A051487 -id:A051487 - cp's OEIS frontend

A346694 Primitive terms of A051487.

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

A051488 Numbers k such that phi(k) < phi(k - phi(k)).

Original entry on oeis.org

30, 60, 66, 120, 132, 138, 174, 210, 240, 246, 264, 276, 318, 330, 348, 420, 480, 492, 498, 510, 528, 534, 552, 630, 636, 660, 678, 690, 696, 786, 840, 870, 910, 960, 984, 996, 1020, 1038, 1056, 1068, 1074, 1104, 1122, 1146, 1260, 1272, 1320, 1330, 1356

Offset: 1

R. K. Guy

If p is a Sophie Germain prime greater than 3 and n is a natural number then 2^n*3*p is in the sequence. That is because if m = 2^n*3*p then phi(m) = 2^n*(p-1) and phi(m - phi(m)) = phi(2^n*3*p - 2^n*(p-1)) = phi(2^n*(2p+1)) = 2^n*p so phi(m) < phi(m-phi(m)) and m is in the sequence. - Farideh Firoozbakht, Jun 19 2005

Erdős (1980) proposed the problem to prove that this sequence is infinite and has an asymptotic density 0. Grytczuk et al. (2001) proved that this sequence is infinite with an upper asymptotic density < 0.45637. - Amiram Eldar, May 22 2021

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 209.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Paul Erdős, Problem P. 294, Canad. Math. Bull., Vol. 23, No. 4 (1980), p. 505.
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. A051487, A005384.

Cf. A000010, A051953.

a051488 n = a051488_list !! (n-1)
a051488_list = [x | x <- [2..], let t = a000010 x, t < a000010 (x - t)]
-- Reinhard Zumkeller, Apr 12 2014

Select[Range[1360], EulerPhi[ # ] < EulerPhi[ # - EulerPhi[ # ]] &] (* Farideh Firoozbakht, Jun 19 2005 *)

More terms from James Sellers

A108569 Numbers n such that phi(n) = phi(n + phi(n)).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 110, 128, 220, 256, 440, 506, 512, 550, 880, 1012, 1024, 1100, 1760, 1830, 2024, 2048, 2162, 2200, 2750, 3422, 3520, 3660, 4048, 4096, 4114, 4324, 4400, 4746, 5490, 5500, 5566, 6806, 6844, 7040, 7320, 7782, 8096, 8192, 8228, 8648, 8800, 9150, 9492

Offset: 1

Farideh Firoozbakht, Jul 05 2005

nonn

If n is an even term of this sequence then 2n is also in the sequence. This is because phi(2n) = 2*phi(n) = 2*phi(n+phi(n)) = phi(2n+ 2*phi(n)) = phi(2n+phi(2n)). If n is an even term of this sequence then for each natural number m, 2^m*n is in the sequence. For example, since 4 is in the sequence 2^n for each n, n>1 is in the sequence. If p is a Sophie Germain prime greater than 3 then n = 2*p*(2p+1) is in the sequence because phi(n+phi(n)) = phi(2*p*(2p+1)+2*p*(p-1)) = phi(6p^2) = 2*p*(p-1) = phi(n). Conjecture: Except for the first term all terms are even.

If n is in the sequence and the natural number m divides gcd(phi(n),n) then for all nonnegative integers k, m^k*n are in the sequence. For example 110 is in the sequence and 10 divides gcd(phi(110),110), so 11*10^k for all natural numbers k are in the sequence. - Farideh Firoozbakht, Dec 12 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..384

Cf. A005384, A051487.

[n: n in [1..10000] | EulerPhi(n) eq EulerPhi(n + EulerPhi(n))]; // Vincenzo Librandi, Nov 13 2014

with(numtheory): A108569:=n->`if`(phi(n) = phi(n+phi(n)), n, NULL): seq(A108569(n), n=1..10^4); # Wesley Ivan Hurt, Nov 12 2014

Select[Range[11000], EulerPhi[ # ]==EulerPhi[ # + EulerPhi[ # ]]&]

select(n->eulerphi(n) == eulerphi(n + eulerphi(n)), vector(10000, i, i)) \\ Michel Marcus, Nov 13 2014

A185165 Numbers n such that lambda(n) = lambda(n - lambda(n)).

Original entry on oeis.org

2, 6, 8, 20, 42, 75, 90, 117, 154, 156, 189, 220, 363, 385, 490, 525, 702, 775, 777, 845, 975, 990, 1050, 1183, 1276, 1300, 1505, 1587, 1628, 1742, 1806, 1824, 1860, 1905, 1911, 2436, 2496, 2523, 2541, 2793, 2860, 2943, 3660, 3720, 3800, 3960, 4309, 5043, 5060, 5390, 5540, 5994, 6069, 6160, 6664, 6845, 8127, 8268, 8325, 8427

Offset: 1

Michel Lagneau, Mar 31 2011

nonn

Lambda is the function in A002322. If there are infinitely many Sophie Germain primes (conjecture), then this sequence is infinite. Proof: The numbers of the form 3p^2 are in a subsequence if p and 2p+1 are both prime with p > 3, because from the property that lambda(3p^2) = p(p-1) and lambda (p(2p+1)) = p(p-1), if m = 3p^2 then lambda(m-phi(m)) = lambda (3p^2 - p(p-1)) = lambda(p(2p+1)) = p(p-1) = lambda(m).

			75 is in the sequence because lambda(75) = 20, lambda(75 - 20) = lambda(55) = 20.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002322, A005384.

Cf. A051487 (numbers n such that phi(n) = phi(n - phi(n))).

Select[Range[20000], CarmichaelLambda[ #] == CarmichaelLambda[ # - CarmichaelLambda[#] ] &]

A188466 Numbers n such that lambda(n) = lambda(n + lambda(n)).

Original entry on oeis.org

1, 4, 6, 16, 36, 55, 78, 105, 124, 144, 171, 200, 253, 325, 406, 465, 666, 689, 715, 741, 915, 930, 990, 1027, 1081, 1136, 1240, 1421, 1448, 1610, 1653, 1711, 1752, 1764, 1800, 1827, 2211, 2352, 2448, 2667, 2800, 2835, 3403, 3600, 3619, 3620, 3660, 3900, 4840, 4970, 5253, 5264, 5513, 5671, 5886, 6100, 6328, 8001, 8112

Offset: 1

Michel Lagneau, Apr 01 2011

nonn

Lambda is the function (A002322). If there are infinitely many Sophie Germain primes (conjecture), then this sequence is infinite. Proof: The numbers of the form p(2p+1) are in a subsequence if p and 2p+1 are both prime with p > 3, because from the property that lambda(p(2p+1)) = p(p-1), if m = p(2p+1) then lambda(m+phi(m)) = lambda (p(2p+1) + p(p-1)) = lambda(3p^2) = p(p-1) = lambda(m).

			36 is in the sequence because lambda(36) = 6, and lambda(36 + 6) = lambda(42) = 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002322, A005384.
Cf. A185165: Numbers n such that lambda(n)= lambda(n - lambda(n)).
Cf. A051487: Numbers n such that phi(n) = phi(n - phi(n)).
Cf. A108569: Numbers n such that phi(n) = phi(n + phi(n)).

[1] cat [n: n in [2..8140] | CarmichaelLambda(n) eq CarmichaelLambda(n+CarmichaelLambda(n))];  // Bruno Berselli, Apr 10 2011

Select[Range[20000], CarmichaelLambda[ #] == CarmichaelLambda[ # + CarmichaelLambda[#]  ] &]

lambda(n) = lcm(znstar(n)[2]);
isok(n) = lambda(n) == lambda(n+lambda(n)); \\ Michel Marcus, May 12 2018

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).

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A346694 Primitive terms of A051487.

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A051488 Numbers k such that phi(k) < phi(k - phi(k)).

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A108569 Numbers n such that phi(n) = phi(n + phi(n)).

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A185165 Numbers n such that lambda(n) = lambda(n - lambda(n)).

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A188466 Numbers n such that lambda(n) = lambda(n + lambda(n)).

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

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