A035504 -id:A035504 - cp's OEIS frontend

A016777 a(n) = 3*n + 1.

1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178, 181, 184, 187

Offset: 0

N. J. A. Sloane, Dec 11 1996

Numbers k such that the concatenation of the first k natural numbers is not divisible by 3. E.g., 16 is in the sequence because we have 123456789101111213141516 == 1 (mod 3).
Ignoring the first term, this sequence represents the number of bonds in a hydrocarbon: a(#of carbon atoms) = number of bonds. - Nathan Savir (thoobik(AT)yahoo.com), Jul 03 2003
n such that Sum_{k=0..n} (binomial(n+k,n-k) mod 2) is even (cf. A007306). - Benoit Cloitre, May 09 2004
Hilbert series for twisted cubic curve. - Paul Barry, Aug 11 2006
If Y is a 3-subset of an n-set X then, for n >= 3, a(n-3) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
a(n) = A144390 (1, 9, 23, 43, 69, ...) - A045944 (0, 5, 16, 33, 56, ...). From successive spectra of hydrogen atom. - Paul Curtz, Oct 05 2008
Number of monomials in the n-th power of polynomial x^3+x^2+x+1. - Artur Jasinski, Oct 06 2008
A145389(a(n)) = 1. - Reinhard Zumkeller, Oct 10 2008
Union of A035504, A165333 and A165336. - Reinhard Zumkeller, Sep 17 2009
Hankel transform of A076025. - Paul Barry, Sep 23 2009
From Jaroslav Krizek, May 28 2010: (Start)
a(n) = numbers k such that the antiharmonic mean of the first k positive integers is an integer.
A169609(a(n-1)) = 1. See A146535 and A169609. Complement of A007494.
See A005408 (odd positive integers) for corresponding values A146535(a(n)). (End)
Apart from the initial term, A180080 is a subsequence; cf. A180076. - Reinhard Zumkeller, Aug 14 2010
Also the maximum number of triangles that n + 2 noncoplanar points can determine in 3D space. - Carmine Suriano, Oct 08 2010
A089911(4*a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013
The number of partitions of 6*n into at most 2 parts. - Colin Barker, Mar 31 2015
For n >= 1, a(n)/2 is the proportion of oxygen for the stoichiometric combustion reaction of hydrocarbon CnH2n+2, e.g., one part propane (C3H8) requires 5 parts oxygen to complete its combustion. - Kival Ngaokrajang, Jul 21 2015
Exponents n > 0 for which 1 + x^2 + x^n is reducible. - Ron Knott, Oct 13 2016
Also the number of independent vertex sets in the n-cocktail party graph. - Eric W. Weisstein, Sep 21 2017
Also the number of (not necessarily maximal) cliques in the n-ladder rung graph. - Eric W. Weisstein, Nov 29 2017
Also the number of maximal and maximum cliques in the n-book graph. - Eric W. Weisstein, Dec 01 2017
For n>=1, a(n) is the size of any snake-polyomino with n cells. - Christian Barrientos and Sarah Minion, Feb 27 2018
The sum of two distinct terms of this sequence is never a square. See Lagarias et al. p. 167. - Michel Marcus, May 20 2018
It seems that, for any n >= 1, there exists no positive integer z such that digit_sum(a(n)*z) = digit_sum(a(n)+z). - Max Lacoma, Sep 18 2019
For n > 2, a(n-2) is the number of distinct values of the magic constant in a normal magic triangle of order n (see formula 5 in Trotter). - Stefano Spezia, Feb 18 2021
Number of 3-permutations of n elements avoiding the patterns 132, 231, 312. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
Erdős & Sárközy conjecture that a set of n positive integers with property P must have some element at least a(n-1) = 3n - 2. Property P states that, for x, y, and z in the set and z < x, y, z does not divide x+y. An example of such a set is {2n-1, 2n, ..., 3n-2}. Bedert proves this for large enough n. (This is an upper bound, and is exact for all known n; I have verified it for n up to 12.) - Charles R Greathouse IV, Feb 06 2023
a(n-1) = 3*n-2 is the dimension of the vector space of all n X n tridiagonal matrices, equals the number of nonzero coefficients: n + 2*(n-1) (see Wikipedia link). - Bernard Schott, Mar 03 2023

			G.f. = 1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 16*x^5 + 19*x^6 + 22*x^7 + ... - _Michael Somos_, May 27 2019

W. Decker, C. Lossen, Computing in Algebraic Geometry, Springer, 2006, p. 22.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.1 Terminology, p. 264.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Hacène Belbachir, Toufik Djellal, and Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
Benjamin Bedert, On a problem of Erdős and Sárközy about sequences with no term dividing the sum of two larger terms, arXiv preprint, arXiv:2301.07065 [math.NT], 2023.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Paul Erdős and András Sárközy, On the divisibility properties of sequences of integers, Proc. London Math. Soc. (3), 21 (1970), pp. 97-101.
Leonhard Euler, Observatio de summis divisorum p. 9.
Leonhard Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, see p. 9.
L. B. W. Jolley, Summation of Series, Dover, 1961, pp. 16, 38.
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
J. C. Lagarias, A. M. Odlyzko, and J. B. Shearer, On the density of sequences of integers the sum of no two of which is a square. I. Arithmetic progressions, Journal of Combinatorial Theory. Series A, 33 (1982), pp. 167-185.
T. Mansour, Permutations avoiding a set of patterns from S_3 and a pattern from S_4, arXiv:math/9909019 [math.CO], 1999.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
Eric Weisstein's World of Mathematics, Book Graph
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Ladder Rung Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Wikipedia, Tridiagonal matrix.
Chengcheng Yang, A Problem of Erdös Concerning Lattice Cubes, arXiv:2011.15010 [math.CO], 2020. See Table p. 27.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000290, A001477, A005408, A007306, A007494, A016789, A016933.
Cf. A017569, A035504, A045944, A058183, A076025, A089911, A144390.
Cf. A145389, A146535, A165333, A165336, A169609, A180076, A180080.
Cf. A214550, A238731.
Cf. A007559 (partial products), A051536 (lcm).
First differences of A000326.
Row sums of A131033.
Complement of A007494. - Reinhard Zumkeller, Oct 10 2008
Some subsequences: A002476 (primes), A291745 (nonprimes), A135556 (squares), A016779 (cubes).

a016777 = (+ 1) . (* 3)
a016777_list = [1, 4 ..]  -- Reinhard Zumkeller, Feb 28 2013, Feb 10 2012

[3*n+1 : n in [1..70]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

Range[1, 199, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
(* Start from Eric W. Weisstein, Sep 21 2017 *)
3 Range[0, 70] + 1
Table[3 n + 1, {n, 0, 70}]
LinearRecurrence[{2, -1}, {1, 4}, 70]
CoefficientList[Series[(1 + 2 x)/(-1 + x)^2, {x, 0, 70}], x]
(* End *)

A016777(n):=3*n+1$
makelist(A016777(n),n,0,30); /* Martin Ettl, Oct 31 2012 */

a(n)=3*n+1 \\ Charles R Greathouse IV, Jul 28 2015

[3*n+1 for n in range(1,71)] # G. C. Greubel, Mar 15 2024

G.f.: (1+2*x)/(1-x)^2.
a(n) = A016789(n) - 1.
a(n) = 3 + a(n-1).
Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) + log(2)). [Jolley, p. 16, (79)] - Benoit Cloitre, Apr 05 2002
(1 + 4*x + 7*x^2 + 10*x^3 + ...) = (1 + 2*x + 3*x^2 + ...)/(1 - 2*x + 4*x^2 - 8*x^3 + ...). - Gary W. Adamson, Jul 03 2003
E.g.f.: exp(x)*(1+3*x). - Paul Barry, Jul 23 2003
a(n) = 2*a(n-1) - a(n-2); a(0)=1, a(1)=4. - Philippe Deléham, Nov 03 2008
a(n) = 6*n - a(n-1) - 1 (with a(0) = 1). - Vincenzo Librandi, Nov 20 2010
Sum_{n>=0} 1/a(n)^2 = A214550. - R. J. Mathar, Jul 21 2012
a(n) = A238731(n+1,n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-5)^k. - Philippe Deléham, Mar 05 2014
Sum_{i=0..n} (a(i)-i) = A000290(n+1). - Ivan N. Ianakiev, Sep 24 2014
From Wolfdieter Lang, Mar 09 2018: (Start)
a(n) = denominator(Sum_{k=0..n-1} 1/(a(k)*a(k+1))), with the numerator n = A001477(n), where the sum is set to 0 for n = 0. [Jolley, p. 38, (208)]
G.f. for {n/(1 + 3*n)}_{n >= 0} is (1/3)*(1-hypergeom([1, 1], [4/3], -x/(1-x)))/(1-x). (End)
a(n) = -A016789(-1-n) for all n in Z. - Michael Somos, May 27 2019

Better description from T. D. Noe, Aug 15 2002

Partially edited by Joerg Arndt, Mar 11 2010

A046197 Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits.

Original entry on oeis.org

0, 1, 153, 370, 371, 407

Offset: 1

Richard C. Schroeppel

Suppose n has d digits; then the sum of the cubes of its digits is <= 729d and n >= 10^(d-1). So d <= 5. It is now easy to check that the numbers shown are the only solutions. [Corrected by M. F. Hasler, Apr 12 2015]

This is row n=3 of A252648. - M. F. Hasler, Apr 12 2015

			1^3 + 5^3 + 3^3 = 153. 3^3+7^3 +0^3 = 370.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 153, p. 50, Ellipses, Paris 2008.
G. H. Hardy, A Mathematician's Apology, Cambridge, 1967.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 60-62.
J. Shallit, Number theory and formal languages, in Emerging applications of number theory (Minneapolis, MN, 1996), 547-570, IMA Vol. Math. Appl., 109, Springer, New York, 1999.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 140.

H. Lehning, La migration des nombres vers le bonheur, Tangente: L'aventure mathématique, pp. 27 No. 108 Jan-Feb 2006 Pole Paris.

Cf. A005188, A023052, A046156.
Cf. A165330, A035504, A008585, A165333, A165334, A165335.
Cf. A052455, A052464, A124068, A124069, A226970, A003321.

Select[Range[0,407],Total[IntegerDigits[#]^3]==# &] (* Stefano Spezia, Sep 08 2024 *)

for(n=0,10^5,A055012(n)==n&&print1(n",")) \\ M. F. Hasler, Apr 12 2015

A055012(a(n))=a(n); A165331(a(n))=0; subset of A031179. - Reinhard Zumkeller, Sep 17 2009

A031179 Numbers having period-1 3-digitized sequences.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 83, 84, 85

Offset: 1

Eric W. Weisstein

Equivalently, numbers that eventually reach a fixed point under "x -> sum of cubes of digits of x". - Reinhard Zumkeller, Sep 17 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Digitaddition

Complement of A165336.
Union of A035504, A008585, A165333, A165334, and A165335.
Cf. A031178.

Missing a(0)=0 inserted by Sean A. Irvine, Apr 14 2020

A165333 Numbers that eventually reach the fixed point 370 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

7, 19, 34, 37, 43, 58, 67, 70, 73, 76, 85, 88, 91, 109, 118, 124, 139, 142, 145, 148, 154, 157, 166, 169, 175, 178, 181, 184, 187, 190, 193, 196, 214, 223, 226, 232, 241, 247, 259, 262, 268, 274, 277, 286, 295, 304, 307, 319, 322, 334, 340, 343, 346, 355, 358

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 370;

Subsequence of A031179 and of A016777; a(n) mod 3 = 1.

			a(3)=34: 34 -> 3^3+4^3=91 -> 9^3+1=730 -> 7^3+3^3+0=370.

Cf. A031179, A046197, A035504, A008585, A165334, A165335.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ ,All]]; Select[Range[358], Last[Trajectory[#]] == 370&] (* Ant King, May 24 2013 *)

A165334 Numbers that eventually reach the fixed point 371 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 50, 53, 56, 59, 62, 65, 68, 71, 80, 83, 86, 92, 95, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179, 182, 185, 188, 191

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 371;
Subsequence of A031179;
complement of A165335 with respect to A016789; a(n) mod 3 = 2.

			a(10)=29: 29 -> 2^3+9^3=737 -> 2*7^3+3^3=713 -> 7^3+1+3^3=371.

Cf. A031179, A046197, A035504, A008585, A165333.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f, n, UnsameQ ,All]]; Select[Range[191], Last[Trajectory[#]]==371 &] (* Ant King, May 24 2013 *)

A165335 Numbers that eventually reach the fixed point 407 under "x -> sum of cubes of digits of x" (see A055012).

Original entry on oeis.org

47, 74, 77, 89, 98, 407, 449, 470, 494, 578, 587, 668, 686, 704, 707, 740, 758, 770, 785, 788, 809, 857, 866, 875, 878, 887, 890, 908, 944, 980, 1124, 1139, 1142, 1148, 1157, 1175, 1178, 1184, 1187, 1193, 1214, 1241, 1319, 1367, 1376, 1391, 1412, 1418, 1421

Offset: 1

Reinhard Zumkeller, Sep 17 2009

A165330(a(n)) = 407.

			a(4)=89: 89 -> 8^3+9^3=1241 -> 1+2^3+4^3+1=74 -> 7^3+4^3=407.

Subsequence of A031179.

Cf. A046197, A035504, A008585, A165333.

f[n_] := Plus@@(IntegerDigits[n]^3); Trajectory[n_] := Most[NestWhileList[f,n,UnsameQ,All]]; Select[Range[1421], Last[Trajectory[#]]==407 &] (* Ant King, May 24 2013 *)
Select[Range[1500],FixedPoint[Total[IntegerDigits[#]^3]&,#,100]==407&] (* Harvey P. Dale, Apr 17 2020 *)

Complement of A165334 with respect to A016789; a(n) mod 3 = 2.

A219111 Numbers that eventually reach 1 under "x -> sum of 4th power of digits of x".

Original entry on oeis.org

1, 10, 100, 1000, 10000, 11123, 11132, 11213, 11231, 11312, 11321, 12113, 12131, 12311, 13112, 13121, 13211, 21113, 21131, 21311, 23111, 31112, 31121, 31211, 32111, 44688, 44868, 44886, 46488, 46848, 46884, 48468, 48486, 48648, 48684, 48846, 48864, 64488

Offset: 1

Michel Lagneau, Nov 12 2012

			11123 is in the sequence because 1^4 +1^4+1^4+2^4+3^4 = 100 and 1^4 + 0^4 + 0^4 = 1.
2 is not in the sequence because  2 --> 16 --> 1297 -->8979 -->19619 -->14420 -->529 -->7202 -->2433 -->434 -->593 -->7267 -->6114 -->1554 -->1507 -->3027 -->2498 -->10929 -->13139 -->6725 -->4338 -->4514 -->1138 -->4179 -->9219 -->13139 which repeats with period 7, so never reaches 1.

Cf. A007770, A035504.

Select[Range[50000], FixedPoint[Total[IntegerDigits[#]^4]&, #, 10]==1&]

A219112 Numbers that eventually reach 1 under "x -> sum of 5th power of digits of x".

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 1111222, 1112122, 1112212, 1112221, 1121122, 1121212, 1121221, 1122112, 1122121, 1122211, 1211122, 1211212, 1211221, 1212112, 1212121, 1212211, 1221112, 1221121, 1221211, 1222111, 2111122, 2111212, 2111221, 2112112

Offset: 1

Michel Lagneau, Nov 12 2012

			1112122 is in the sequence because 1^5 +1^5+1^5+2^5+1^5 +2^5+2^5 = 100 and 1^5 + 0^5 + 0^5 = 1.

Cf. A007770, A035504, A219111.

Select[Range[50000], FixedPoint[Total[IntegerDigits[#]^5]&, #, 10]==1&]

A383646 Smallest number that takes n steps to reach 1 under iteration of sum-of-cubes-of-digits map.

Original entry on oeis.org

1, 10, 112, 1189, 778, 13477, 2388889999999999999999

Offset: 0

Shyam Sunder Gupta, May 11 2025

These could also be called the smallest cubic happy numbers of height n.
Subsequent terms are too large to display in full.
a(7) = 1127 * 10^3276941015089163237 - 1 (1126 followed by 3276941015089163237 nines).
a(8) = 35678 * 10^((a(7) - 1054)/729) - 1.

			a(2) = 112, because 112 takes 2 steps to reach 1 (112 --> 1 + 1 + 8 = 10 --> 1 + 0 = 1).

H. G. Grundman and E. A. Teeple, Heights of happy numbers and cubic happy numbers, Fib Quart. 41 (4) (2003) 301-306.
Shyam Sunder Gupta, Happy Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 7, 209-222.

Cf. A007770, A001273, A035504.

cp's OEIS Frontend

A016777 a(n) = 3*n + 1.

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A046197 Fixed points for operation of repeatedly replacing a number with the sum of the cubes of its digits.

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A031179 Numbers having period-1 3-digitized sequences.

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A165333 Numbers that eventually reach the fixed point 370 under "x -> sum of cubes of digits of x" (see A055012).

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A165334 Numbers that eventually reach the fixed point 371 under "x -> sum of cubes of digits of x" (see A055012).

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A165335 Numbers that eventually reach the fixed point 407 under "x -> sum of cubes of digits of x" (see A055012).

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A219111 Numbers that eventually reach 1 under "x -> sum of 4th power of digits of x".

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A219112 Numbers that eventually reach 1 under "x -> sum of 5th power of digits of x".

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