A260961 -id:A260961 - cp's OEIS frontend

A260962 Numbers k such that phi(k) = phi'(k'), where phi(k) is the Euler totient function of k and k' is the arithmetic derivative of k.

8, 26, 122, 351, 31195, 47201, 51243, 118265, 300985, 472491, 672147, 673863, 850969, 931383, 1440625, 3000927, 3669213, 3740755, 4688645, 4822143, 4864175, 11224565, 13897079, 13949343, 16362857, 16744355, 18844265, 19536205, 35580099, 38656975, 42056215, 46294105

Offset: 1

Paolo P. Lava, Aug 06 2015

			Arithmetic derivative of 26 is 15, phi(15) = 8 and 8' = 12 that is equal to phi(26).

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000010, A003415, A190402, A260961.

with(numtheory):P:=proc(q) local a,b,n,p;
for n from 1 to q do a:=phi(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]));
b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]);
if phi(n)=b then print(n); fi; od; end: P(10^9);

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range@ 100000, f@ EulerPhi@ f@ # == EulerPhi@ # &] (* Michael De Vlieger, Aug 07 2015, after Michael Somos at A003415 *)

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).

Original entry on oeis.org

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

Marius A. Burtea, May 07 2022

nonn

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.

			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.

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

f:=func; [p:p in [3..550]|not IsPrime(p) and  #Divisors(Floor(f(p))) eq Floor(f(#Divisors(p)))];

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

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 *)

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

cp's OEIS Frontend

A260962 Numbers k such that phi(k) = phi'(k'), where phi(k) is the Euler totient function of k and k' is the arithmetic derivative of k.

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