A088730 -id:A088730 - cp's OEIS frontend

A023212 Primes p such that 4*p+1 is also prime.

3, 7, 13, 37, 43, 67, 73, 79, 97, 127, 139, 163, 193, 199, 277, 307, 373, 409, 433, 487, 499, 577, 619, 673, 709, 727, 739, 853, 883, 919, 997, 1033, 1039, 1063, 1087, 1093, 1123, 1129, 1297, 1327, 1423, 1429, 1453, 1543, 1549, 1567, 1579, 1597, 1663, 1753

Offset: 1

David W. Wilson

nonn

If p > 3 is a Sophie Germain prime (A005384), p cannot be in this sequence, because all Germain primes greater than 3 are of the form 6k - 1, and then 4p + 1 = 3*(8k-1). - Enrique Pérez Herrero, Aug 15 2011
a(n), except 3, is of the form 6k+1. - Enrique Pérez Herrero, Aug 16 2011
According to Beiler: the integer 2 is a primitive root of all primes of the form 4p + 1 with p prime. - Martin Renner, Nov 06 2011
Chebyshev showed that 2 is a primitive root of all primes of the form 4p + 1 with p prime. - Jonathan Sondow, Feb 04 2013
Also solutions to the equation: floor(4/A000005(4*n^2+n)) = 1. - Enrique Pérez Herrero, Jan 12 2013
Prime numbers p such that p^p - 1 is divisible by 4*p + 1. - Gary Detlefs, May 22 2013
It appears that whenever (p^p - 1)/(4*p + 1) is an integer, then this integer is even (see previous comment). - Alexander R. Povolotsky, May 23 2013
4p + 1 does not divide p^n + 1 for any n. - Robin Garcia, Jun 20 2013
Primes in this sequence of the form 4k+1 are listed in A113601. - Gary Detlefs, May 07 2019
There are no numbers with last digit 1 in this list (i.e., members of A030430) because primes p == 1 (mod 10) lead to 5|(4p+1) such that 4p+1 is not prime. - R. J. Mathar, Aug 13 2019

Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 102, nr. 5.
P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Grigory Ryabov, On schurity of the dihedral group D_(2p), arXiv:2308.14209 [math.GR], 2023.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.

Cf. A001122, A005384, A043297, A088730.
Cf. A005098, A090866.
Cf. A182265, A182434.

[n: n in [0..1000] | IsPrime(n) and IsPrime(4*n+1)]; // Vincenzo Librandi, Nov 20 2010

isA023212 := proc(n)
    isprime(n) and isprime(4*n+1) ;
end proc:
for n from 1 to 1800 do
    if isA023212(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 26 2013

Select[Range[2000], PrimeQ[#] && PrimeQ[4# + 1] &] (* Alonso del Arte, Aug 15 2011 *)
Join[{3}, Select[Range[7, 2000, 6], PrimeQ[#] && PrimeQ[4# + 1] &]] (* Zak Seidov, Jan 21 2012 *)
Select[Prime[Range[300]],PrimeQ[4#+1]&] (* Harvey P. Dale, Oct 17 2021 *)

forprime(p=2,1800,if(Mod(p,4*p+1)^p==1, print1(p", \n"))) \\ Alexander R. Povolotsky, May 23 2013

Sum_{n>=1} 1/a(n) is in the interval (0.892962433, 1.1616905) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021

Name edited by Michel Marcus, Nov 27 2020

A125135 Triangle read by rows in which row n lists prime factors of p^p - 1 where p = prime(n).

Original entry on oeis.org

3, 2, 13, 2, 2, 11, 71, 2, 3, 29, 4733, 2, 5, 15797, 1806113, 2, 2, 3, 53, 264031, 1803647, 2, 2, 2, 2, 10949, 1749233, 2699538733, 2, 3, 3, 109912203092239643840221, 2, 11, 461, 1289, 831603031789, 1920647391913

Offset: 1

N. J. A. Sloane, Jan 21 2007

			Triangle begins:
3;
2, 13;
2, 2, 11, 71;
2, 3, 29, 4733;
2, 5, 15797, 1806113;
2, 2, 3, 53, 264031, 1803647;
2, 2, 2, 2, 10949, 1749233, 2699538733;
2, 3, 3, 109912203092239643840221;
2, 11, 461, 1289, 831603031789, 1920647391913;
2, 2, 7, 59, 16763, 84449, 2428577, 14111459, 58320973, 549334763;
...
n=4: p=7, 7^7-1 = 823542 = 2*3*29*4733 gives row 4.

Sam Wagstaff, Factorizations of p^p - 1 for most p < 180

Cf. A006486, A088730, A125136, A212552, A214812.

for p in [ n : n in [1..100] | IsPrime(n) ] do "\nDoing p =", p; n := p^p -1; Factorisation(n); end for; // John Cannon

T:= n-> (p-> sort(map(i-> i[1]$i[2], ifactors(p^p-1)[2]))[])(ithprime(n)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 20 2022

A125137 a(n) = p^p + 1, where p = prime(n).

Original entry on oeis.org

5, 28, 3126, 823544, 285311670612, 302875106592254, 827240261886336764178, 1978419655660313589123980, 20880467999847912034355032910568, 2567686153161211134561828214731016126483470, 17069174130723235958610643029059314756044734432, 10555134955777783414078330085995832946127396083370199442518

Offset: 1

N. J. A. Sloane, Jan 21 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..77

See A125136 for factorizations. Cf. A088730, A125135.

Cf. A104131, A114846.

[p^p+1: p in PrimesUpTo(40)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime[n]^Prime[n] + 1, {n , 20}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = A051674(n)+1. - R. J. Mathar, Apr 23 2007

A125136 Triangle read by rows in which row n gives list of prime factors of p^p + 1 where p = prime(n).

Original entry on oeis.org

5, 2, 2, 7, 2, 3, 521, 2, 2, 2, 113, 911, 2, 2, 3, 23, 89, 199, 58367, 2, 7, 13417, 20333, 79301, 2, 3, 3, 45957792327018709121, 2, 2, 5, 108301, 1049219, 870542161121, 2, 2, 2, 3, 47, 139, 1013, 1641281, 52626071, 1522029233, 2, 3, 5, 233, 6864997

Offset: 1

N. J. A. Sloane, Jan 21 2007

Product over the n-th row of the table is A051674(n) + 1. The number of elements in the n-th row is A115973(n). - R. J. Mathar, Jan 22 2007

(p + 1) divides p^p + 1 for odd prime p. - Alexander Adamchuk, Jan 22 2007

			Rows read
  5;
  2, 2, 7;
  2, 3, 521;
  2, 2, 2, 113, 911;
  2, 2, 3, 23, 89, 199, 58367;
  2, 7, 13417, 20333, 79301;
  2, 3, 3, 45957792327018709121;
  2, 2, 5, 108301, 1049219, 870542161121;
  2, 2, 2, 3, 47, 139, 1013, 1641281, 52626071, 1522029233;
  2, 3, 5, 233, 6864997, 9487923853, 5639663878716545087233;
  2, 2, 2, 2, 2, 373, 1613, 62869, 145577, 35789156484227, 2706690202468649;
  etc.

Sam Wagstaff, Factorizations of p^p + 1 for most p < 180

Cf. A088730, A125135.

Cf. A007571 = largest factor of n^n + 1.

pfs := proc(n) local ifs,a,e,b ; ifs := ifactors(n)[2] ; a := [] ; for b from 1 to nops(ifs) do for e from 1 to op(2,op(b,ifs)) do a := [op(a),op(1,op(b,ifs))] ; od ; od ; RETURN(a) ; end; A125136 := proc(nmax) local a,p,n,pp ; a := [] ; p := 2 ; while nops(a) < nmax do a := [op(a),op(pfs(p^p+1))] ; p := nextprime(p) ; od ; RETURN(a) ; end; A125136(40) ; # R. J. Mathar, Jan 22 2007

lpf[n_]:=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]; lpf/@(#^#+1&/@ Prime[Range[10]])//Flatten (* Harvey P. Dale, Oct 18 2020 *)

More terms from Alexander Adamchuk and R. J. Mathar, Jan 22 2007

A307653 a(n) = Sum_{d|n} mu(d) * d^d.

Original entry on oeis.org

1, -3, -26, -3, -3124, 46626, -823542, -3, -26, 9999996872, -285311670610, 46626, -302875106592252, 11112006824734470, 437893890380856224, -3, -827240261886336764176, 46626, -1978419655660313589123978, 9999996872, 5842587018385982521380300852

Offset: 1

Seiichi Manyama, Apr 20 2019

sign

			a(6) = 1 - 2^2 - 3^3 + 6^6 = 46626.

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A000040, A008683, A023900, A088730, A307654, A321222.

Array[DivisorSum[#, MoebiusMu[#]*#^# &] &, 21] (* Michael De Vlieger, Apr 21 2019 *)

PARI
```
{a(n) = sumdiv(n, d, moebius(d)*d^d)}
```

a(prime(n)^m) = 1 - prime(n)^prime(n) = -A088730(n) for m > 0.

G.f.: Sum_{k>=1} mu(k)*k^k*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 20 2019

A088807 Number of distinct prime factors of p^p - 1 where p = prime(n).

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 4, 3, 6, 9, 4, 9, 7, 6, 5, 7, 5, 7, 9, 7, 12, 8, 6, 8, 8, 11, 8, 6, 7, 10, 9, 7, 13

Offset: 1

Cino Hilliard, Nov 23 2003

Cf. A000040, A088730, A344870.

PrimeNu/@Table[p^p-1,{p,Prime[Range[30]]}] (* The program takes a long time to run. *) (* Harvey P. Dale, Sep 05 2020 *)

omegaptop(n,m) = { sr=0; forprime(x=2,n, y=omega(x^x-m); print1(y","); sr += 1.0/y; ); print(); }

from sympy import factorint, prime
def a(n): p = prime(n); return len(factorint(p**p-1).values())
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, May 27 2022

a(n) = A344870(A000040(n)). - Amiram Eldar, Jul 04 2024

More terms from Ray Chandler, Feb 21 2004

Name clarified, offset and data corrected and a(27)-a(33) added by Amiram Eldar, Jul 04 2024

A368107 Prime powers p^m such that p | m.

Original entry on oeis.org

4, 16, 27, 64, 256, 729, 1024, 3125, 4096, 16384, 19683, 65536, 262144, 531441, 823543, 1048576, 4194304, 9765625, 14348907, 16777216, 67108864, 268435456, 387420489, 1073741824, 4294967296, 10460353203, 17179869184, 30517578125, 68719476736, 274877906944, 282429536481

Offset: 1

Michael De Vlieger, Jan 15 2024

Proper subset of A072873, which in turn is a proper subset of A342090.
This sequence represents the prime power block in A072873 and A342090.
A342090 \ {a(n)} is in A126706.
A072873 \ {{1} U {a(n)}} is in A286708, in turn a proper subset of A001694.
Contains A051674.

			This sequence contains prime powers of the following form:
  2^2, 2^4, i.e., 2^k such that k is even.
  3^3, 3^6, 3^9, i.e., 3^k such that 3 | k.
  5^5, 5^10, 5^15, i.e., 5^k such that 5 | k, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..3351

Cf. A001694, A051674, A072873, A088730, A126706, A246547, A286708, A342090.

N:= 10^13: # for terms <= N
R:= NULL:
for i from 1 do
  p:= ithprime(i);
  if p^p > N then break fi;
  R:= R, seq(p^k,k=p..floor(log[p](N)), p);
od:
sort([R]); # Robert Israel, Jan 16 2024

nn = 10^12; i = 1; p = 2; While[p^p <= nn, p = NextPrime[p] ];
MapIndexed[Set[S[First[#2]], #1] &, Prime@ Range@ PrimePi[p] ];
Union@ Reap[
    While[j = S[i];
    While[S[i]^j < nn,
      Sow[S[i]^j]; j += S[i] ]; j > 2,
    i++] ][[-1, 1]]

import heapq
from itertools import islice
from sympy import nextprime
def agen(): # generator of terms
    v, h, m, nextp = 4, [(4, 2)], 4, 3
    while True:
        v, p = heapq.heappop(h)
        yield v
        if v >= m:
            m = nextp**nextp
            heapq.heappush(h, (m, nextp))
            nextp = nextprime(nextp)
        heapq.heappush(h, (v*p**p, p))
print(list(islice(agen(), 31))) # Michael S. Branicky, Jan 16 2024

from sympy import integer_nthroot, primefactors
def A368107(n):
    def f(x):
        c = n+x
        for k in range(1,x.bit_length()):
            m = integer_nthroot(x,k)[0]
            c -= sum(1 for p in primefactors(k) if p<=m)
        return c
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/A088730(n) = 0.372116188498... . - Amiram Eldar, Jan 20 2024

A376431 a(n) is the least odd prime factor of prime(n)^prime(n)-1.

Original entry on oeis.org

3, 13, 11, 3, 5, 3, 10949, 3, 11, 7, 3, 3, 5, 3, 23, 13, 29, 3, 3, 5, 3, 3, 41, 11, 3, 5, 3, 53, 3, 7, 3, 5, 17, 3, 37, 3, 3, 3, 83, 43, 89, 3, 5, 3, 7, 3, 3, 3, 113, 3, 29, 7, 3, 5, 21589, 131, 67, 3, 3, 5, 3, 73, 3, 5, 3, 79, 3, 3, 173, 3, 11, 179, 3, 3, 3, 191

Offset: 1

Hugo Pfoertner, Sep 27 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..6542

Cf. A078701, A088730, A376432.

a376431(n) = my(pp=prime(n)^prime(n)-1); forprime (p=3, oo, if(pp%p==0, return(p)))

a(n) = A078701(A088730(n)). - Michel Marcus, Sep 27 2024

A088816 Numbers of the form p^p - 2 where p is prime.

Original entry on oeis.org

2, 25, 3123, 823541, 285311670609, 302875106592251, 827240261886336764175, 1978419655660313589123977, 20880467999847912034355032910565, 2567686153161211134561828214731016126483467

Offset: 1

Cino Hilliard, Nov 23 2003

nonn

Sum of reciprocals = 0.5403214192032919719402677251

#^#-2&/@Prime[Range[10]] (* Harvey P. Dale, Sep 22 2014 *)

ptop(n,m) = { sr=0; forprime(x=2,n, y=x^x-m; print1(y","); sr += 1.0/y; ); print(); print(sr) }

a(n) = A088730(n)-1=A088731(n)+1. - R. J. Mathar, Apr 26 2007

cp's OEIS Frontend

A023212 Primes p such that 4*p+1 is also prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A125135 Triangle read by rows in which row n lists prime factors of p^p - 1 where p = prime(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

A125137 a(n) = p^p + 1, where p = prime(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A125136 Triangle read by rows in which row n gives list of prime factors of p^p + 1 where p = prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A307653 a(n) = Sum_{d|n} mu(d) * d^d.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A088807 Number of distinct prime factors of p^p - 1 where p = prime(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A368107 Prime powers p^m such that p | m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple