A061819 -id:A061819 - cp's OEIS frontend

A008585 a(n) = 3*n.

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177

Offset: 0

N. J. A. Sloane

If n != 1 and n^2+2 is prime then n is a member of this sequence. - Cino Hilliard, Mar 19 2007
Multiples of 3. Positive members of this sequence are the third transversal numbers (or 3-transversal numbers): Numbers of the 3rd column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 3rd column in the square array A057145. - Omar E. Pol, May 02 2008
Numbers n for which polynomial 27*x^6-2^n is factorizable. - Artur Jasinski, Nov 01 2008
1/7 in base-2 notation = 0.001001001... = 1/2^3 + 1/2^6 + 1/2^9 + ... - Gary W. Adamson, Jan 24 2009
A165330(a(n)) = 153 for n > 0; subsequence of A031179. - Reinhard Zumkeller, Sep 17 2009
A011655(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
A215879(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2012
Moser conjectured, and Newman proved, that the terms of this sequence are more likely to have an even number of 1s in binary than an odd number. The excess is an undulating multiple of n^(log 3/log 4). See also Coquet, who refines this result. - Charles R Greathouse IV, Jul 17 2013
Integer areas of medial triangles of integer-sided triangles.
Also integer subset of A188158(n)/4.
A medial triangle MNO is formed by joining the midpoints of the sides of a triangle ABC. The area of a medial triangle is A/4 where A is the area of the initial triangle ABC. - Michel Lagneau, Oct 28 2013
From Derek Orr, Nov 22 2014: (Start)
Let b(0) = 0, and b(n) = the number of distinct terms in the set of pairwise sums {b(0), ... b(n-1)} + {b(0), ... b(n-1)}. Then b(n+1) = a(n), for n > 0.
Example: b(1) = the number of distinct sums of {0} + {0}. The only possible sum is {0} so b(1) = 1. b(2) = the number of distinct sums of {0,1} + {0,1}. The possible sums are {0,1,2} so b(2) = 3. b(3) = the number of distinct sums of {0,1,3} + {0,1,3}. The possible sums are {0, 1, 2, 3, 4, 6} so b(3) = 6. This continues and one can see that b(n+1) = a(n). (End)
Number of partitions of 6n into exactly 2 parts. - Colin Barker, Mar 23 2015
Partial sums are in A045943. - Guenther Schrack, May 18 2017
Number of edges in a maximal planar graph with n+2 vertices, n > 0 (see A008486 comments). - Jonathan Sondow, Mar 03 2018
Also numbers such that when the leftmost digit is moved to the unit's place the result is divisible by 3. - Stefano Spezia, Jul 08 2025

			G.f.: 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
John Graham-Cumming, The hollow triangular numbers are divisible by three (2013)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 315
Tanya Khovanova, Recursive Sequences
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Wikipedia, Maximal planar graphs
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Row / column 3 of A004247 and of A325820.
Cf. A016957, A057145, A139600, A139606, A001651 (complement), A032031 (partial products), A190944 (binary), A061819 (base 4).
Cf. A031179, A008486, A008591, A017197, A017233, A045943.

a008585 = (* 3)
a008585_list = iterate (+ 3) 0  -- Reinhard Zumkeller, Feb 19 2013

[3*n: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[0, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

makelist(3*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=3*n \\ Charles R Greathouse IV, Jun 28 2013

G.f.: 3*x/(1-x)^2. - R. J. Mathar, Oct 23 2008
a(n) = A008486(n), n > 0. - R. J. Mathar, Oct 28 2008
G.f.: A(x) - 1, where A(x) is the g.f. of A008486. - Gennady Eremin, Feb 20 2021
a(n) = Sum_{k=0..inf} A030308(n,k)*A007283(k). - Philippe Deléham, Oct 17 2011
E.g.f.: 3*x*exp(x). - Ilya Gutkovskiy, May 18 2016
From Guenther Schrack, May 18 2017: (Start)
a(3*k) = a(a(k)) = A008591(n).
a(3*k+1) = a(a(k) + 1) = a(A016777(n)) = A017197(n).
a(3*k+2) = a(a(k) + 2) = a(A016789(n)) = A017233(n). (End)

Partially edited by Joerg Arndt, Mar 11 2010

A061818 Multiples of 2 containing only digits 0,1,2.

Original entry on oeis.org

0, 2, 10, 12, 20, 22, 100, 102, 110, 112, 120, 122, 200, 202, 210, 212, 220, 222, 1000, 1002, 1010, 1012, 1020, 1022, 1100, 1102, 1110, 1112, 1120, 1122, 1200, 1202, 1210, 1212, 1220, 1222, 2000, 2002, 2010, 2012, 2020, 2022, 2100, 2102, 2110, 2112, 2120

Offset: 0

Amarnath Murthy, May 28 2001

			102 is a term containing digits less than 3.

Cf. A061819, etc. Ternary representation of A007494.

Select[2*Range[0,1100],Max[IntegerDigits[#]]<3&] (* Harvey P. Dale, Jun 21 2025 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 30 2001

A061820 Multiples of 4 containing only digits 0,...,4.

Original entry on oeis.org

0, 4, 12, 20, 24, 32, 40, 44, 100, 104, 112, 120, 124, 132, 140, 144, 200, 204, 212, 220, 224, 232, 240, 244, 300, 304, 312, 320, 324, 332, 340, 344, 400, 404, 412, 420, 424, 432, 440, 444, 1000, 1004, 1012, 1020, 1024, 1032, 1040, 1044, 1100, 1104, 1112

Offset: 0

Amarnath Murthy, May 28 2001

			140 is a term containing digits less than 5.

Cf. A061818, A061819.

Select[4Range[0,400],Max[IntegerDigits[#]]<5&] (* Harvey P. Dale, Dec 20 2012 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 30 2001

A061821 Multiples of 5 containing only digits 0,...,5.

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 300, 305, 310, 315, 320, 325, 330, 335, 340, 345, 350, 355, 400, 405, 410, 415, 420, 425, 430

Offset: 1

Amarnath Murthy, May 28 2001

			145 is a term containing digits less than 6.

A061818, A061819, A061820.

Select[5Range[0,90],Max[IntegerDigits[#]]<6&] (* Harvey P. Dale, May 02 2011 *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 30 2001

A061822 Multiples of 6 containing only digits 0,...,6.

Original entry on oeis.org

0, 6, 12, 24, 30, 36, 42, 54, 60, 66, 102, 114, 120, 126, 132, 144, 150, 156, 162, 204, 210, 216, 222, 234, 240, 246, 252, 264, 300, 306, 312, 324, 330, 336, 342, 354, 360, 366, 402, 414, 420, 426, 432, 444, 450, 456, 462, 504, 510, 516, 522, 534, 540, 546, 552

Offset: 0

Amarnath Murthy, May 28 2001

			156 is a term containing digits less than 7.

A061818, A061819, A061820, A061821.

More terms from Larry Reeves (larryr(AT)acm.org), May 30 2001

A061823 Multiples of 7 containing only digits 0,...,7.

Original entry on oeis.org

0, 7, 14, 21, 35, 42, 56, 63, 70, 77, 105, 112, 126, 133, 140, 147, 154, 161, 175, 203, 210, 217, 224, 231, 245, 252, 266, 273, 301, 315, 322, 336, 343, 350, 357, 364, 371, 406, 413, 420, 427, 434, 441, 455, 462, 476, 504, 511, 525, 532, 546, 553, 560, 567, 574

Offset: 1

Amarnath Murthy, May 28 2001

			147 =7*21, is a term containing digits less than 8.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

A061818, A061819, A061820, A061821, A061822.

Select[7*Range[0,2000],Max[IntegerDigits[#]]<=7&] (* Enrique Pérez Herrero, May 13 2012 *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 30 2001

A061824 Multiples of 8 containing only the digits 0, ..., 8.

Original entry on oeis.org

0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 400, 408, 416, 424, 432, 440, 448, 456, 464

Offset: 1

Amarnath Murthy, May 28 2001

Subsequence of A008590. - Michel Marcus, Nov 28 2014

			288 = 8*36 is a term containing no digit 9.
The first multiple of 8 that is not here is 96.

Cf. A008590, A061818, A061819, A061820, A061821, A061822, A061823.

lista(nn) = {for (n=0, nn, d = vecsort(digits(m=8*n)); if (!vecsearch(d, 9), print1(m, ", ")););} \\ Michel Marcus, Nov 28 2014

More terms from Larry Reeves (larryr(AT)acm.org), May 30 2001

Offset changed to 1 by Michel Marcus, Nov 28 2014

cp's OEIS Frontend

A008585 a(n) = 3*n.

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A061818 Multiples of 2 containing only digits 0,1,2.

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A061820 Multiples of 4 containing only digits 0,...,4.

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A061821 Multiples of 5 containing only digits 0,...,5.

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A061822 Multiples of 6 containing only digits 0,...,6.

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A061823 Multiples of 7 containing only digits 0,...,7.

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A061824 Multiples of 8 containing only the digits 0, ..., 8.

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