A022843 -id:A022843 - cp's OEIS frontend

A226898 Hooley's Delta function: maximum number of divisors of n in [u, eu] for all u. (Here e is Euler's number 2.718... = A001113.)

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 2, 2, 2, 1, 2, 1, 3, 1, 4, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 2, 2

Offset: 1

Charles R Greathouse IV, Jun 21 2013

This function measures the tendency of divisors of a number to cluster.
Tenenbaum (1985) proves that a(1) + ... + a(n) < n exp(c sqrt(log log n log log log n)) for some constant c > 0 and all n > 16. In particular, the average order of a(n) is O((log n)^k) for any k > 0.
Maier & Tenenbaum show that (log log n)^(g + o(1)) < a(n) < (log log n)^(log 2 + o(1)) for almost all n, with g = log 2/log((1-1/log 27)/(1-1/log 3)) = 0.338....
For generalizations, see de la Bretèche & Tenenbaum, Brüdern, Hall & Tenenbaum, and Caballero.

			The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. For u = 3, {3, 4, 6, 8} are in [3, 3e] = [3, 8.15...] and thus a(24) = 4.

R. R. Hall and G. Tenenbaum, On the average and normal orders of Hooley's ∆-function, J. London Math. Soc. (2), Vol. 25, No. 3 (1982), pp. 392-406.
R. R. Hall and G. Tenenbaum, Divisors. Cambridge Tracts in Mathematics, 90. Cambridge University Press, Cambridge, 1988.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. de la Bretèche and G. Tenenbaum, Oscillations localisées sur les diviseurs, J. Lond. Math. Soc. 2 85:3 (2012), pp. 669-693.
Régis de la Bretèche and Gérald Tenenbaum, Two upper bounds for the Erdős--Hooley Delta-function, arXiv preprint (2022). arXiv:2210.13897 [math.NT]
Jörg Brüdern, Daniel's twists of Hooley's Delta function, Contributions in Analytic and Algebraic Number Theory, Springer Proceedings in Mathematics 9 (2012), pp 31-82.
Paul Erdős, On abundant-like numbers, Canad. Math. Bull. 17 (1974), pp. 599-602.
Paul Erdős and Jean-Louis Nicolas, Méthodes probabilistes et combinatoires en théorie des nombres, Bulletin des Sciences Mathématiques 2 (1976), pp. 301-320.
P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math. France 103 (1975), pp. 65-90.
R. R. Hall and G. Tenenbaum, The average orders of Hooley's Δ_r-functions, Mathematika 31:1 (1984), pp. 98-109.
R. R. Hall and G. Tenenbaum, The average orders of Hooley's Δ_r-functions, II, Compositio Math. 60 (1986), pp. 163-186.
C. Hooley, On a new technique and its applications to the theory of numbers, Proc. London Math. Soc. 3 38:1 (1979), pp. 115-151.
Dimitris Koukoulopoulos and Terence Tao, A note on the mean value of the Erdős-Hooley Delta function, arXiv preprint (2023). arXiv:2306.08615 [math.NT]
Helmut Maier and Gérald Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), pp. 121-128.
Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors, J. London Math. Soc. 2 31:3 (1985), pp. 393-400.
Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors. II., Math. Proc. Cambridge Philos. Soc. 147:3 (2009), pp. 513-540.
J.-L. Nicolas, Méthodes probabilistes et combinatoires en théorie des nombres, Séminaire Delange-Pisot-Poitou. Théorie des nombres, Tome 17 (1975-1976) no. 1, Exposé no. 9, p. 1.
J. M. Rodríguez Caballero, Symmetric Dyck Paths and Hooley's Δ-Function, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
Gérald Tenenbaum, Sur la concentration moyenne des diviseurs, Commentarii Mathematici Helvetici 60:1 (1985), pp. 411-428.
Index entries for "core" sequences

Partial sums are A226901. Cf. A226899, A226900, A027750, A022843.

a226898 = maximum . map length .
   map (\ds@(d:_) -> takeWhile (<= e' d) ds) . init . tails . a027750_row
   where e' = floor . (* e) . fromIntegral; e = exp 1
-- Reinhard Zumkeller, Jul 06 2013

with(numtheory):
a:= n-> (l-> max(seq(nops(select(x-> is(x<=exp(1)*l[i]), l))-i+1,
        i=1..nops(l))))(sort([divisors(n)[]])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 21 2013

a[n_] := Module[{d = Divisors[n], m = 1}, For[i = 1, i < Length[d], i++, t = E*d[[i]]; m = Max[ Sum[ Boole[d[[j]] < t], {j, i, Length[d]}], m]]; m]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 08 2013, after Pari *)

a(n)=my(d=divisors(n),m=1);for(i=1,#d-1, my(t=exp(1)*d[i]); m=max(sum(j=i,#d,d[j]

a(n)=my(d=divisors(n),r,t); for(i=1,#d\2, t=setsearch(d, d[i]*exp(1)\1,1); t=if(t, t-i, setsearch(d,d[i]*exp(1)\1)+1-i); if(t>r, r=t)); r \\ Charles R Greathouse IV, Mar 01 2018

from sympy import divisors, exp
def a(n):
    d = divisors(n)
    m = 1
    for i in range(len(d) - 1):
        t = exp(1)*d[i]
        m = max(sum(1 for j in range(i, len(d)) if d[j]Indranil Ghosh, Jul 19 2017

a(mn) <= d(m)a(n) where d(n) is A000005.

The average order is between log log x and (log log x)^(11/4); the lower bound is due to Hall & Tenenbaum (1988) and the upper bound to Koukoulopoulos & Tao. - Charles R Greathouse IV, Jun 26 2023

A023123 Signature sequence of e (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 5, 3, 6, 1, 4, 7, 2, 5, 8, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8, 11, 3, 6, 9, 1, 12, 4, 7, 10, 2, 13, 5, 8, 11, 3, 14, 6, 9, 1, 12, 4, 15, 7, 10, 2, 13, 5, 16, 8, 11, 3, 14, 6, 17, 9, 1, 12, 4, 15, 7, 18, 10, 2, 13, 5, 16, 8, 19, 11, 3, 14, 6, 17, 9, 20, 1, 12, 4, 15, 7, 18, 10, 21, 2

Offset: 1

Clark Kimberling

If one deletes the first occurrence of 1, the first occurrence of 2, the first occurrence of 3, etc., then the sequence is unchanged. - Brady J. Garvin, Sep 11 2024

Any signature sequence A is closely related to the partial sums of the corresponding homogeneous Beatty sequence: Let Q(d) = d + the sum from g=0 to g=d-1 of floor(theta * g) and Qinv(i) = the maximum integer d such that Q(d) <= i. If there is some d for which Q(d) = i, then A_i = 1. Otherwise, A_i = A_{i - Qinv(i)} + 1. - Brady J. Garvin, Sep 13 2024

Clark Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
Clark Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A023124, A022843

Quiet[Block[{$ContextPath}, Needs["Combinatorica`"]], {General::compat}]
theta = E;
sums = {0};
cached = <||>;
A023123[i_] := Module[{term, path, base},
  While[sums[[-1]] < i,
    term = sums[[-1]] + Floor[theta * (Length[sums] - 1)] + 1;
    AppendTo[sums, term];
    cached[term] = 1
  ];
  path = {i};
  While[Not[KeyExistsQ[cached, path[[-1]]]],
    AppendTo[path, path[[-1]] - Combinatorica`BinarySearch[sums, path[[-1]]] + 3/2];
  ];
  base = cached[path[[-1]]];
  MapIndexed[(cached[#1] = base + Length[path] - First[#2]) &, path];
  cached[i]
];
Print[Table[A023123[i], {i, 1, 100}]]; (* Brady J. Garvin, Sep 13 2024 *)

from bisect import bisect
from sympy import floor, E
theta = E
sums = [0]
cached = {}
def A023123(i):
    while sums[-1] < i:
        term = sums[-1] + floor(theta * (len(sums) - 1)) + 1
        sums.append(term)
        cached[term] = 1
    path = [i]
    while path[-1] not in cached:
        path.append(path[-1] - bisect(sums, path[-1]) + 1)
    base = cached[path[-1]]
    for offset, vertex in enumerate(reversed(path)):
        cached[vertex] = base + offset
    return cached[i]
print([A023123(i) for i in range(1, 1001)])  # Brady J. Garvin, Sep 13 2024

The a(47) term was missing. Corrected by T. D. Noe, Aug 12 2008

A184855 Numbers m such that prime(m) is of the form (k*e); complement of A184858.

Original entry on oeis.org

1, 3, 6, 8, 10, 14, 17, 19, 21, 24, 25, 27, 31, 35, 37, 38, 40, 41, 51, 52, 53, 56, 57, 58, 59, 62, 63, 67, 68, 69, 71, 76, 82, 86, 91, 98, 102, 107, 113, 114, 116, 126, 127, 130, 131, 135, 136, 143, 145, 146, 147, 153, 155, 158, 159, 163, 168, 170, 171, 176, 177, 181, 185, 186, 187, 192, 193, 195, 196, 197, 199, 200, 202, 206, 208, 210, 214, 216, 218, 219, 222, 225, 226, 230, 232, 234, 237, 240, 243, 244, 248, 249, 252, 254

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

			See A077545.

Cf. A077545, A184858.

r=E; s=r/(r-1);
a[n_]:=Floor [n*r];
b[n_]:=Floor [n*s];
Table[a[n], {n, 1, 120}]  (* A022843 *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1, a[n]]], {n, 1, 600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2, n]], {n, 1, 600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3, n]], {n, 1, 300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4, b[n]]], {n, 1, 600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5, n]], {n, 1, 600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6, n]], {n, 1, 300}]; t6
(* List t1 matches A077545; list t2 matches A062409;
lists t3-t6 match A184855-A184858. *)

A276853 Beatty sequence for 2*e.

Original entry on oeis.org

0, 5, 10, 16, 21, 27, 32, 38, 43, 48, 54, 59, 65, 70, 76, 81, 86, 92, 97, 103, 108, 114, 119, 125, 130, 135, 141, 146, 152, 157, 163, 168, 173, 179, 184, 190, 195, 201, 206, 212, 217, 222, 228, 233, 239, 244, 250, 255, 260, 266, 271, 277, 282, 288, 293, 299

Offset: 0

Clark Kimberling, Sep 24 2016

A bisection of A022843.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Beatty sequences

Cf. A022843, A276860, A276876.

z = 500; r = 2 E; b = Table[Floor[k*r], {k, 0, z}] (* A276853 *)

a(n) = floor(2*e*n).

A276860 First differences of the Beatty sequence A276853 for 2*e.

Original entry on oeis.org

5, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 5, 6, 5, 6, 5, 6, 5

Offset: 1

Clark Kimberling, Sep 24 2016

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A022843, A276876.

z = 500; r = 2*E; b = Table[Floor[k*r], {k, 0, z}]; (* A276853 *)
Differences[b] (* A276860 *)

a(n) = floor(n*r) - floor(n*r - r), where r = 2e, n >= 1.

A062409 Numbers k such that floor(e*k) is prime.

Original entry on oeis.org

1, 2, 5, 7, 11, 16, 22, 25, 27, 33, 36, 38, 47, 55, 58, 60, 64, 66, 86, 88, 89, 97, 99, 100, 102, 108, 113, 122, 124, 128, 130, 141, 155, 163, 172, 192, 205, 216, 227, 228, 236, 258, 261, 270, 272, 280, 283, 303, 305, 309, 314, 325, 334, 342, 345, 356, 367, 373

Offset: 1

Jason Earls, Jul 08 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A022843, A077545.

Select[Range[400],PrimeQ[Floor[E #]]&] (* Harvey P. Dale, Feb 12 2015 *)

je=[]; for(n=0,1000, if(isprime(floor(exp(1)*n)),je=concat(je,n),)); je

{ default(realprecision, 50); n=0; e=exp(1); for (m=1, 10^5, if (isprime(floor(e*m)), write("b062409.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 07 2009

A108599 Self-inverse integer permutation induced by Beatty sequences for e and e/(e-1).

Original entry on oeis.org

2, 1, 5, 8, 3, 10, 13, 4, 16, 6, 19, 21, 7, 24, 27, 9, 29, 32, 11, 35, 12, 38, 40, 14, 43, 46, 15, 48, 17, 51, 54, 18, 57, 59, 20, 62, 65, 22, 67, 23, 70, 73, 25, 76, 78, 26, 81, 28, 84, 86, 30, 89, 92, 31, 95, 97, 33, 100, 34, 103, 106, 36, 108, 111, 37, 114, 39, 116, 119, 41

Offset: 1

Reinhard Zumkeller, Jun 11 2005

nonn

Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to Beatty sequences

Cf. A108591.

a(A022843(n))=A054385(n) and a(A054385(n))=A022843(n).

A276875 Sums-complement of the Beatty sequence for e.

Original entry on oeis.org

1, 4, 7, 12, 15, 18, 23, 26, 31, 34, 37, 42, 45, 50, 53, 56, 61, 64, 69, 72, 75, 80, 83, 88, 91, 94, 99, 102, 105, 110, 113, 118, 121, 124, 129, 132, 137, 140, 143, 148, 151, 156, 159, 162, 167, 170, 175, 178, 181, 186, 189, 194, 197, 200, 205, 208, 211, 216

Offset: 1

Clark Kimberling, Sep 27 2016

See A276871 for a definition of sums-complement and guide to related sequences.

			The Beatty sequence for e is A022843 = (0,2,5,8,10,13,16,...), with difference sequence s = A276859 = (2,3,3,2,3,3,3,2,3,3,2,3,3,3,2,...).  The sums s(j)+s(j+1)+...+s(k) include (2,3,5,6,8,9,10,12,13,...), with complement (1,4,7,12,15,...).

Index entries for sequences related to Beatty sequences

Cf. A022843, A276859, A276871.

z = 500; r = E; b = Table[Floor[k*r], {k, 0, z}]; (* A022843 *)
t = Differences[b]; (* A276859 *)
c[k_, n_] := Sum[t[[i]], {i, n, n + k - 1}];
u[k_] := Union[Table[c[k, n], {n, 1, z - k + 1}]];
w = Flatten[Table[u[k], {k, 1, z}]]; Complement[Range[Max[w]], w]  (* A276875 *)

A247964 Beatty sequence for e^(1/3): a(n) = floor(n*(e^(1/3))).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 48, 50, 51, 53, 54, 55, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93

Offset: 0

Sarah Nathanson, Oct 01 2014

nonn

The Beatty complement is given in A248522. - M. F. Hasler, Oct 07 2014

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A248522, A022843, A061402, A092041, A098005, A198268.

static int a(int n) {return (int) (n*Math.pow(Math.E, (1.0/3))); }

Floor[Range[0,100]*Exp[1/3]] (* Paolo Xausa, Jul 16 2024 *)

a(n)=n\exp(-1/3) \\ M. F. Hasler, Oct 07 2014

A248522 Beatty sequence for 1/(1-exp(-1/3)): a(n) = floor(n/(1-exp(-1/3))).

Original entry on oeis.org

3, 7, 10, 14, 17, 21, 24, 28, 31, 35, 38, 42, 45, 49, 52, 56, 59, 63, 67, 70, 74, 77, 81, 84, 88, 91, 95, 98, 102, 105, 109, 112, 116, 119, 123, 126, 130, 134, 137, 141, 144, 148, 151, 155, 158, 162, 165, 169, 172, 176, 179, 183, 186, 190, 194, 197, 201

Offset: 1

M. F. Hasler, Oct 07 2014

Beatty complement of A247964.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A247964, A022843, A061402, A092041, A098005, A198268.

Floor[Range[100]/(1 - Exp[-1/3])] (* Paolo Xausa, Jul 16 2024 *)

PARI
```
a(n)=n\(1-exp(-1/3))
```

cp's OEIS Frontend

A226898 Hooley's Delta function: maximum number of divisors of n in [u, eu] for all u. (Here e is Euler's number 2.718... = A001113.)

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A023123 Signature sequence of e (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

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A184855 Numbers m such that prime(m) is of the form (k*e); complement of A184858.

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A276853 Beatty sequence for 2*e.

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A276860 First differences of the Beatty sequence A276853 for 2*e.

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A062409 Numbers k such that floor(e*k) is prime.

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PARI

A108599 Self-inverse integer permutation induced by Beatty sequences for e and e/(e-1).

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A276875 Sums-complement of the Beatty sequence for e.

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