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 1-4 of 4 results.

A076365 Duplicate of A070971.

Original entry on oeis.org

3, 4, 15, 6, 105, 30, 1155, 770, 36465, 210, 15015, 6006
Offset: 1

Views

Author

Keywords

A048669 The Jacobsthal function g(n): maximal gap in a list of all the integers relatively prime to n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 3, 2, 2, 4, 2, 4, 3, 4, 2, 4, 2, 4, 2, 4, 2, 6, 2, 2, 3, 4, 3, 4, 2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 6, 2, 4, 3, 2, 3, 6, 2, 4, 3, 6, 2, 4, 2, 4, 3, 4, 3, 6, 2, 4, 2, 4, 2, 6, 3, 4, 3, 4, 2, 6, 3, 4, 3, 4, 3, 4, 2, 4, 3, 4, 2, 6, 2, 4, 5
Offset: 1

Views

Author

Jan Kristian Haugland

Keywords

Comments

Equivalently, g(n) is the least integer such that among any g(n) consecutive integers i, i+1, ..., i+g(n)-1 there is at least one which is relatively prime to n.
The definition refers to all integers, not just those in the range 1..n-1.
Differs from A070194 by 1 at the primes. - T. D. Noe, Mar 21 2007
Jacobsthal's function is used in the proofs of Recamán's and Pomerance's conjectures on P-integers--see A192224. - Jonathan Sondow, Jun 14 2014

Examples

			g(6)=4 because the gap between 1 and 5, both being relatively prime to 6, is maximal and 5-1 = 4.
g(7)=2, because the numbers relatively prime to 7 are 1,2,3,4,5,6,8,9,10,..., and the biggest gap is 2. Similarly a(p) = 2 for any prime p. - _N. J. A. Sloane_, Sep 08 2012

References

  • E. Jacobsthal, Uber Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist, I, II, III. Norske Vid. Selsk. Forh., 33, 1960, 117-139.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Pages 33-34.
  • E. Westzynthius, Uber die Verteilung der Zahlen, die zu der n ersten Primzahlen teilerfremd sind, Comm. Phys. Math. Helsingfors 25 (1931), 1-37.

Links

Crossrefs

Essentially same as A049298. See A132468 for another version.
Cf. A048670, A070971, A007947, A038566, A192224, A285183.

Programs

  • Haskell
    a048669 n = maximum $ zipWith (-) (tail ts) ts where
   ts = a038566_row n ++ [n + 1]
-- Reinhard Zumkeller, Oct 01 2012
  • Mathematica
    g[n_] := Module[{L = 1, m = 1}, For[k = 2, k <= n+1, k++, If[GCD[k, n] == 1, If[L+m < k, m = k-L]; L = k]]; m]; Table[g[n], {n, 1, 105}] (* Jean-François Alcover, Sep 03 2013, after M. F. Hasler *)
Table[Max[Differences[Select[Range[110],CoprimeQ[#,n]&]]],{n,110}] (* Harvey P. Dale, Jan 10 2022 *)
  • PARI
    A048669(n)=my(L=1,m=1);for(k=2,n+1,gcd(k,n)>1 && next;L+mM. F. Hasler, Sep 08 2012

Formula

From N. J. A. Sloane, Apr 19 2017 (Start):
g(n) = g(Rad(n)) (cf. A007947). So in studying g(n) we may focus on the case when n is a product of w (say) distinct primes.
g(n) <= 2^w for all w [Kanold].
g(n) <= 2^(1/w) for all w >= e^50 [Kanold].
For some unknown X, g(n) <= X*(w*log(w))^2 for all w [Iwaniec].
(End)
g(n) << (log(n))^2, as proved by Iwaniec. - Charles R Greathouse IV, Sep 08 2012.

Extensions

Edited, changed symbol to g(n), added references pertaining to bounds. - N. J. A. Sloane, Apr 19 2017

A076366 Number of numbers for which the count of nonprimes (i.e., 1 and composites) in their reduced residue set equals n.

Original entry on oeis.org

10, 6, 6, 4, 4, 7, 3, 4, 3, 7, 4, 4, 0, 6, 5, 1, 4, 3, 7, 4, 7, 2, 3, 3, 2, 2, 6, 5, 2, 2, 0, 6, 4, 3, 5, 4, 5, 3, 1, 3, 3, 4, 4, 6, 2, 3, 1, 6, 1, 6, 3, 6, 1, 4, 4, 4, 1, 1, 3, 6, 3, 2, 4, 4, 1, 1, 2, 4, 6, 0, 3, 4, 3, 5, 4, 1, 2, 8, 2, 5, 6, 2, 2, 5, 1, 4, 2, 4, 7, 2, 1, 2, 6, 1, 3, 5, 2, 3, 5, 3
Offset: 1

Views

Author

Labos Elemer, Oct 10 2002

Keywords

Examples

			A048864(x) = 13: S = {},                                a(13) =  0;
A048864(x) = 16: S = {144},                             a(16) =  1;
A048864(x) = 22: S = {57,92},                           a(22) =  2;
A048864(x) = 7:  S = {13,34,50},                        a(7)  =  3;
A048864(x) = 4:  S = {15,22,54,84},                     a(4)  =  4;
A048864(x) = 15: S = {35,64,68,156,240},                a(15) =  5;
A048864(x) = 2:  S = {5,10,14,20,42,60},                a(2)  =  6;
A048864(x) = 6:  S = {11,21,32,40,72,78,210},           a(6)  =  7;
A048864(x) = 78: S = {133,177,268,440,490,552,870,990}, a(78) =  8;
A048864(x) = 1:  S = {1,2,3,4,6,8,12,18,24,30},         a(1)  = 10; See A048597.

Links

Crossrefs

Cf. A000010, A000720, A001221, A048597, A048864, A048865, A070971, A072022, A072023, A074915.

Programs

  • PARI
    listn(nn) = {my(v = vector(10^5, n, eulerphi(n) - (primepi(n) - omega(n)))); vector(nn, k, if (#(w=Vec(select(x->(x==k), v, 1))) == 0, 0, #w));} \\ Michel Marcus, Feb 23 2020

Formula

a(n) = Card{x; A048864(x) = n}; a(n)=0 if supposedly no such number exists (see A072023).

A128759 Least k such that the Jacobsthal function A048669(k) = n.

Original entry on oeis.org

1, 2, 15, 6, 105, 30, 1155, 770, 36465, 210, 15015, 6006, 255255, 2310, 8580495, 102102, 4849845, 72930, 20056049013, 74364290, 5898837945, 30030, 3234846615, 881790, 195282582495, 510510, 218257003965, 20281170, 100280245065, 17160990, 934482952262145, 6614136163635
Offset: 1

Views

Author

T. D. Noe, Mar 28 2007

Keywords

Comments

Except for a(1) and a(2), the same as A070971. It appears that a(2n)=2a(n) for odd n. Because the primorial numbers (A002110) yield record values of the Jacobsthal function, we have a(A048670(n))=A002110(n). Note that numbers in this sequence up to n=18 have the form p#, p#/2, p#/q, or p#/(2q), where p and q are primes with 2

Crossrefs

Cf. A048669, A070971, A002110, A048670.

Extensions

a(19) onwards from Don Reble, Oct 17 2013
Showing 1-4 of 4 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.