 2097152, 2097152, 4194304, 4194304

Offset: 0

Reinhard Zumkeller, Feb 24 2009

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,8).

Cf. A001018, A008588, A013730, A016921, A016933, A016945, A016957, A016969, A067412, A103333.

LinearRecurrence[{0,0,0,0,0,8},{1, 2, 2, 3, 5, 8},45] (* Stefano Spezia, May 29 2024 *)

a(n+6) = 8*a(n).
a(6*k) = 8^k; a(A008588(n))=A001018(n);
a(6*k+1) = a(6*k+2) = 2*8^k; a(A016921(n))=a(A016933(n))=A013730(n);
a(6*k+3) = 3*8^k; a(A016945(n))=A103333(n+1);
a(6*k+4) = 5*8^k; a(A016957(n))=A067412(n+1);
a(6*k+5) = 8^(k+1); a(A016969(n))=A001018(n+1).
G.f.: (1 + 2*x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5)/((1 - 2*x^2)*(1 + 2*x^2 + 4*x^4)). - Stefano Spezia, May 29 2024

a(43)-a(44) from Stefano Spezia, May 29 2024

A328632 Numbers k such that A276086(k) == 1 (mod 6), where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 12, 24, 30, 42, 54, 60, 72, 84, 90, 102, 114, 120, 132, 144, 150, 162, 174, 180, 192, 204, 216, 228, 246, 258, 276, 288, 306, 318, 336, 348, 366, 378, 396, 408, 420, 432, 444, 450, 462, 474, 480, 492, 504, 510, 522, 534, 540, 552, 564, 570, 582, 594, 600, 612, 624, 636, 648, 666, 678, 696, 708, 726, 738, 756, 768, 786, 798, 816

Offset: 1

Antti Karttunen, Oct 27 2019

nonn

Numbers k >= 0 for which A328578(k) = A257993(A276086(A276086(k))) = 2, where A276086 converts the primorial base expansion of k into its prime product form, and A257993 returns the index of the least prime not present in its argument. - The original, equivalent definition.
Numbers k for which A276087(k) is an even number, but not a multiple of three.
All terms are multiples of 6, and thus apart from the initial zero, this is a subsequence of A328587, numbers k for which A257993(A276086(A276086(k))) is less than A257993(k).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Row 2 of A328631.
After the initial zero, setwise difference A328587 \ A328762. Also setwise difference A008588 \ A358843.
Positions of 1's in A358840 and A358841 (characteristic function), positions of 2's in A328578.
Cf. A257993, A276086, A328578, A358845 (= a(n)/6).
Cf. also A328317.

A257993(n) = { for(i=1,oo,if(n%prime(i),return(i))); }
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328578(n) = A257993(A276086(A276086(n)));
isA328632(n) = (2==A328578(n));

isA326832(n) = A358841(n);
A358841(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (1==(m%6)); }; \\ Antti Karttunen, Dec 12 2022

{k | A358840(k) == 1}. - Antti Karttunen, Dec 02 2022

Definition replaced with a simpler one and the original definition moved to the comments section by Antti Karttunen, Dec 03 2022

A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 36, 42, 54, 48, 60, 84, 66, 78, 72, 126, 90, 102, 108, 114, 96, 120, 150, 138, 162, 132, 144, 168, 246, 156, 180, 186, 240, 204, 192, 216, 198, 210, 174, 258, 252, 222, 234, 228, 318, 282, 264, 276, 342, 306, 294, 312, 270, 354, 372

Offset: 1

Alexei Kourbatov, May 05 2020

nonn

Contains A268928 as a subsequence. First differs from A268928 at a(5)=24.
Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

			The first two primes of the form 6k-1 are 5 and 11, so a(1)=11-5=6. The next primes of this form are 17, 23, 29; the gaps 17-11 = 23-17 = 29-23 have size 6 which already occurred before; so nothing is added to the sequence. The next prime of this form is 41 and the gap 41-29=12 has not occurred before, so a(2)=12.

Alexei Kourbatov, Table of n, a(n) for n = 1..160
Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.

Cf. A007528, A014320, A058320, A268928, A330853, A334544, A334545.

isFirstOcc=vector(9999,j,1); s=5; forprime(p=11,1e8,if(p%6!=5,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

a(n) = A334545(n) - A334544(n).

A359409 Integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly four elements.

Original entry on oeis.org

18, 24, 36, 54, 66, 72, 78, 84, 102, 108, 114, 132, 138, 144, 156, 162, 168, 174, 186, 192, 198, 204, 216, 222, 228, 234, 246, 258, 264, 276, 282, 288, 294, 306, 312, 318, 324, 336, 342, 348, 354, 366, 372, 378, 384, 396, 402, 408, 414, 432, 438, 444, 456, 462, 468, 486

Offset: 1

Bernard Schott, Jan 23 2023

nonn

These 4 elements are not necessarily consecutive primes.
A342309(d) gives the first element of the smallest AP with 4 elements whose common difference is a(n) = d.
All the terms are multiples of 6 (A008588) but are not multiples of 5 and also must not belong to A206039; indeed, terms d' in A206039 correspond to the largest possible arithmetic progression (AP) of primes that have exactly five elements with this common difference d'.

			d = 18 is a term because the largest possible APs of primes with common difference d = 18 have all 4 elements; the first such APs start with 5, 43, 53, ... The smallest one is (5, 23, 41, 59) then 77 is composite.
d = 24 is another term because the largest possible APs of primes with common difference d = 24 have all 4 elements; the first such APs start with 59, 79, 349, ... The smallest one is (59, 83, 107, 131) then 155 is composite.

Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A123556, A342309.
Subsequence of A008588.
Largest AP of prime numbers with k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), this sequence (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7).

isok(d) = (d%5) && !(d%6) && !(isprime(5+d) && isprime(5+2*d) && isprime(5+3*d) && isprime(5+4*d)); \\ Michel Marcus, Jan 23 2023

m is a term iff A123556(m) = 4.

A169823 Multiples of 60.

Original entry on oeis.org

0, 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020, 1080, 1140, 1200, 1260, 1320, 1380, 1440, 1500, 1560, 1620, 1680, 1740, 1800, 1860, 1920, 1980, 2040, 2100, 2160, 2220, 2280, 2340, 2400, 2460, 2520, 2580, 2640, 2700

Offset: 0

N. J. A. Sloane, May 29 2010

Numbers that are divisible by all of 1, 2, 3, 4, 5, 6.

Erich Friedman, What's Special About This Number? (See the entry for "60")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A005843, A008585, A008586, A008587, A008588, A008592, A008594.

Cf. A008597, A008602, A169825, A169827, A249674.

Range[0, 2700, 60] (* Vladimir Joseph Stephan Orlovsky, Jul 12 2011 *)

a(n)=60*n \\ Charles R Greathouse IV, Mar 19 2017

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 60*x/(x-1)^2.
E.g.f.: 60*x*exp(x).
a(n) = 60*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A249674(n) = 3*A008602(n) = 4*A008597(n) = 5*A008594(n) = 6*A008592(n) = 10*A008588(n) = 12*A008587(n) = 15*A008586(n) = 20*A008585(n) = 30*A005843(n) = 60*A001477(n) = A169827(n)/14 = A169825(n)/7. (End)

A212701 Main transitions in systems of n particles with spin 3.

Original entry on oeis.org

6, 84, 882, 8232, 72030, 605052, 4941258, 39530064, 311299254, 2421216420, 18643366434, 142367525496, 1079620401678, 8138676874188, 61040076556410, 455765904954528, 3389758918099302, 25124095510618356, 185639150161791186, 1367867422244777160, 10053825553499112126

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=7 (see formula), corresponding to spin S=(b-1)/2=3.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (14,-49).

Cf. A001787, A212697, A212698, A212699, A212700, A212702, A212703, A212704 (b = 2, 3, 4, 5, 6, 8, 9, 10).

Cf. A000420, A008588, A027473.

LinearRecurrence[{14,-49},{6,84},20] (* Harvey P. Dale, Aug 02 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212701.txt", n, " ", mtrans(n, 7)))

Vec(6*x/(7*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=7.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 14*a(n-1) - 49*a(n-2) for n > 2.
G.f.: 6*x/(7*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 6*x*exp(7*x).
a(n) = 6*A027473(n) = A008588(n)*A000420(n-1). (End)

A225326 Partition numbers of the form 6k.

Original entry on oeis.org

30, 42, 792, 1002, 2436, 5604, 37338, 105558, 614154, 4087968, 8118264, 15796476, 133230930, 384276336, 2841940500, 3163127352, 4835271870, 7346629512, 18440293320, 30388671978, 45060624582, 107438159466, 142798995930, 684957390936, 1820701100652

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008588 and A000041.

			30 is in the sequence because 6*5 = 30 and 30 is a partition number: p(9) = A000041(9) = 30.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000041, A008588, A052001, A072871, A087183, A225324, A225325, A225327, A225358, A225360, A127544, A225361

Select[PartitionsP[Range[300]], Mod[#, 6] == 0 &] (* T. D. Noe, May 05 2013 *)

for(n=9,1e3,t=numbpart(n);if(t%6,,print1(t", "))) \\ Charles R Greathouse IV, May 08 2013

a(n) = 6*A217726(n). - Omar E. Pol, May 08 2013

a(8)-a(25) from T. D. Noe, May 05 2013

cp's OEIS Frontend

A087409 Multiples of 6 with digits grouped in pairs and leading zeros omitted.

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A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

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A138750 a(n) = ceiling(n/2) if n == 2 (mod 3), a(n) = 2n otherwise.

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A157176 a(n+1) = a(n - n mod 2) + a(n - n mod 3), a(0) = 1.

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A328632 Numbers k such that A276086(k) == 1 (mod 6), where A276086 is the primorial base exp-function.

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A334543 First occurrences of gaps between primes 6k - 1: gap sizes.

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A359409 Integers d such that the largest possible arithmetic progression (AP) of primes with common difference d has exactly four elements.

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PARI

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A169823 Multiples of 60.

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