A349323 a(1)=1, a(2)=2, a(3)=4. Thereafter, for n>=3, a(n+1) is the smallest unused k such that d(k) is prime to both d(a(n)) and d(a(n-2)), but not to d(a(n-1)), where d is the divisor counting (tau) function A000005.
1, 2, 4, 3, 9, 5, 25, 6, 36, 7, 49, 8, 100, 10, 121, 11, 144, 13, 16, 14, 81, 12, 625, 15, 1296, 17, 324, 19, 169, 21, 196, 22, 225, 23, 256, 24, 289, 26, 361, 27, 400, 29, 441, 30, 484, 31, 529, 33, 576, 34, 64, 35, 729, 18, 5184, 20, 2401, 28, 10000, 32, 11664
Offset: 1
Keywords
Examples
a(1)=1, a(2)=2, a(3)=4, with number of divisors 1,2,3 respectively. a(4) must be 3 because d(3)=2, which is prime to d(a(3))=d(4)=3 and to d(a(1))=d(1)=1 but it is not prime to d(a(2))=d(2)=2, and 3 is the least unused number with this property.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..10000, with a color function showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, where magenta represents powerful numbers that are not prime powers. Squares comprise the upper trajectory, whereas the lower trajectory contains relatively few squares.
Programs
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Mathematica
Nest[Block[{a = #1, i = #2, j = #3, k = #4, m = 3}, While[Nand[FreeQ[a, m], CoprimeQ[#, i], ! CoprimeQ[#, j], CoprimeQ[#, k]] &@DivisorSigma[0, m], m++]; Append[#1, m]] & @@ Join[{#}, DivisorSigma[0, #[[-3 ;; -1]]]] &, {1, 2, 4}, 58] (* Michael De Vlieger, Jan 15 2022 *) -
PARI
isok(k, ndx, ndy, ndz, set) = {if (!setsearch(set, k), my(ndk=numdiv(k)); (gcd(ndx,ndk)==1) && (gcd(ndy,ndk)!=1) && (gcd(ndz,ndk)==1););} lista(nn) = {my(x=1, y=2, z=4, list=List([x,y,z]), set = Set(list)); for (n=4, nn, my(k=1, ndx=numdiv(x), ndy=numdiv(y), ndz=numdiv(z)); while (!isok(k, ndx, ndy, ndz, set), k++); listput(list, k); set = Set(list); x=y; y=z; z=k;); Vec(list);} \\ Michel Marcus, Jan 16 2022
Extensions
More terms from Michael De Vlieger, Dec 24 2021
Comments