Kevin J. Gomez - cp's OEIS frontend

A308419 Stopping time for Recamán-like iteration of each n: a(0) = n, a(k) = a(k-1) - k if positive and not already in the sequence, a(k) = a(k-1) + k if not already in the sequence, otherwise stop.

24, 24, 13, 21, 3, 3, 3, 15, 6, 6, 6, 15, 12, 9, 9, 9, 16, 20, 15, 12, 12, 12, 8, 10, 12, 20, 15, 15, 15, 10, 15, 24, 22, 26, 18, 18, 18, 11, 13, 18, 29, 28, 27, 21, 21, 21, 15, 13, 19, 17, 25, 31, 23, 24, 24, 24, 16, 18, 20, 21

Offset: 0

Kevin J. Gomez, May 25 2019

nonn

a(0) is the index of the first repeated value in Recamán's sequence (A005132).

a(n) appears to grow like sqrt(2n).

			For n = 8, the Recamán-like sequence generated is 8, 7, 5, 2, 6, 1; the sequence halts after a(8) = 6 terms since 1 - 6 = -5 is negative and 1 + 6 = 7 is already in the sequence.

Iteration rule nearly identical to A005132.

A334219 is essentially the same sequence.

def seqr(n):
    sequence = [n]
    i = 1
    while True:
        if n - i > 0 and n - i not in sequence:
            n -= i
            sequence.append(n)
        elif n + i not in sequence:
            n += i
            sequence.append(n)
        else:
            break
        i += 1
    return len(sequence)
print([seqr(n) for n in range(1000)])

a(n) >= ceiling((sqrt(1 + 8*n)-1)/2). - Markel Zubia, May 03 2025

A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

Original entry on oeis.org

3, 7, 13, 19, 31, 36, 37, 43, 61, 73, 79, 97, 103, 127, 139, 157, 163, 181, 223, 229, 241, 271, 283, 307, 337, 349, 373, 379, 409, 421, 433, 439, 457, 499, 547, 577, 607, 631, 643, 673, 691, 709, 733, 751, 787, 811, 829, 853, 877, 919, 937, 967

Offset: 1

Kevin J. Gomez, Sep 10 2015

The only composite term less than 10^11 is 36. - Giovanni Resta, Sep 14 2015

			3 is in the sequence since phi(13) = phi(3) + 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)
Wikipedia, Euler's totient function.

Cf. A000010, A023203.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262085 (k=8), this sequence (k=10).

[n: n in [1..1000] | EulerPhi(n+10) eq EulerPhi(n)+10]; // Vincenzo Librandi, Sep 11 2015

Select[Range@1000, EulerPhi@(# + 10) == EulerPhi[#] + 10 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 10) == eulerphi(n) + 10 \\ Anders Hellström, Sep 11 2015

A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

Original entry on oeis.org

3, 5, 11, 23, 24, 29, 36, 42, 48, 50, 53, 56, 59, 71, 72, 80, 89, 101, 102, 125, 131, 132, 149, 173, 176, 191, 230, 233, 248, 263, 269, 359, 368, 389, 401, 431, 449, 464, 479, 491, 563, 569, 593, 599, 638, 653, 656, 683, 701, 719, 743, 761, 821, 848, 911, 929, 983

Offset: 1

Kevin J. Gomez, Sep 10 2015

Sequence includes numbers n such that n and n + 8 are both prime (A023202).
Sequence also includes numbers n equal to 8*(a Mersenne prime) (cf A000668).
Sequence also includes n such that n/16 and n/8 + 1 are both odd primes.
Contains more composites than sequences A262084 and A262086. This is most likely due to the fact that 8 is a power of 2, as in A001838.

			3 since phi(11) = phi(3) + 8 (3 and 11 are both prime).
24 is a solution since phi(32) = phi(24) + 8 (24 is 8 * 3; 3 is a Mersenne prime).

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010.

Cf. A001838 (k=2), A056772 (k=4), A262084 (k=6), A262086 (k=10).

[n: n in [1..1000] | EulerPhi(n+8) eq EulerPhi(n)+8]; // Vincenzo Librandi, Sep 11 2015

select(t -> numtheory:-phi(t+8) = numtheory:-phi(t)+8, [$1..1000]); # Robert Israel, Mar 04 2016

Select[Range@1000, EulerPhi@(# + 8)== EulerPhi[#] + 8 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 8) == eulerphi(n) + 8 \\ Anders Hellström, Sep 11 2015

[n for n in (1..1000) if euler_phi(n+8) == euler_phi(n)+8] # Bruno Berselli, Mar 04 2016

A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

Original entry on oeis.org

5, 7, 11, 13, 17, 21, 23, 31, 37, 40, 41, 47, 53, 56, 61, 67, 73, 83, 88, 97, 98, 101, 103, 107, 131, 136, 151, 152, 156, 157, 167, 173, 191, 193, 223, 227, 233, 237, 248, 251, 257, 263, 271, 277, 296, 307, 311, 328, 331, 347, 353, 367, 373, 376, 383, 433, 443

Offset: 1

Kevin J. Gomez, Sep 10 2015

The majority of solutions can be predicted by known properties of the equality. There are several solutions that do not fit these parameters.
An odd natural number m is a solution if m and m + 6 are both prime (sexy primes) (A023201).
Among the solutions for even natural numbers are all m = 8*p with odd primes p such that 4*p+3 is a prime number. Proof: From A000010 we can learn that the formula phi(p*2) = floor(((2 + p - 1) mod p)/(p - 1)) + p - 1 is known. If we define p = 4*q+3 and m = 8*q and insert, we will obtain phi(8*q+6) = 4*q+2. Also it is known that phi(8*q) = 4*q-4 if q is any odd prime. - Thomas Scheuerle, Dec 20 2024

			5 is a term since phi(5+6) = 10 = 6 + 4 = phi(5) + 6.

Peter Kagey, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010, A023201.

Cf. A001838 (k=2), A056772 (k=4), A262085 (k=8), A262086 (k=10).

[n: n in [1..500] | EulerPhi(n+6) eq EulerPhi(n)+6]; // Vincenzo Librandi, Sep 11 2015

Select[Range@500, EulerPhi@(# + 6)== EulerPhi[#] + 6 &] (* Vincenzo Librandi, Sep 11 2015 *)

is(n)=eulerphi(n + 6) == eulerphi(n) + 6 \\ Anders Hellström, Sep 11 2015

[n for n in [1..1000] if euler_phi(n+6)==euler_phi(n)+6] # Tom Edgar, Sep 10 2015

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User: Kevin J. Gomez

A308419 Stopping time for Recamán-like iteration of each n: a(0) = n, a(k) = a(k-1) - k if positive and not already in the sequence, a(k) = a(k-1) + k if not already in the sequence, otherwise stop.

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A262086 Numbers k such that phi(k + 10) = phi(k) + 10, where phi(k) = A000010(k) is Euler's totient function.

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A262085 Numbers n such that phi(n + 8) = phi(n) + 8 where phi(n) = A000010(n) is Euler's totient function.

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A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage