A023514 -id:A023514 - cp's OEIS frontend

A057690 Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

3, 3, 4, 4, 3, 4, 4, 5, 4, 6, 3, 4, 4, 6, 5, 5, 3, 4, 6, 3, 6, 5, 5, 4, 4, 5, 6, 4, 4, 8, 5, 4, 5, 5, 5, 3, 4, 6, 4, 6, 4, 8, 3, 5, 6, 4, 7, 5, 4, 5, 7, 4, 6, 4, 6, 6, 6, 3, 12, 4, 5, 5, 6, 3, 4, 4, 4, 5, 5, 4, 7, 6, 4, 5, 9, 5, 3, 4, 4, 6, 3, 8, 4, 6, 5, 6, 3, 5, 6, 6, 8, 5, 5, 6, 7, 5, 5, 4, 3, 4, 5, 5, 5, 5, 4

Offset: 2

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

Note that not all cycles for the iteration starting with p contain the number 1; a(60), for the prime 281, is the first example of this. Its iterates are: 281, 78962, 39481, 3037, 853398, 426699, 142233, 47411, 6773, 521, 146402, 73201, 1031, 289712, 144856, 72428, 36214, 18107, 953, 267794, 133897, with the last 12 terms cycling. Another example is provided by 2543, the 372nd prime. - T. D. Noe, Apr 02 2008

			For n=4, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.

Michel Marcus, Table of n, a(n) for n = 2..10000

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057691.

Cf. A023514.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

f(m, p) = {forprime(q=2, precprime(p-1), if (! (m % q), return (m/q));); m*p+1;}
a(n) = {my(p=prime(n), x=p, list = List()); listput(list, x); while (1, x = f(x, p); for (i=1, #list, if (x == list[i], return (#list - i + 1));); listput(list, x););} \\ Michel Marcus, Jan 12 2021

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, traj, seen = P, list(primerange(2, P)), [], set()
    while x not in seen:
        traj.append(x)
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return len(traj) - traj.index(x)
print([a(n) for n in range(2, 107)]) # Michael S. Branicky, Dec 11 2023

a(n) = A023514(n)+1 if the cycle contains the number 1. - Jon Maiga, Jan 12 2021

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

Corrected by T. D. Noe, Apr 02 2008

A008335 Number of distinct primes dividing p+1 as p runs through the primes.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001221, A008864, A023514.

for i from 1 to 500 do if isprime(i) then print(nops(factorset(i+1))); fi; od;

a[n_] := PrimeNu[Prime[n]+1]; Array[a, 100] (* Amiram Eldar, Sep 10 2024 *)

a(n) = omega(prime(n)+1); \\ Michel Marcus, Mar 29 2016

a(n) = A001221(A008864(n)). - Michel Marcus, Mar 29 2016

A210934 Sum of prime factors of prime(n)+1 (counted with multiplicity).

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 8, 9, 9, 10, 10, 21, 12, 15, 11, 11, 12, 33, 21, 12, 39, 13, 14, 13, 16, 22, 19, 13, 18, 24, 14, 18, 28, 16, 15, 25, 81, 45, 16, 34, 15, 22, 15, 99, 19, 16, 57, 17, 26, 30, 21, 16, 24, 17, 48, 20, 16, 25, 141, 52, 75, 19, 22, 22, 159, 58, 87

Offset: 1

Paolo P. Lava, Mar 30 2012

nonn

From an idea of Michael B. Porter.

			prime(8) = 19, and 19+1 = 20 = 2*2*5, so a(8) = 2+2+5 = 9.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A023508, A023514, A210936

with(numtheory);
P:=proc(i)
local a,k,n;
for n from 1 to i do
  a:=ifactors(ithprime(n)+1)[2]; print(add(a[k][1]*a[k][2],k=1..nops(a)));
od; end:
# alternative
A210934 := proc(n)
    local p,pplus,f ;
    p := ithprime(n) ;
    pplus := ifactors(p+1)[2] ;
    add(op(1,f)*op(2,f),f=pplus) ;
end proc:
seq(A210934(n),n=1..300) ; # R. J. Mathar, May 25 2022

a(n) = A001414(A008864(n)). - Michel Marcus, Oct 05 2013

A210936 Sum of prime factors of prime(n)-1 (counted with multiplicity).

Original entry on oeis.org

0, 2, 4, 5, 7, 7, 8, 8, 13, 11, 10, 10, 11, 12, 25, 17, 31, 12, 16, 14, 12, 18, 43, 17, 13, 14, 22, 55, 13, 15, 15, 20, 23, 28, 41, 15, 20, 14, 85, 47, 91, 15, 26, 15, 18, 19, 17, 42, 115, 26, 35, 26, 16, 17, 16, 133, 71, 16, 30, 18, 52, 77, 25, 38, 22, 83

Offset: 1

Paolo P. Lava, Mar 30 2012

nonn

From an idea of Michael B. Porter.

			prime(10) = 29, and 29-1 = 28 = 2*2*7, so a(10) = 2+2+7 = 11.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A023508, A023514, A210934

with(numtheory);
P:=proc(i)
local a,k,n;
for n from 1 to i do
  a:=ifactors(ithprime(n)-1)[2]; print(add(a[k][1]*a[k][2],k=1..nops(a)));
od; end:
# alternative
A210936 := proc(n)
        local p,pplus,f ;
        p := ithprime(n) ;
        pplus := ifactors(p-1)[2] ;
        add(op(1,f)*op(2,f),f=pplus) ;
end proc:
seq(A210936(n),n=1..300) ; # R. J. Mathar, May 25 2022

a(n) = A001414(A006093(n)). - Michel Marcus, Oct 05 2013

A254787 a(n)= least initial term of n consecutive primes {p(m),..,p(m+n-1)} such that all numbers {1+p(m),..,1+p(m+n-1)} are a product of the same number k primes, where p(m) is m-th prime A000040(m).

Original entry on oeis.org

2, 3, 139, 557, 3821, 53609, 179659, 7190917, 3599100, 16687573, 20394197, 101439558

Offset: 1

Zak Seidov, Feb 15 2015

nonn

Corresponding values of m are: 1,2,34,102,530,5462,16309,488739,3599100,308495917,20394197,101439558
Corresponding values of k are: 1,2,4,4,5,4,5,4,5,4,5,5
A023514(i=m,..,m+n-1) are of the same value.

			a(2)=3, because {p(2),p(3)}={3,5}, and {4,6}={2^2,2*3}, k=2;
a(3)=139, because {p(34),p(35),p(36)}={139,149,151}, and {140,150,152}={2^2*5*7,2*3*5^2,2^3*19}, k=4.

Cf. A000040, A023514.

A283652 Primes p such that bigomega(p+1) = 20.

Original entry on oeis.org

5505023, 8847359, 13271039, 17915903, 22118399, 24379391, 27131903, 29859839, 31981567, 32440319, 34406399, 36863999, 37486591, 38273023, 42205183, 46448639, 48496639, 54001663, 57016319, 60948479, 61439999, 62128127, 62705663, 67184639

Offset: 1

Zak Seidov, Mar 12 2017

nonn

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A023514, A118883.

p = 4128767; While[p<=57016319, If[PrimeOmega[1 + p] == 20, Print[p,", "]]; p = NextPrime[p + 2]] (* Indranil Ghosh, Mar 13 2017, after the PARI program from the author *)
Select[Prime[Range[4*10^6]],PrimeOmega[#+1]==20&] (* Harvey P. Dale, Jul 05 2020 *)

{p=4128767; while(p<=57016319, if(bigomega(1+p)==20, print1(p ",")); p=nextprime(p+2))}

A255164 Numbers m such that, starting with prime(m), 10 consecutive primes plus 1 are a product of the same number k of primes (i.e., are k-almost primes).

Original entry on oeis.org

16687573, 18163180, 19945279, 21433861, 32241571, 41642534, 61124701, 84985671, 99125673, 120180818, 132409582, 136276974, 139516858, 149714850, 152735870, 160041096, 161934847, 172578057, 177536370, 177733590, 185207739

Offset: 1

Zak Seidov, Feb 15 2015

nonn

{A023514(m),..,A023514(m+9)} are of the same value k.

			a(1)=16687573 because A023514(16687573),..,A023514(16687573+9) are all equal to 4, or
prime(16687573,..,16687573+9)=
{308495917, 308495947, 308495953, 308496017, 308496043, 308496049, 308496091, 308496157, 308496169, 308496173}, and
308495918=2*23*103*65111, 308495948=2^2*79*976253,
308495954=2*13*127*93427, 308496018=2*3*43*1195721,
308496044=2^2*139*554849, 308496050=2*5^2*6169921,
308496092=2^2*131*588733, 308496158=2*37*317*13151,
308496170=2*5*421*73277, 308496174=2*3*7*7345147.

Cf. A023514, A254787.

cp's OEIS Frontend

A057690 Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

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A008335 Number of distinct primes dividing p+1 as p runs through the primes.

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A210934 Sum of prime factors of prime(n)+1 (counted with multiplicity).

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Maple

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A210936 Sum of prime factors of prime(n)-1 (counted with multiplicity).

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Maple

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A254787 a(n)= least initial term of n consecutive primes {p(m),..,p(m+n-1)} such that all numbers {1+p(m),..,1+p(m+n-1)} are a product of the same number k primes, where p(m) is m-th prime A000040(m).

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A283652 Primes p such that bigomega(p+1) = 20.

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Programs

Mathematica

PARI

A255164 Numbers m such that, starting with prime(m), 10 consecutive primes plus 1 are a product of the same number k of primes (i.e., are k-almost primes).

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