A353702 Composite k such that tau(k') = (tau(k))', where tau(k) is the number of divisors of k (A000005) and k' is the arithmetic derivative of k (A003415).

12, 15, 21, 26, 27, 33, 38, 57, 62, 69, 74, 85, 88, 93, 106, 108, 129, 133, 134, 145, 166, 177, 178, 205, 213, 217, 218, 226, 237, 248, 249, 253, 254, 262, 265, 278, 309, 314, 328, 362, 375, 376, 393, 398, 417, 422, 424, 445, 459, 466, 469, 488, 489, 493, 502

Offset

1,1

Comments

Since for any prime number p, p' = 1 and (tau(p))' = 2' = 1 = tau(1) = tau (p'), the sequence requires only composite numbers that satisfy the given relation.

For p in A092109 the number m = 3*p is a term. Indeed, (tau(m))' = (tau(3*p))' = 4' = 4 and tau(m') = tau((3*p)') = tau(p + 3) = 4, so m is a term.

If p is in A045536 then p, p + 2 and 2*p + 1 are prime numbers and m = 3*(2*p + 1) is a term. Indeed, tau(m') = tau((3*(2*p + 1))') = tau(2*p + 4) = tau(2*(p+2)) = 4 and (tau(m))' = (tau((3*(2*p + 1)))' = 4' = 4, so m is a term.

If k is in A174100 then the numbers 2*k + 1 and 6*k + 1 are prime numbers and the numbers m = 2*(6*k + 1) is a term. Indeed, (tau(m))' = (tau(2*(6*k + 1)) )' = 4' = 4 and tau(m') = tau(2*(6*k + 1))') = tau(6*k + 3) = tau(2*(2*k + 1)) = 4, so m is a term.

Links

Table of n, a(n) for n=1..55.

Example

12' = 16, (tau(12)) = 6' = 5 and tau(12') = tau(16) = tau(2^4) = 5, so 12 is a term.
15' = 8, (tau(15))’ = 4' = 4 and tau(15') = tau(8) = tau(2^3) = 4, so 15 is a term.

Maple

isA353702 := proc(n)
    if not isprime(n) and numtheory[tau](A003415(n)) = A003415( numtheory[tau](n) ) then
        true ;
    else
        false;
    end if;
end proc:
for n from 2 to 1000 do
    if isA353702(n) then
        printf("%d, ", n) ;
    end if;
end do: # R. J. Mathar, May 05 2023

Mathematica

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[500], CompositeQ[#] && DivisorSigma[0, d[#]] == d[DivisorSigma[0, #]] &] (* Amiram Eldar, May 07 2022 *)

Prog

(Magma) f:=func<h |h eq 1 select 0 else  h*(&+[Factorisation(h)[i][2] / Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [p:p in [3..550]|not IsPrime(p) and  #Divisors(Floor(f(p))) eq Floor(f(#Divisors(p)))];
(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = (k>1) && !isprime(k) && numdiv(ad(k)) == ad(numdiv(k)); \\ Michel Marcus, May 08 2022

Crossrefs

Cf. A000005, A003415, A045536, A092109, A174100, A260961.

Sequence in context: A162826 A162820 A267195 * A335859 A259040 A158190

Adjacent sequences: A353699 A353700 A353701 * A353703 A353704 A353705

Keyword

nonn

Author

Marius A. Burtea, May 07 2022

Status

approved