A054540 -id:A054540 - cp's OEIS frontend

A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, 77, 82, 87, 97, 99, 102, 106, 114, 126, 131, 138, 145, 148, 155, 175, 177, 180, 182, 189, 197, 206, 209, 219, 221, 236, 238, 241, 243, 253, 258, 260, 273, 282, 309, 316, 319, 324, 339

Offset: 1

N. J. A. Sloane

Ulam conjectured that this sequence has density 0. However, calculations up to 6.759*10^8 (Jud McCranie) indicate that the density hovers near 0.074.
A plot of the first 3 million terms shows that they lie very close to the straight line 13.51*n, so even if we cannot prove it, we believe we now know how this sequence grows (see the plots in the links below). - N. J. A. Sloane, Sep 27 2006
After a few initial terms, the sequence settles into a regular pattern of dense clumps separated by sparse gaps, with period 21.601584+. This pattern continues up to at least a(n) = 5*10^6. (This comment is just a qualitative statement about the wavelike distribution of Ulam numbers, not meant to imply that every period includes Ulam numbers.) - David W. Wilson
_Don Knuth_ (Sep 26 2006) remarks that a(4952)=64420 and a(4953)=64682 (a gap of more than ten "dense clumps"); and there is a gap of 315 between a(18857) and a(18858).
1,2,3,47 are the only values of x < 6.759*10^8 such that x and x+1 are both Ulam numbers. - Jud McCranie, Jun 08 2001.  This holds through the first 28 billion Ulam numbers - Jud McCranie, Jan 07 2016.
From Jud McCranie on David W. Wilson's illustration, Jun 20 2008: (Start)
The integers are shown from left to right, top to bottom, with a dot where there is an Ulam number. I think his plot is 216 wide. The local density of Ulam numbers goes in waves with a period of 21.6+, so his plot shows ten cycles.
When they are arranged that way you can see the waves. The crests of the density waves don't always have Ulam numbers there but the troughs are practically void of Ulam numbers. I noticed that the ratio of that period (21.6+) to the frequency of Ulam numbers (1 in 13.52) is very close to 8/5. (End)
a(50000000) = 675904508. - Jud McCranie, Feb 29 2012
a(100000000) = 1351856726. - Jud McCranie, Jul 31 2012
a(1000000000) = 13517664323. - Jud McCranie, Aug 28 2015
a(28000000000) = 378485625853 - Philip Gibbs & Jud McCranie, Sep 09 2015
3 (=1+2) and 131 (=62+69) are the only two Ulam numbers in the first 28 billion Ulam numbers that are the sum of two consecutive Ulam numbers. - Jud McCranie, Jan 09 2016
Named after the Polish-American scientist Stanislaw Ulam (1909-1984). - Amiram Eldar, Jun 08 2021

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.2.
Richard K. Guy, Unsolved Problems in Number Theory, C4.
Donald E. Knuth, The Art of Computer Programming, Volume 4A, Section 7.1.3.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 116.
Marvin C. Wunderlich, The improbable behavior of Ulam's summation sequence, pp. 249-257 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
David Zeitlin, Ulam's sequence {U_n}, U_1=1, U_2=2, is a complete sequence, Notices Amer. Math. Soc., 22 (No. 7, 1975), Abstract 75T-A267, p. A-707.

Jud McCranie, Table of n, a(n) for n = 1..10000
Richard A. Becker, Plot of residuals a(n) - 13.5167*n for n <= 3000000, postscript file [uses Jud McCranie's values of a(n)].
Richard A. Becker, Plot of residuals a(n) - 13.5167*n for n <= 3000000, pdf file [uses Jud McCranie's values of a(n)].
Thomas Bloom, Problem 342, Erdős Problems.
Henning Fernau and Kshitij Gajjar, The Space Complexity of Sum Labelling, arXiv:2107.12973 [cs.DS], 2021.
Steven R. Finch, Ulam s-Additive Sequences. [From Steven Finch, Apr 20 2019]
Steven R. Finch, Stolarsky-Harborth Constant.
Steven R. Finch, Stolarsky-Harborth Constant. [From the Wayback machine]
Philip Gibbs, An Efficient Way to Compute Ulam Numbers.
Philip Gibbs, A Conjecture for Ulam Sequences.
Philip Gibbs and Judson McCranie, The Ulam Numbers up to One Trillion, (2017).
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).
Alois P. Heinz, Ulam spiral of Ulam numbers.
Donald E. Knuth, Downloadable programs
Noah Kravitz and Stefan Steinerberger, Ulam Sequences and Ulam Sets, arXiv:1705.01883 [math.CO], 2017.
Borys Kuca, Structures in Additive Sequences, arXiv:1804.09594 [math.NT], 2018. See p. 3.
Jud McCranie, Density 10^2 to 10^12
Jud McCranie, Density 10^4 to 10^12
Jud McCranie, Density 10^6 to 10^12
J. W. Moon, R. K. Guy, and N. J. A. Sloane, Correspondence, 1973.
Ed Pegg, Jr., Graph of 10^6 terms of a(n) - 13.5*n.
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #8. [Annotated and scanned copy]
R. Queneau, Sur les suites s-additives, J. Combin. Theory, A12 (1972), 31-71.
Bernardo Recamán, Questions on a sequence of Ulam, Amer. Math. Monthly, Vol. 80, No. 8 (1973), pp. 919-920.
Ivano Salvo and Agnese Pacifico, Computing Integer Sequences: Filtering vs Generation (Functional Pearl), arXiv:1807.11792 [cs.PL], 2018.
James Schmerl and Eugene Spiegel, The regularity of some 1-additive sequences, J. Combin. Theory. Ser. A, Vol. 66, No. 1 (1994), pp. 172-175. Math. Rev. 95h:11010.
Arseniy Sheydvasser, The Ulam Sequence of the Integer Polynomial Ring, arXiv:1910.00109 [math.NT], 2019.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282).
Arseniy (Senia) Sheydvasser, The Ulam Sequence of Linear Integer Polynomials, J. Int. Seq., Vol. 24 (2021), Article 21.10.8.
Stefan Steinerberger, A hidden signal in the Ulam sequence, Research Report YALEU/DCS/TR-1508, Yale University, 2015.
Stefan Steinerberger, A hidden signal in the Ulam sequence, Figure 2 from Research Report YALEU/DCS/TR-1508, Yale University, 2015.
Stefan Steinerberger, The Ulam Sequence, blog post, April 12, 2016.
Stanislaw M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]
Stanislaw Ulam, Combinatorial analysis in infinite sets and some physical theories, SIAM Rev., Vol. 6, No. 4 (1964), pp. 343-355. Reprinted in Selected Papers, MIT Press, see p. 393.
Eric Weisstein's World of Mathematics, Ulam Sequence.
Wikipedia, Ulam number.
David W. Wilson, Plot of initial terms, showing their quasiperiodicity as vertical bars. The image width was chosen to include approximately 10 periods. For an explanation of this picture, see Comments above.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.
David Zeitlin, Ulam's sequence {U_n}, U_1=1, U_2=2, is a complete sequence, Notices Amer. Math. Soc., 22 (No. 7, 1975), Abstract 75T-A267, p. A-707. (Annotated scanned copy)
Index entries for Ulam numbers

Cf. A002859 (version beginning 1,3), A054540, A003667, A001857, A007300, A117140, A214603.
First differences: A072832, A072540.
Cf. A080287, A080288, A004280 (if distinct removed from definition).
Cf. A199016, A199017, A033629, A274522.
See also the density plots in A080573 and A285884.

a002858 n = a002858_list !! (n-1)
a002858_list = 1 : 2 : ulam 2 2 a002858_list
ulam :: Int -> Integer -> [Integer] -> [Integer]
ulam n u us = u' : ulam (n + 1) u' us where
   u' = f 0 (u+1) us'
   f 2 z _                         = f 0 (z + 1) us'
   f e z (v:vs) | z - v <= v       = if e == 1 then z else f 0 (z + 1) us'
                | z - v `elem` us' = f (e + 1) z vs
                | otherwise        = f e z vs
   us' = take n us
-- Reinhard Zumkeller, Nov 03 2011

function isUlam(u, n, h, i, r)
    h == 2 && return false
    ur = u[r]; ui = u[i]
    ur <= ui && return h == 1
    if ur + ui > n
        r -= 1
    elseif ur + ui < n
        i += 1
    else
        h += 1; i += 1; r -= 1
    end
    isUlam(u, n, h, i, r)
end
function UlamList(len)
    u = Array{Int, 1}(undef, len)
    u[1] = 1; u[2] = 2; i = 2; n = 2
    while i < len
        n += 1
        if isUlam(u, n, 0, 1, i)
            i += 1
            u[i] = n
        end
    end
    return u
end
println(UlamList(59)) # Peter Luschny, Apr 07 2019

UlamList := proc(len) local isUlam, nextUlam, behead; behead := u -> u[2..numelems(u)]; isUlam := proc(n, h, u, r) local hu, tu, hr, tr; hu := u[1]; hr := r[1]; if h = 2 then return false fi; if hr <= hu then return evalb(h = 1) fi; if (hr + hu) = n then tu := behead(u); tr := behead(r); return isUlam(n, h+1, tu, tr) fi; if (hr + hu) < n then tu := behead(u): return isUlam(n, h, tu, r) fi; tr := behead(r); isUlam(n, h, u, tr) end: nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len-1 then return [op(u), n] fi; nextUlam(n+1, [op(u), n], [n, op(r)]) else nextUlam(n+1, u, r) fi end: nextUlam(3, [1, 2], [2, 1]) end:
UlamList(59); # Peter Luschny, Apr 05 2019

Ulam4Compiled = Compile[{{nmax, _Integer}, {init, _Integer, 1}, {s, _Integer}}, Module[{ulamhash = Table[0, {nmax}], ulam = init}, ulamhash[[ulam]] = 1; Do[ If[Quotient[Plus @@ ulamhash[[i - ulam]], 2] == s, AppendTo[ulam, i]; ulamhash[[i]] = 1], {i, Last[init] + 1, nmax}]; ulam]]; ulams = Ulam4Compiled[355, {1, 2}, 1]
(* Second program: *)
ulams = {1, 2}; Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {100}]; ulams (* Jean-François Alcover, Sep 08 2011 *)
findUlams[s_List, j_Integer] := Block[{k = s[[-1]] + 1, ss = Plus @@@ Subsets[s, {j}]}, While[ Count[ss, k] != 1, k++]; Append[s, k]]; ulams = Nest[findUlams[#, 2] &, {1, 2}, 70] (* Robert G. Wilson v, Jul 05 2014 *)

aupto(N)= my(seen=vector(N), U=[]); seen[1]=seen[2]=1; for(i=1,N, if(1==seen[i], for(j=1,#U, my(sum=i+U[j]); if(sum>N, break); seen[sum]++); U=concat(U,i))); U \\ Ruud H.G. van Tol, Dec 29 2022

def isUlam(n, h, u, r):
    if h == 2: return False
    hu = u[0]; hr = r[0]
    if hr <= hu: return h == 1
    if hr + hu > n: r = r[1:]
    elif hr + hu < n: u = u[1:]
    else: h += 1; r = r[1:]; u = u[1:]
    return isUlam(n, h, u, r)
def UlamList(length):
    u = [1, 2]; r = [2, 1]; n = 2
    while len(u) < length:
        n += 1
        if isUlam(n, 0, u[:], r[:]):
            u.append(n); r.insert(0, n)
    return u
print(UlamList(59)) # Peter Luschny, Apr 06 2019

More terms from Jud McCranie

A001149 A self-generating sequence: a(1)=1, a(2)=2, a(n+1) chosen so that a(n+1)-a(n-1) is the first number not obtainable as a(j)-a(i) for 1<=i

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 17, 26, 34, 45, 54, 67, 81, 97, 115, 132, 153, 171, 198, 228, 256, 288, 323, 357, 400, 439, 488, 530, 581, 627, 681, 732, 790, 843, 908, 963, 1029, 1085, 1152, 1213, 1284, 1346, 1418, 1484, 1561, 1630, 1710, 1785, 1867, 1945, 2034, 2116

Offset: 1

N. J. A. Sloane

nonn

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).

Manfred Scheucher, Table of n, a(n) for n = 1..2000
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Manfred Scheucher, Python Script

Cf. A005282, A054540.

Description corrected and moved to name line by Franklin T. Adams-Watters, Nov 01 2009

More terms from Manfred Scheucher, Jul 01 2015

A060526 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of six simple musical tones: 8/7 5/4 4/3 3/2 8/5 7/4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 12, 15, 19, 21, 22, 31, 53, 84, 87, 94, 99, 118, 130, 140, 171, 270, 410, 441, 612, 935, 966, 1053, 1106, 1277, 1547, 1578, 2954, 3125, 3566, 6691, 9816, 11664, 14789, 18355, 39835, 48545, 54624, 58190, 59768, 63334, 81689, 84814

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 01 2001

nonn

The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 84814.

The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

			84 = 53 + the previous term 31. Again, 291152 = 103169 + the previous terms (84814 + 81689 + 11664 + 9816).

Cf. A054540, A060525, A060257.

Recurrence: the next term equals the current term plus one or more of the previous terms: a(n+1) = a(n) + a(n-x)... + a(n-y)... +a(n-z)..., etc.

A060525 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to four of the simple ratios of musical harmony: 5/4, 4/3, 3/2 and 8/5.

Original entry on oeis.org

1, 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, 12276, 16572, 20868, 25164, 48545, 69413, 73709, 78005, 151714, 229719, 689157, 792326, 944040, 1022045, 1173759, 1251764, 2733247, 3985011

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 01 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to over 3985011. The self-accumulating nature of this sequence fails once, between the third and fourth terms. The sequence therefore does not meet the rigorous definition of 'impeccable' recurrence. The otherwise perfect recurrence in this sequence is of the type seen in A054540.

The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

Cf. A054540, A060526, A060527.

A060527 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of 8 musical tones: 8/7 16/11 5/4 4/3 3/2 8/5 11/8 7/4.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 22, 26, 31, 41, 63, 72, 87, 109, 161, 202, 224, 270, 494, 612, 742, 764, 836, 1012, 1084, 1106, 1308, 1417, 1578, 3426, 4843, 6421, 6691, 10698, 12276, 18355, 19461, 21039, 22887, 25046, 26894, 31737, 33585, 35163

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 01 2001

nonn

The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 35163.

The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

			109 = 87 + the previous term 22. Again, 184417 = 121524 + the previous terms (54624 and 6691 and 1578).

Cf. A054540, A060525, A060526.

Recurrence: the next term equals the current term plus one or more of the previous terms. a(n+1) = a(n) + a(n-x)... + a(n-y)... + a(n-z)..., etc.

A060528 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 29, 41, 53, 200, 253, 306, 359, 665, 8286, 8951, 9616, 10281, 10946, 11611, 12276, 12941, 13606, 14271, 14936, 15601, 31867, 79335, 111202, 190537, 5446238, 5636775, 5827312, 6017849, 6208386, 6398923, 6589460, 6779997, 6970534, 7161071

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 12 2001

The sequence was found by a computer search of all the equal divisions of the octave from 1 to over 6589460. This is not a perfect recurrent sequence because its self-accumulating nature fails between the 9th and 10th terms, between the 14th and 15th terms, and between the 30th and 31st terms. The examples of recurrence which are present in this sequence are of the same type that is seen in sequences A054540, A060526 and A060527. The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts. - corrected by K. G. Stier, Jan 29 2015

Also the denominators of increasingly better rational approximations to log(3)/log(2) = 1.5849625... (see A020857). The respective numerators are A254351. The reason why the sequence's "self-accumulating nature fails between the 9th and 10th terms, the 14th and 15th terms and the 30th and 31st terms" (see original comment) is simply that 84/53, 1054/665 and 301994/190537 are very good approximations, thus followed by a jump. (E.g., this phenomenon can also be seen in the numerators and denominators of rational approximations to Pi.). - K. G. Stier, Jan 29 2015

Cf. A054540, A060525, A060526, A060527, A254351.

A005664 is a subsequence, A206788 is a supersequence.

x:bfloat(log(3)/log(2)),fpprec:100, errold:2,for denominator:1 thru 10000 do (numerator:round(x*denominator), errnew:abs(x-numerator/denominator), if errnew < errold then (errold:errnew, print(denominator))); /* K. G. Stier, Jan 29 2015 */

lista(nn) = {d = 2; v = log(3)/log(2); for (den=1, nn, num = round(v*den); newd = abs(v-num/den); if (newd < d, print1(den, ", "); d = newd;););} \\ after Maxima, Michel Marcus, Feb 28 2015

Incorrect term 571611 removed by K. G. Stier, Jan 29 2015

More terms from Jon E. Schoenfield, Feb 06 2015

A117538 Locations of the increasing peak values of the integral of the absolute value of the Riemann zeta function between successive zeros on the critical line. This can also be defined in terms of the Z function; if t and s are successive zeros of a renormalized Z function, z(x) = Z(2 Pi x/log(2)), then take the integral between t and s of |z(x)|. For each successively higher value of this integral, the corresponding term of the integer sequence is r = (t+s)/2 rounded to the nearest integer.

Original entry on oeis.org

2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973

Offset: 0

Gene Ward Smith, Mar 27 2006

The fractional parts of the numbers r = (t+s)/2 above are very unevenly distributed. For all of the values in the table, the integers are in fact the unique integers contained in the interval of zeros [t, s] of z(x). An interesting challenge to anyone wishing to do computations related to the zeta function would be to find the first counterexample, where in fact the peak value interval did not contain the corresponding integer. Perhaps even more than the peak values of the zeta function themselves, these integrals are extremely closely related to relatively good equal divisions of the octave in music theory.

Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
Paris, R. B. and Kaminski, D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001

Andrew Odlyzko, The first 100,000 zeros of the Riemann zeta function, accurate to within 3*10^(-9)
Wikipedia, Z function
Index entries for zeta function.

Cf. A117536, A117537, A117539, A054540.

Extended by T. D. Noe, Apr 21 2010

A060529 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of three complementary pairs of simple musical tones: 7/6 and 12/7, 6/5 and 5/3 and 7/5 and 10/7.

Original entry on oeis.org

1, 2, 3, 4, 12, 14, 15, 18, 19, 23, 27, 45, 68, 72, 99, 171, 346, 445, 517, 616, 688, 787, 1133, 1304, 3912, 7136, 8440, 9744, 11048, 12352, 18355, 19659, 20963, 22267, 26795, 28099, 29403, 30707, 40451, 41755, 69854, 71158, 72462, 143620, 216082

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 12 2001

nonn

The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 216082. The self-accumulating nature of this sequence fails once, between the fourth and fifth terms. The sequence therefore does not meet the rigorous definition of 'impeccable' recurrence. The otherwise perfect recurrence in this sequence is of the type seen in sequences A054540, A060526 and A060527. The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

A054540, A060525, A060526, A060527, A060528, A001149, A000045.

A117536 Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i2(Pi/log(2))*t)) for increasing real t.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691

Offset: 0

Gene Ward Smith, Mar 27 2006

These correspond to increasing peaks of the absolute value of the Riemann zeta function along the critical line. If Z'(s)=0 is a positive zero of the derivative of Z, then |Z(s)| is the peak value.
The fractional parts of these values are not randomly distributed; r = log(2) * s(n) / (2*Pi) shows a very strong tendency to be near an integer.
It would be interesting to have theorems on the distribution of the fractional part of the "r" above, for which the Riemann hypothesis would surely be needed. It would be particularly interesting to know if the absolute value's fractional part is constrained to be less than some bound, such as 0.25. This computation could be pushed much farther by someone using a better algorithm, for instance the Riemann-Siegel formula and better computing resources. The computations were done using Maple's accurate but very slow zeta function evaluation. They are correct as far as they go, but do not go very far. The terms of the sequence have an interpretation in terms of music theory; the terms which appear in it, 12, 19, 22 and so forth, are equal divisions of the octave which do relatively well approximating intervals given by rational numbers with small numerators and denominators.
This sequence was extended by examining the peaks of |zeta(0.5+xi)| between each the first million zeros of the zeta function. These record peaks occur between zeros that are relatively far apart. The fractional part of r decreases as the magnitude of r increases. - T. D. Noe, Apr 19 2010

			The function f(m) = |zeta(1/2 + i*2*(Pi/log(2))*m)| has a local maximum f(m') ~ 3.66 at m' ~ 5.0345, which corresponds to a(5)=round(m)=5. The peak at f(6.035) ~ 2.9 is smaller, and after two more smaller local maxima, there is a larger peak at f(6.9567) ~ 4.167, whence a(6)=7.

H. M. Edwards, Riemann's Zeta-Function, Academic Press, 1974.
K. Ramachandra, On the Mean-Value and Omega-Theorems for the Riemann Zeta-Function, Springer-Verlag, 1995.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986.

Andrew Odlyzko, Tables of the zeros of the Riemann zeta function.
Wikipedia, Z function
Index entries for zeta function.

Cf. A117537, A117538, A117539, A079630, A088749, A088750, A054540.

{my(c=I/log(2)*2*Pi,f(n)=abs(zeta(.5+n*c)), m=0,
find(x,d,e=1e-6)=my(y=f(x)); while(y<(y=f(x+=d)) || eM. F. Hasler, Jan 26 2012

Extended by T. D. Noe, Apr 19 2010

A007335 MU-numbers: next term is uniquely the product of 2 earlier terms.

Original entry on oeis.org

2, 3, 6, 12, 18, 24, 48, 54, 96, 162, 192, 216, 384, 486, 768, 864, 1458, 1536, 1944, 3072, 3456, 4374, 6144, 7776, 12288, 13122, 13824, 17496, 24576, 31104, 39366, 49152, 55296, 69984, 98304, 118098, 124416, 157464, 196608, 221184, 279936

Offset: 1

N. J. A. Sloane

All terms are 3-smooth. - Reinhard Zumkeller, Aug 13 2015

Empirically, this sequence corresponds to numbers of the form 2^v * 3^w with v = 1 or w = 1 or v and w both odd (see illustration in Links section). - Rémy Sigrist, Feb 16 2023

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 359.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).
Rémy Sigrist, Scatterplot of the 2-adic valuation of a(n) vs the 3-adic valuation of a(n) for n = 1..50000
Robert G. Wilson v, Note, n.d.

Subsequence of A000423.

Cf. A054540, A003586, A261255.

a007335 n = a007335_list !! (n-1)
a007335_list = 2 : 3 : f [3, 2] (singleton 6 1) where
   f xs m | v == 1    = y : f (y : xs) (g (map (y *) xs) m')
          | otherwise = f xs m'
          where g [] m = m
                g (z:zs) m = g zs $ insertWith (+) z 1 m
                ((y,v),m') = deleteFindMin m
-- Reinhard Zumkeller, Aug 13 2015

function isMU(u, n, h, i, r)
    ur = u[r]; ui = u[i]
    ur <= ui && return h
    if ur * ui > n
        r -= 1
    elseif ur * ui < n
        i += 1
    else
        h && return false
        h = true; i += 1; r -= 1
    end
    isMU(u, n, h, i, r)
end
function MUList(len)
    u = Array{Int, 1}(undef, len)
    u[1] = 2; u[2] = 3; i = 2; n = 2
    while i < len
        n += 1
        if isMU(u, n, false, 1, i)
            i += 1
            u[i] = n
        end
    end
    return u
end
MUList(41) |> println # Peter Luschny, Apr 07 2019

s={2,3}; Do[n=Select[ Table[s[[j]] s[[k]], {j, Length@s}, {k, j+1, Length@s}] // Flatten // Sort // Split, #[[1]] > s[[-1]] && Length[#] == 1 &][[1,1]]; AppendTo[s, n], {39}]; s  (* Jean-François Alcover, Apr 22 2011 *)
Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Times @@ # &, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {2, 3}, 39] (* Michael De Vlieger, Nov 16 2017 *)

a(n) = A003586(A261255(n)). - Reinhard Zumkeller, Aug 13 2015

Conjecture: Sum_{n>=1} 1/a(n) = 181/144. - Amiram Eldar, Jul 31 2022

cp's OEIS Frontend

A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

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A001149 A self-generating sequence: a(1)=1, a(2)=2, a(n+1) chosen so that a(n+1)-a(n-1) is the first number not obtainable as a(j)-a(i) for 1<=i

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A060526 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of six simple musical tones: 8/7 5/4 4/3 3/2 8/5 7/4.

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A060525 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to four of the simple ratios of musical harmony: 5/4, 4/3, 3/2 and 8/5.

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A060527 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of 8 musical tones: 8/7 16/11 5/4 4/3 3/2 8/5 11/8 7/4.

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A060528 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2.

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A060529 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of three complementary pairs of simple musical tones: 7/6 and 12/7, 6/5 and 5/3 and 7/5 and 10/7.

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A117536 Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i*2*(Pi/log(2))*t)) for increasing real t.

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A117536 Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i2(Pi/log(2))*t)) for increasing real t.