A285457 -id:A285457 - cp's OEIS frontend

A086550 Smallest k such that tau(k) - tau(k-1) = n, where tau(k) = number of divisors of k, or 0 if no such number exists.

3, 2, 6, 50, 12, 36, 24, 400, 48, 1850, 60, 144, 120, 1600, 168, 576, 180, 1296, 240, 4356, 630, 2304, 360, 900, 960, 9216, 1008, 40000, 720, 20736, 840, 5184, 1800, 46656, 1260, 36864, 1680, 7056, 3024, 986050, 2880, 3600, 6480, 82944, 2520, 193600, 3360

Offset: 0

Amarnath Murthy, Aug 28 2003

nonn

Conjecture: No term is zero.

a(2k+1) is either a square or one more than a square. - David Wasserman, Mar 24 2005

			a(3) = 50 as tau(50) - tau(49) = 6 - 3 = 3.

Giovanni Resta, Table of n, a(n) for n = 0..1000 (first 500 terms from Donovan Johnson)

Cf. A285457.

With[{tau=Partition[DivisorSigma[0,Range[10^6]],2,1]},Flatten[ Table[ Position[ #[[2]]-#[[1]]&/@tau,n,1,1],{n,0,50}]]]+1 (* Harvey P. Dale, Aug 20 2017 *)

/* finds first 100 terms */ nn=vector(100); nd1=1; for(k=2, 24285184, nd2=numdiv(k); d=nd2-nd1; if(d>0, if(d<=100, if(nn[d]==0, nn[d]=k))); nd1=nd2); for(n=1, 100, write("b086550.txt", n " " nn[n])) /* Donovan Johnson, Sep 25 2013 */

Corrected and extended by David Wasserman, Mar 24 2005

Offset changed to 0, and a(0) added by Giovanni Resta, Apr 28 2017

A285787 Least number k such that the absolute value of the difference between the number of prime factors, with multiplicity, of k and k-1 is equal to n.

Original entry on oeis.org

3, 2, 8, 17, 32, 97, 128, 257, 769, 2048, 4097, 6144, 8192, 40961, 73728, 65537, 131072, 524289, 524288, 3145728, 6291456, 8388608, 18874368, 50331648, 113246209, 167772161, 268435457, 805306368, 1610612737, 2147483649, 2147483648, 17179869184, 21474836480

Offset: 0

Paolo P. Lava, Apr 26 2017

nonn

a(n) <= A051900(n), with equality for n=3,5,7,8,13,15. - Robert Israel, Apr 26 2017

			a(9) = 2048 because 2047 = 23 * 89, 2048 = 2^11 and 11 - 2 = 9.

Giovanni Resta, Table of n, a(n) for n = 0..40

Cf. A001222, A051900, A076191, A285457.

with(numtheory): P:=proc(q) local a,b,k,v; v:=array(0..100);
for k from 0 to 100 do v[k]:=0; od; a:=0;
for k from 2 to q do b:=bigomega(k); if v[abs(b-a)]=0 then v[abs(b-a)]:=k; fi; a:=b; od; k:=0;
while v[k]>0 do print(v[k]); k:=k+1; od; print(); end: P(10^6);

s = PrimeOmega@ Range[10^6]; 1 + First /@ Values@ KeySort@ PositionIndex@ Flatten@ Map[Abs@ Differences@ # &, Partition[s, 2, 1]] (* Michael De Vlieger, Apr 26 2017, Version 10 *)

Least solutions of the equation abs(A001222(k) - A001222(k-1)) = n.

a(24)-a(32) from Giovanni Resta, Apr 26 2017

cp's OEIS Frontend

A086550 Smallest k such that tau(k) - tau(k-1) = n, where tau(k) = number of divisors of k, or 0 if no such number exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions