A039654 -id:A039654 - cp's OEIS frontend

A039652 Becomes prime after n iterations of f(x) = phi(x)+1 (least inverse of A039651).

2, 1, 15, 35, 69, 255, 535, 949, 1957, 2513, 2923, 4531, 17701, 22957, 54589, 79421, 80029, 84493, 98581, 102827, 115243, 239111, 291149, 310813, 362621, 398893, 598341, 801923, 838307, 1063493, 1079833, 1123813, 1311121, 1329403, 1582439

Offset: 0

David W. Wilson

Of the terms n <= 66, all are semiprimes except those for n = 0, 1, 5, and 19. Why? - T. D. Noe, Oct 17 2013

T. D. Noe, Table of n, a(n) for n = 0..66

Cf. A039649, A039650, A039651, A039652, A039653, A039654, A039655, A039656.

nn = 34; t = Table[0, {nn}]; found = 0; n = 0; While[found < nn, n++; len = Length[NestWhileList[EulerPhi[#] + 1 &, n, UnsameQ, All]] - 2; If[len <= nn && t[[len]] == 0, t[[len]] = n; found++]]; t = Join[{2}, t] (* T. D. Noe, Oct 17 2013 *)

A039656 Becomes prime after n iterations of f(x) = sigma(x)-1 (least inverse of A039655).

Original entry on oeis.org

2, 6, 4, 27, 12, 9, 121, 301, 930, 484, 578, 441, 1273, 468, 4863, 3171, 9216, 8373, 19692, 19416, 25442, 13440, 19230, 16641, 16804, 83161, 100652, 226181, 203400, 133200, 419248, 380979, 744796, 553296, 634710, 539476, 505584, 674416, 634206

Offset: 0

David W. Wilson

nonn

Records: 2, 6, 27, 121, 301, 930, 1273, 4863, 9216, 19692, 25442, 83161, 100652, 226181, 419248, 744796, 3739690, 4238314, etc. - Robert G. Wilson v, Sep 23 2017
Indices of records: 0, 1, 3, 6, 7, 8, 12, 14, 16, 18, 20, 25, 26, 27, 30, 32, 46, 47, 48, 49, 50, 56, 57, 58, 59, 61, 63, 65, 67, 76, 77, 78, 82, 83, 84, 85, etc. - Robert G. Wilson v, Sep 23 2017
Checked through a(138)=60780636903. - Hugo Pfoertner, Nov 15 2017

Hugo Pfoertner, Table of n, a(n) for n = 0..138 (terms up to 66 from David W. Wilson, 67-100 from Robert G. Wilson v and Charles R Greathouse IV).

Cf. A039649, A039650, A039651, A039652, A039653, A039654, A039655, A292114, A292115.

f[n_] := Plus @@ Divisors@ n -1; g[n_] := Length@ NestWhileList[f@# &, n, !PrimeQ@# &] - 1; t[] := -1; k = 2; While[k < 10000001, a = g[k]; If[ t@ a == -1, t@ a = k; Print[{a, k}]]; k++]; t@# & /@ Range[0, 50] (* _Robert G. Wilson v, Sep 22 2017 *)

A291301 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to the product of the first n primes, or -1 if no prime is ever reached.

Original entry on oeis.org

2, 11, 71, 743, 6911, 117239, 2013983, 34836479, 921086711, 33596203871, 18754852859999, 1306753691335679, 2795529813471359, 200489563747397471, 7143750592470475271, 146095655504943513599, 161739770170976834876927, 543475838478389870591999, 317180662337566737324195839

Offset: 1

N. J. A. Sloane, Aug 31 2017

nonn

A subsequence of A039654.

			2*3*5*7*11*13 = 30030 -> 96767 -> 111359 -> 117239 takes three steps to reach a prime, so a(6) = 117239.

Chai Wah Wu, Table of n, a(n) for n = 1..35

Cf. A039654, A000203, A002110, A291302 (number of steps).

p[n_]:=Times@@Prime/@Range[n]; f[n_]:=DivisorSigma[1,n]-1;
a[n_]:=Last[NestWhileList[f,p[n],CompositeQ]]; a/@Range[20] (* Ivan N. Ianakiev, Sep 01 2017 *)

from sympy import primorial, isprime, divisor_sigma
def A291301(n):
    m = primorial(n)
    while not isprime(m):
        m = divisor_sigma(m) - 1
    return m # Chai Wah Wu, Aug 31 2017

a(10)-a(19) from Chai Wah Wu, Aug 31 2017

A291302 a(n) = number of steps to reach a prime when x -> sigma(x)-1 is repeatedly applied to the product of the first n primes, or -1 if no prime is ever reached.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 1, 3, 4, 46, 57, 7, 9, 17, 1, 45, 1, 33, 8, 10, 4, 3, 32, 6, 47, 17, 21, 41, 17, 12, 11, 10, 31, 74, 25, 99, 11

Offset: 1

N. J. A. Sloane, Aug 31 2017

			2*3*5*7*11*13 = 30030 -> 96767 -> 111359 -> 117239 takes three steps to reach a prime, so a(6) = 3.

Cf. A039654, A039653, A291301 (the prime reached).

A291302 := proc(n)
    local a,x ;
    a := 0 ;
    x := mul(ithprime(i),i=1..n) ;
    while not isprime(x) do
        x := numtheory[sigma](x)-1 ;
        a := a+1 ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 12 2017

p[n_]:=Times@@Prime/@Range[n];f[n_]:=DivisorSigma[1,n]-1;
a[n_]:=Length[NestWhileList[f,p[n],CompositeQ]]-1;a/@Range[34] (* Ivan N. Ianakiev, Sep 01 2017 *)

from sympy import primorial, isprime, divisor_sigma
def A291302(n):
    m, c = primorial(n), 0
    while not isprime(m):
        m = divisor_sigma(m) - 1
        c += 1
    return c # Chai Wah Wu, Aug 31 2017

a(11)-a(35) from Chai Wah Wu, Aug 31 2017

a(36)-a(38) from Ivan N. Ianakiev, Sep 01 2017

A292114 List of numbers n such that A039655(n) reaches a new record high.

Original entry on oeis.org

2, 4, 9, 121, 301, 441, 468, 3171, 8373, 13440, 16641, 16804, 83161, 100652, 133200, 367428, 395640, 459680, 701823, 3739690, 4238314, 6698616, 9014248, 12301860, 16956850, 22230514, 54889200, 60676144, 84983056, 116648892, 128942664

Offset: 1

N. J. A. Sloane, Sep 22 2017

nonn

Naively, one might have expected these numbers to have some other distinguishing property (primorials, perhaps), but that seems not to be the case.

Increasingly many of the values are of the form m*p with a (large) prime p and a smooth m, often m = 2^k (for a(n), n = 12, 14, 21, 23, 26, 28, 29, ...) or m = 2^k*3^k' (n = 7, 9, 19, 22, 30, ...) or m = 2^k*5^k' (n = 20, 25, ...). I conjecture that almost all terms are even. Also, for most terms (n = 1, 2, 3, 4, 5, 7, 10, 12, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 29, 30, 31, ...), either a(n)-1 or a(n)+1 has at most 2 prime divisors. - M. F. Hasler, Sep 25 2017

Hugo Pfoertner, Table of n, a(n) for n = 1..48
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Cf. A039654, A039655, A292115.

m=-n=1;until(print1(n","),until(A039655(n++)>m,);m=A039655(n)) \\ M. F. Hasler, Sep 25 2017

More terms from Hugo Pfoertner, Sep 22 2017

A323075 The fixed point reached when map x -> 1+(x-(largest divisor d < x)) is iterated, starting from x = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 3, 7, 5, 7, 3, 11, 7, 13, 5, 11, 7, 17, 3, 19, 11, 11, 7, 23, 13, 11, 5, 19, 11, 29, 7, 31, 17, 23, 3, 29, 19, 37, 11, 19, 11, 41, 7, 43, 23, 31, 13, 47, 11, 43, 5, 29, 19, 53, 11, 31, 29, 19, 7, 59, 31, 61, 17, 43, 23, 53, 3, 67, 29, 47, 19, 71, 37, 73, 11, 29, 19, 67, 11, 79, 41, 31, 7, 83, 43, 47, 23, 59, 31, 89, 13

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

After a(1) = 1, the fixed point reached is always a prime. Question: Do all odd primes occur infinitely often?

Yes. All odd primes occur infinitely often. A060681(2*k) = k + 1 and so for each k > 1 there exists an integer m such that a(m) = p where p is an odd prime. - David A. Corneth, Jan 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..20669

Cf. A000040, A000079, A000918, A060681, A323076, A323079, A323164, A323165 (ordinal transform).

Cf. also A039650, A039654.

{1}~Join~Array[FixedPoint[1 + (# - Divisors[#][[-2]]) &, #] &, 89, 2] (* Michael De Vlieger, Jan 04 2019 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323075(n) = { my(nn = 1+A060681(n)); if(nn==n,n,A323075(nn)); };

If n == (1+A060681(n)), then a(n) = n, otherwise a(n) = a(1+A060681(n)).
a(2^k * p - 2^(k+1) + 2) = a(A000079(k) * p - A000918(k+1)) = p for k >= 0. - David A. Corneth, Jan 08 2019
a(1) = 1, and for n > 1, a(n) = A000040(A323164(n)). - Antti Karttunen, Jan 08 2019

A291776 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

Original entry on oeis.org

3, 7, 23, 31, 103, 127, 431, 911, 1847, 6719, 10487, 8191, 56999, 41399, 135647, 131071, 560159, 524287, 1999871, 3982271, 5909759, 17512991, 46092239, 46335599, 164460119, 186592247, 736727807, 3926707199, 4146049487, 2147483647, 8994904463, 11132323439

Offset: 2

N. J. A. Sloane, Aug 31 2017

nonn

			For n=9, 2^n-1 = 511 with iterates 511->591->791->911, and 911 is the first prime, so a(7)=911.

Lars Blomberg, Table of n, a(n) for n = 2..270

Cf. A039654, A039655, A291301, A291302, A291777.

P(x) = {for(c=0,10^6,if(isprime(x),return(x),x=sigma(x)-1));-1}
vector(200,n,P(2^(n+1)-1)) \\ Lars Blomberg, Sep 01 2017

Added a(7) and a(13)-a(33) from Lars Blomberg, Sep 01 2017

A291777 a(n) = number of steps to reach a prime when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 3, 2, 9, 2, 0, 7, 3, 4, 0, 2, 0, 1, 4, 1, 4, 2, 3, 4, 2, 12, 22, 8, 0, 3, 3, 4, 3, 1, 2, 2, 3, 3, 4, 3, 13, 2, 16, 3, 8, 3, 14, 17, 9, 37, 4, 7, 4, 7, 11, 4, 3, 14, 0, 14, 8, 1, 6, 8, 73, 26, 10, 1, 32, 6, 10, 2, 6, 2, 33, 2, 4, 52, 12, 16

Offset: 2

N. J. A. Sloane, Aug 31 2017

nonn

			For n=9, 2^n-1 = 511 with iterates 511->591->791->911, and 911 is the first prime, so a(7)=3.

Lars Blomberg, Table of n, a(n) for n = 2..270

Cf. A039654, A039655, A291301, A291302, A291776.

C(x) = {for(c=0,10^5,if(isprime(x),return(c),x=sigma(x)-1));-1}
vector(200,n,C(2^(n+1)-1)) \\ Lars Blomberg, Sep 01 2017

a(13)-a(82) from Lars Blomberg, Sep 01 2017

A292115 Records in A039655.

Original entry on oeis.org

0, 2, 5, 6, 7, 11, 13, 15, 17, 21, 23, 24, 25, 26, 29, 40, 42, 44, 45, 46, 47, 54, 55, 56, 57, 58, 60, 64, 66, 72, 74, 75, 76, 79, 80, 81, 100, 104, 107, 117, 119, 120, 127, 128, 131, 134, 135, 137, 138

Offset: 1

Hugo Pfoertner and N. J. A. Sloane, Sep 22 2017

Cf. A039654, A039655, A039656, A292114.

A291293 Sequence mod 5 defined by Baldini-Eschgfäller coupled dynamical system (f,lambda,alpha) with f(k) = A000203(k)-1, lambda(y) = 3y+2 mod 5 for y in Y = {0,1,2,3,4}, and alpha(k) = k mod 5 for k in Omega = {primes}.

Original entry on oeis.org

2, 3, 2, 0, 0, 2, 0, 0, 3, 1, 1, 3, 1, 1, 2, 2, 4, 4, 0, 0, 1, 3, 4, 2, 0, 3, 2, 4, 0, 1, 3, 3, 1, 3, 0, 2, 4, 2, 4, 1, 2, 3, 1, 3, 0, 2, 1, 2, 1, 0, 3, 3, 0, 0, 0, 4, 4, 4, 3, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 1, 0, 3, 1, 1, 4, 2, 3, 4, 1, 4, 3, 3, 1, 3, 0

Offset: 2

N. J. A. Sloane, Aug 30 2017

nonn

This sequence assumes that the Erdos conjecture is true, that iterating k -> sigma(k)-1 always reaches a prime (cf. A039654).

Lucilla Baldini, Josef Eschgfäller, Random functions from coupled dynamical systems, arXiv preprint arXiv:1609.01750 [math.CO], 2016. See Example 3.6.

Cf. A000203, A039654, A291291, A262684.

Let f(k) = A000203(k)-1 = sigma(k) - 1, lambda(y) = 3y+2 mod 5 for y in Y = {0,1,2,3,4}, and alpha(k) = k mod 5 for k in Omega = {primes}. Here sigma is the sum of divisors function A000203.

Then a(n) for n >= 2 is defined by a(n) = alpha(n) if n in Omega, and otherwise by a(n) = lambda(a(f(n))).

cp's OEIS Frontend

A039652 Becomes prime after n iterations of f(x) = phi(x)+1 (least inverse of A039651).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A039656 Becomes prime after n iterations of f(x) = sigma(x)-1 (least inverse of A039655).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A291301 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to the product of the first n primes, or -1 if no prime is ever reached.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A291302 a(n) = number of steps to reach a prime when x -> sigma(x)-1 is repeatedly applied to the product of the first n primes, or -1 if no prime is ever reached.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A292114 List of numbers n such that A039655(n) reaches a new record high.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A323075 The fixed point reached when map x -> 1+(x-(largest divisor d < x)) is iterated, starting from x = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A291776 a(n) = prime that is eventually reached when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A291777 a(n) = number of steps to reach a prime when x -> sigma(x)-1 is repeatedly applied to 2^n-1, or -1 if no prime is ever reached.

Views

Author

Keywords

Examples

Links

Crossrefs