A051488 -id:A051488 - cp's OEIS frontend

A333232 Terms of A051488 that do not belong to A083207.

5865, 7395, 10005, 15045, 28815, 37995, 45645, 50235, 99705, 134895, 170085, 275655, 310845, 347565, 391935, 436305, 470235, 486795, 521985, 530265, 590295, 613785, 627555, 635205, 658155, 662745, 707115, 791265, 797385, 830415, 835635, 873885, 887655, 979455, 994755

Offset: 1

Ivan N. Ianakiev, Mar 12 2020

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.

Cf. A000010, A051488, A083207.

(* First 200000 terms of A051488 *)
a051488=Select[Range[200000],EulerPhi[#]T. D. Noe at A083207 *)
zQ[n_]:=Module[{d=Divisors[n],t,ds,x},ds=Plus@@d;If[Mod[ds,2]>0,False,t=CoefficientList[Product[1+x^i,{i,d}],x];t[[1+ds/2]]>0]]; t2=Select[t1,!zQ[#]&]

Terms a(12) and beyond from Giovanni Resta, Mar 12 2020

A051487 Numbers k such that phi(k) = phi(k - phi(k)).

Original entry on oeis.org

2, 6, 12, 24, 48, 96, 150, 192, 300, 384, 600, 726, 750, 768, 1200, 1452, 1500, 1536, 2310, 2400, 2904, 3000, 3072, 3174, 3750, 4620, 4800, 5046, 5808, 5874, 6000, 6090, 6144, 6348, 6930, 7500, 7986, 9240, 9600, 10086, 10092, 10374, 11550, 11616, 11748, 12000

Offset: 1

R. K. Guy

This sequence is infinite, in fact 3*2^n is a subsequence because if m = 3*2^n then phi(m-phi(m)) = phi(3*2^n-2^n) = 2^n = phi(m). Also, if p is a Sophie Germain prime greater than 3 then for each natural number n, 2^n*3*p^2 is in the sequence. Note that there exist terms of this sequence like 750 or 2310 that they aren't of either of these forms. - Farideh Firoozbakht, Jun 19 2005
If n is an even term greater than 2 in this sequence then 2n is also in the sequence. Because for even numbers m we have phi(2m) = 2*phi(m) so phi(2n) = 2*phi(n) = 2*phi(n-phi(n)) and since n is an even number greater than 2, n-phi(n) is even so 2*phi(n-phi(n)) = phi(2n-2*phi(n)) = phi(2n-phi(2n)) hence phi(2n) = phi(2n-phi(2n)) and 2n is in the sequence. Conjecture: All terms of this sequence are even. - Farideh Firoozbakht, Jul 04 2005
If n is in the sequence and the natural number m divides gcd(n,phi(n)) then m*n is in the sequence. The facts that I have found about this sequence earlier (Jun 19 2005 and Jul 04 2005) are consequences of this. If p is a Sophie Germain prime greater than 3, k>1 and k & n are natural numbers then 2^n*3*p^k are in the sequence. - Farideh Firoozbakht, Dec 10 2005
Numbers n such that phi(n) = phi(n + phi(n)) includes all n = 2^k. - Jonathan Vos Post, Oct 25 2007

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Aleksander Grytczuk, Florian Luca and Marek Wojtowicz, A conjecture of Erdős concerning inequalities for the Euler totient function, Publ. Math. Debrecen, Vol. 59, No. 1-2, (2001), pp. 9-16.

Cf. A005384, A051488.

Cf. A000010, A051953.

a051487 n = a051487_list !! (n-1)
a051487_list = [x | x <- [2..], let t = a000010 x, t == a000010 (x - t)]
-- Reinhard Zumkeller, Jun 03 2013

Select[Range[11700], EulerPhi[ # ] == EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)

isA051487(n) = eulerphi(n) == eulerphi(n - eulerphi(n)) \\ Michael B. Porter, Dec 07 2009

More terms from James Sellers

A346692 a(n) = phi(n) - phi(n-phi(n)), a(1) = 1.

Original entry on oeis.org

1, 0, 1, 1, 3, 0, 5, 2, 4, 2, 9, 0, 11, 2, 2, 4, 15, 2, 17, 4, 6, 6, 21, 0, 16, 6, 12, 4, 27, -2, 29, 8, 8, 10, 14, 4, 35, 10, 16, 8, 39, 4, 41, 12, 12, 14, 45, 0, 36, 12, 14, 12, 51, 6, 32, 8, 24, 20, 57, -4, 59, 14, 18, 16, 32, -2, 65, 20, 24, 2, 69, 8, 71, 18, 16, 20, 44, 6, 77, 16

Offset: 1

Bernard Schott, Jul 29 2021

sign

P. Erdős conjectured that a(n) > 0 on a set of asymptotic density 1, then Luca and Pomerance proved this conjecture (see link).

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

Paul Erdős, Problem P. 294, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505.
Florian Luca and Carl Pomerance, On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions phi and sigma, Colloq. Math., Vol. 92, No. 1 (2002), pp. 111-130.

Cf. A000010, A054571.

Cf. A051487 (a(n)=0), A051488 (a(n)<0).

with(numtheory):
A := seq(phi(n) - phi(n-phi(n)), n=1..100);

a[n_] := (phi = EulerPhi[n]) - EulerPhi[n - phi]; Array[a, 100] (* Amiram Eldar, Jul 29 2021 *)

a(n) = if (n==1, 1, eulerphi(n) - eulerphi(n-eulerphi(n))); \\ Michel Marcus, Jul 29 2021

from sympy import totient as phi
def a(n):
    if n == 1: return 1
    phin = phi(n)
    return phin - phi(n - phin)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 29 2021

a(n) = A000010(n) - A054571(n).

If p prime, a(p) = p-2, and for k >= 2, a(p^k) = (p-1)^2 * p^(k-2).

A346694 Primitive terms of A051487.

Original entry on oeis.org

6, 150, 726, 750, 2310, 3174, 3750, 5046, 5874, 6090, 6930, 7986, 10086, 10374, 11550, 16854, 18270, 18750, 20790, 24378, 31122, 34650, 41334, 42630, 47526, 54810, 57750, 62370, 63618, 64614, 73002, 76614, 87846, 93366, 93750, 102966, 103950, 127890, 140910, 146334, 146370, 164430

Offset: 1

Bernard Schott, Aug 06 2021

nonn

If k is an even term greater than 2 of A051487 then 2k is another term.
This sequence lists the initial term k_0 of each infinite subsequence that is solution of the equation phi(k) = phi(k - phi(k)).
About 2: one could argue that 2 is primitive since it is not the double of any previous term of A051487, but as 2^k is not solution for n>1, 2 is not primitive.
Each k_0 is of the form k_0 = 6*m with m odd.
If p > 3 is a Sophie Germain prime, then every m = 2*3*p^q, q >=2 is a term because phi(m) = phi(m-phi(m)) = 2*(p-1)*p^(q-1); the first terms that are not of this form are 6, 2310, 5874, ... (see examples).

			a(1) = 6 because every k = 3*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 2^m, and k_0 = 6 is the smallest term of this subsequence of A051487.
a(2) = 150 because every k = 3*5^2*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 5*2^(m+2) and k_0 = 150 is the smallest term of this subsequence of A051487.
a(3) = 726 because every k = 3*11^2*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 5*11*2^(m+1) and k_0 = 726 is the smallest term of this subsequence of A051487.
a(5) = 2310 because every k = 3*5*7*11*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 3*5*2^(m+4) and k_0 = 2310 is the smallest term of this subsequence of A051487.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

Subsequence of A051487.

Cf. A000010, A051488, A051953, A346692.

with(numtheory):
for q from 0 to 13800 do
m := 6*(2*q+1);
if phi(m) = phi(m-phi(m)) then print(m); else fi; od:

isdouble(n, list)= {my(v = Vecrev(list)); for(k=1, #v, if (n == 2*v[k], return(1)););}
lista(nn) = {my(list = List(), listp = List()); for (n=3, nn, if (eulerphi(n) == eulerphi(n - eulerphi(n)), if (!isdouble(n, list), listput(listp, n)); listput(list, n););); Vec(listp);} \\ Michel Marcus, Aug 06 2021

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A333232 Terms of A051488 that do not belong to A083207.

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A051487 Numbers k such that phi(k) = phi(k - phi(k)).

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A346692 a(n) = phi(n) - phi(n-phi(n)), a(1) = 1.

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A346694 Primitive terms of A051487.

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