cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A262502 a(n) = least k such that A261104(k) = n; positions of records in A261104.

Original entry on oeis.org

0, 1, 3, 7, 11, 17, 23, 31, 40, 50, 62, 74, 86, 98, 110, 126, 142, 158, 174, 192, 210, 228, 248, 268, 288, 308, 328, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 738, 768, 798, 828, 860, 892, 924, 956, 988, 1020, 1052, 1084, 1116, 1148, 1180, 1212, 1244, 1280, 1316, 1352, 1388, 1424, 1460, 1496, 1532, 1568, 1604, 1640, 1676, 1716
Offset: 0

Views

Author

Antti Karttunen, Sep 24 2015

Keywords

Comments

a(n+2) should give a safe upper bound for A262503(n), and actually seems to significantly overshoot it when n grows.

Links

Crossrefs

Cf. A261104, A261089, A262503, A262507.
Cf. A262504 (first differences).

Formula

Other identities. For all n >= 0:
A261104(a(n)) = n.

A155043 a(0)=0; for n >= 1, a(n) = 1 + a(n-d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 4, 3, 3, 3, 4, 3, 5, 4, 5, 5, 6, 4, 7, 5, 7, 5, 8, 6, 6, 6, 9, 6, 10, 6, 11, 7, 11, 7, 12, 10, 13, 8, 13, 8, 14, 8, 15, 9, 14, 9, 15, 9, 10, 10, 16, 10, 17, 10, 17, 10, 18, 11, 19, 10, 20, 12, 19, 19, 21, 12, 22, 13, 22, 13, 23, 11, 24, 14, 23, 14, 25, 14, 26, 14, 15, 15
Offset: 0

Views

Author

Ctibor O. Zizka, Jan 19 2009

Keywords

Comments

From Antti Karttunen, Sep 23 2015: (Start)
Number of steps needed to reach zero when starting from k = n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005).
The original name was: a(n) = 1 + a(n-sigma_0(n)), a(0)=0, sigma_0(n) number of divisors of n.
(End)

Links

Crossrefs

Cf. A000005, A049820, A060990, A259934.
Sum of A262676 and A262677.
Cf. A261089 (positions of records, i.e., the first occurrence of n), A262503 (the last occurrence), A262505 (their difference), A263082.
Cf. A262518, A262519 (bisections, compare their scatter plots), A262521 (where the latter is less than the former).
Cf. A261085 (computed for primes), A261088 (for squares).
Cf. A262507 (number of times n occurs in total), A262508 (values occurring only once), A262509 (their indices).
Cf. A263265 (nonnegative integers arranged by the magnitude of a(n)).
Cf. also A263077, A263078, A263079, A263080.
Cf. also A261104, A262680, A262904, A263254, A263259, A263260, A263266, A263270.
Cf. also A004001, A005185.
Cf. A264893 (first differences), A264898 (where repeating values occur).

Programs

  • Haskell
    import Data.List (genericIndex)
a155043 n = genericIndex a155043_list n
a155043_list = 0 : map ((+ 1) . a155043) a049820_list
-- Reinhard Zumkeller, Nov 27 2015
  • Maple
    with(numtheory): a := proc (n) if n = 0 then 0 else 1+a(n-tau(n)) end if end proc: seq(a(n), n = 0 .. 90); # Emeric Deutsch, Jan 26 2009
  • Mathematica
    a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[a@n, {n, 0, 82}] (* Michael De Vlieger, Sep 24 2015 *)
  • PARI
    uplim = 110880; \\ = A002182(30).
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
A155043 = n -> if(!n,n,v155043[n]);
for(n=0, uplim, write("b155043.txt", n, " ", A155043(n)));
\\ Antti Karttunen, Sep 23 2015
  • Python
    from sympy import divisor_count as d
def a(n): return 0 if n==0 else 1 + a(n - d(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 03 2017
  • Scheme
    (definec (A155043 n) (if (zero? n) n (+ 1 (A155043 (A049820 n)))))
;; Antti Karttunen, Sep 23 2015

Formula

From Antti Karttunen, Sep 23 2015 & Nov 26 2015: (Start)
a(0) = 0; for n >= 1, a(n) = 1 + a(A049820(n)).
a(n) = A262676(n) + A262677(n). - Oct 03 2015.
Other identities. For all n >= 0:
a(A259934(n)) = a(A261089(n)) = a(A262503(n)) = n. [The sequence works as a left inverse for sequences A259934, A261089 and A262503.]
a(n) = A262904(n) + A263254(n).
a(n) = A263270(A263266(n)).
A263265(a(n), A263259(n)) = n.
(End)

Extensions

Extended by Emeric Deutsch, Jan 26 2009
Name edited by Antti Karttunen, Sep 23 2015

A070319 a(n) = Max_{k=1..n} tau(k) where tau(x)=A000005(x) is the number of divisors of x.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12
Offset: 1

Views

Author

Benoit Cloitre, May 11 2002

Keywords

Comments

Is this the same as A068509? - David Scambler, Sep 10 2012
They are different even asymptotically: A068509(n)=O(sqrt(n)), while a(n) does not have polynomial growth. One example where the sequences differ: a(625) = 24 < A068509(625). (The inequality is implied by the set {1,2,..,25} where each pair of the elements has lcm <= 625.) - Max Alekseyev, Sep 11 2012
The two sequences first differ when n = 336, due to the set of 21 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 21, 24, 30, 36, 42, 48} where each pair of elements has lcm <= 336, while no positive integer <= 336 has more than 20 divisors. Therefore A068509(336) = 21 and A070319(336) = 20. - William Rex Marshall, Sep 11 2012
Indices of records give A002182. - Omar E. Pol, Feb 18 2023

References

  • Sándor, J., Crstici, B., Mitrinović, Dragoslav S. Handbook of Number Theory I. Dordrecht: Kluwer Academic, 2006, p. 44.
  • S. Wigert, Sur l'ordre de grandeur du nombre des diviseurs d'un entier, Arkiv. for Math. 3 (1907), 1-9.

Links

Crossrefs

Cf. A000005, A002182, A002183, A261100, A261104.

Programs

  • Haskell
    a070319 n = a070319_list !! (n-1)
a070319_list = scanl1 max $ map a000005 [1..]
-- Reinhard Zumkeller, Apr 01 2011
  • Mathematica
    a = {0}; Do[AppendTo[a, Max[DivisorSigma[0, n], a[[n]]]], {n, 120}]; Rest@ a (* Michael De Vlieger, Sep 29 2015 *)
  • PARI
    a(n)=vecmax(vector(n,k,numdiv(k)))
  • PARI
    v=vector(100);v[1]=1;for(n=2,#v,v[n]=max(v[n-1],numdiv(n))); v \\ Charles R Greathouse IV, Sep 12 2012
  • PARI
    A070319(n,m=1,s=2)={for(k=s,n,mM. F. Hasler, Sep 12 2012
  • PARI
    {a=0;for(n=1,100,print1(a=A070319(n,a,n),","))} /* Using this pattern, computation of a(1..10^6) is faster than "normal" computation of a(1..3000). */

Formula

a(n) = exp(log(2) log(n) / log(log(n)) + O(log(n) log(log(log(n))) / (log(log(n)))^2)). (See Sándor reference for more formulas.) - Eric M. Schmidt, Jun 30 2013
a(n) = A002183(A261100(n)). - Antti Karttunen, Jun 06 2017
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.