A174100 -id:A174100 - cp's OEIS frontend

A185026 The first bisection of the sequence A002616 of reduced totients.

1, 2, 3, 3, 5, 6, 2, 8, 9, 3, 11, 10, 9, 14, 15, 5, 6, 18, 6, 20, 21, 6, 23, 21, 8, 26, 10, 9, 29, 30, 3, 6, 33, 11, 35, 36, 10, 15, 39, 27, 41, 8, 14, 44, 6, 15, 18, 48, 15, 50, 51, 6, 53, 54, 18, 56, 22, 6, 24, 55, 20, 50, 63, 21, 65, 9, 18, 68, 69, 23, 30, 14, 21, 74, 75, 24, 30, 78, 26, 33

Offset: 1

Paul Curtz, Dec 24 2012

nonn

Fixed points n where n=a(n) are n=1, 2, 3, 5, 6, 8, 9, 11,... = A005097.

There are pairs of the consecutive terms of the form (3*k,k): 6,2, 9,3, 15,5, 18,6, 33,11, 54,18, 63,21, 69,23, 78,26, ... . It seems that k is the set of all A174100(n), n>1, plus a few exceptions (like the pair 126, 42).

G. C. Greubel, Table of n, a(n) for n = 1..5000

A185026 := proc(n)
        numtheory[lambda](2*n+1)/2 ;
end proc: # R. J. Mathar, Mar 27 2013

Table[ CarmichaelLambda[2*n + 1]/2, {n, 1, 80}] (* Jean-François Alcover, Apr 04 2013 *)

a(n) = A002616(2*n+1).

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

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A185026 The first bisection of the sequence A002616 of reduced totients.

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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).

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