A023195 -id:A023195 - cp's OEIS frontend

A187823 Primes of the form (p^x - 1)/(p^y - 1), where p is prime, y > 1, and y is the largest proper divisor of x.

5, 17, 73, 257, 757, 65537, 262657, 1772893, 4432676798593, 48551233240513, 378890487846991, 3156404483062657, 17390284913300671, 280343912759041771, 319913861581383373, 487014306953858713, 5559917315850179173, 7824668707707203971, 8443914727229480773, 32564717507686012813

Offset: 1

Bernard Schott, Dec 27 2012

nonn

Complement of A023195 relative to A003424.
Only eight primes of this form don't exceed 1.275*10^10 (see Bateman and Stemmler):
(1) three of the form (p^9 - 1)/(p^3 - 1): 73 (p=2), 757 (p=3), 1772893 (p=11);
(2) four of the form (2^x - 1)/(2^y - 1) with x = 2y: 5 (x=4), 17 (x=8), 257 (x=16), 65537 (x=32); and
(3) the prime 262657 = (2^27 - 1)/(2^9 - 1).
Some of these prime numbers are not Brazilian, these are Fermat primes > 3: 5, 17, 257, 65537, so they are in A220627.
The other primes are Brazilian so they are in A085104, example: (p^9 - 1)/(p^3 - 1) = 111_{p^3} with 73 = 111_8, 757 = 111_27, 1772893 = 111_1331, also 262657 = 111_512 [See section V.4 of Quadrature article in Links] (comment improved in Mar 03 2023).
Comments from Don Reble, Jul 28 2022 (Start)
This is an easy sequence that looks hard.
Note that x must be a power of a prime; otherwise (p^x-1)/(p^y-1) has too many cyclotomic factors.
Almost all values are (p^9-1)/(p^3-1). The exceptions below 10^45
    are the Fermat primes 5, 17, 257, 65537 and also
    262657, 4432676798593, 5559917315850179173,
    227376585863531112677002031251,
    467056170954468301850494793701001,
    36241275390490156321975496980895092369525753,
    284661951906193731091845096405947222295673201 (see examples).
(End)

			5 = (2^4 - 1)/(2^2 - 1)= 11_{2^2} = 11_4.
17 = (2^8 - 1)/(2^4 - 1) = 11_{2^4} = 11_16.
257 = (2^16 - 1)/(2^8 - 1) = 11_{2^8} = 11_256.
757 = (3^9 - 1)/(3^3 - 1) = 111_{3^3} = 111_27.
262657 = (2^27 - 1)/(2^9 - 1) = 111_{2^9} = 111_512.
655357 = (2^32 - 1)/(2^16 - 1) = 11_{2^16} = 11_655356.
4432676798593 = (2^49 - 1)/(2^7 - 1) = 1111111_{2^7} = 1111111_128.
5559917315850179173 = (11^27 - 1)/(11^9 - 1) = 111_{11^3} = 111_1331.
227376585863531112677002031251 = (5^49 - 1)/(5^7 - 1) = 1111111_{5^7}.
467056170954468301850494793701001 = (43^25 - 1)/(43^5 - 1) = 11111_{43^5}.
36241275390490156321975496980895092369525753 = (263^27 - 1)/(263^9 - 1).
284661951906193731091845096405947222295673201 = (167^25 - 1)/(167^5 - 1).

Don Reble, Table of n, a(n) for n = 1..50000
Paul T. Bateman and Rosemarie M. Stemmler, Waring's problem for algebraic number fields and primes of the form (p^r-1)/(p^d-1), Illinois J. Math. 6 (1962), pp. 142-156.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Index entries for sequences related to Brazilian numbers.

Equals A003424 \ A023195.

Cf. A085104, A220627.

a(9)-a(16) from Don Reble, Jul 28 2022

a(17)-a(20) from Don Reble, Mar 21 2023

A248963 Prime powers p^m for which sigma(p^2m) is not prime.

Original entry on oeis.org

1, 7, 9, 11, 13, 16, 19, 23, 25, 29, 31, 32, 37, 43, 47, 53, 61, 67, 73, 79, 81, 83, 97, 103, 107, 109, 113, 121, 127, 128, 137, 139, 149, 151, 157, 163, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 257, 263, 269, 271, 277, 281, 283, 307, 311, 313, 317, 331

Offset: 1

M. F. Hasler, Oct 18 2014

nonn

sigma(x) cannot be prime unless x is a square of a prime power, x = p^2m, cf. A055638 and A023194. This sequence lists the complement: prime powers whose square does not have a prime sum of divisors.

Although generally 1 is not considered a prime power, it seemed logical for various good reasons to include the initial term a(1)=1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000961, A055638, A023194, A023195, A062700.

for(n=1,999,isprimepower(n)||next;isprime(sigma(n^2))||print1((n)","))

A248963 = A000961 \ A055638, i.e., the complement of A055638 in A000961.

A331036 Odd values of the sum-of-divisors function sigma (A000203), listed by increasing size and with multiplicity.

Original entry on oeis.org

1, 3, 7, 13, 15, 31, 31, 39, 57, 63, 91, 93, 121, 127, 133, 171, 183, 195, 217, 255, 307, 363, 381, 399, 399, 403, 403, 465, 511, 549, 553, 741, 781, 819, 847, 855, 871, 921, 931, 961, 993, 1023, 1093, 1143, 1209, 1281, 1407, 1651, 1659, 1723, 1729, 1767, 1767, 1815, 1893, 1953

Offset: 1

M. F. Hasler, Jan 08 2020

nonn

See A060657 for the range (without repeated terms) and A152677 for the subsequence of odd values in A000203.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A060657, A023195 (subset of primes), A152677 (subsequence of odd values in A000203), A300869 (repeated terms).

N:= 2000: # for terms <= N
Res:= NULL:
for m from 1 to floor(sqrt(N)) by 2 do
  sm:= numtheory:-sigma(m^2);
  for k from 1 to floor(log[2](N/sm+1)) do
    v:= sm*(2^k-1);
    if v <= N then Res:= Res, v; count:= count+1 fi;
  od
od:
sort([Res]); # Robert Israel, Jan 14 2020

Sort@ Select[DivisorSigma[1, Range@ 2000], OddQ[#] && # < 2000 &] (* Giovanni Resta, Jan 08 2020 *)

list(lim)=select(k->k<=lim, vecsort(apply(sigma, concat(vector(sqrtint(lim\1), i, i^2), vector(sqrtint(lim\2), i, 2*i^2))))) \\ Charles R Greathouse IV, Feb 15 2013 [originally added in A152677]

A377927 Numbers k such that 4^sigma(k) - k is a prime.

Original entry on oeis.org

1, 5, 17, 57, 675, 1329, 1425, 3803, 39617

Offset: 1

J.W.L. (Jan) Eerland, Nov 11 2024

			17 is in the sequence because 4^sigma(17) - 17 = 4^18 - 17 = 68719476719 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651, A367460.

[n: n in[1..10000] | IsPrime((4^SumOfDivisors(n)) - n)];

a[n_] := Select[Range@ n, PrimeQ[4^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[4^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

a(9) from Michael S. Branicky, Nov 11 2024

A243765 Numbers that have all their divisors in A002191 (possible values for sigma(n), A000203).

Original entry on oeis.org

1, 3, 7, 13, 31, 39, 91, 93, 127, 217, 307, 381, 403, 921, 961, 1093, 1209, 1651, 1723, 2149, 2801, 2821, 3279, 3541, 3937, 3991, 4953, 5113, 5169, 7651, 8011, 8191, 8403, 9517, 10303, 10623, 11811, 11973, 12061, 12493, 15339, 17293, 19531, 19607, 22399

Offset: 1

Michel Marcus, Jun 10 2014

nonn

Since 2 does not belong to A002191, all terms are odd.
All primes p that are in A023195 (Prime numbers that are the sum of the divisors of some n), are also in this sequence; and the prime factors of all terms can only belong to A023195.
Up to 10^7, only one term is a prime power: 961=31^2 (being a square, see A038688, A228061 and A243810).

			The divisors of 3 are 1 and 3 that both belong to A002191, 1 as sigma(1) and 3 as sigma(2).
The divisors of 39 are 1, 3, 13 and 39 all of which belong to A002191, 13 as sigma(9) 39 as sigma(18).

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A000203, A002191, A023195.

Cf. A045572 (analog sequence with the sum of proper divisors instead).

N:= 10^6: # to get all terms up to N
A002191:= select(`<=`,{seq(numtheory[sigma](i),i=1..N)},N):
A243765:= select(t -> numtheory[divisors](t) subset A002191, A002191); # Robert Israel, Jun 16 2014

list(lim) = select(n->n<=lim, Set(vector(lim\=1, n, sigma(n))));
isok(n, lists) = {fordiv (n, d, if (!vecsearch(lists, d), return(0))); return(1);}
lista(nn) = {lists = list(nn); for(n=1, nn, if (isok(n, lists), print1(n, ", ")););}

A292446 Numbers k such that sigma((k + 1) / 2) is a prime q.

Original entry on oeis.org

3, 7, 17, 31, 49, 127, 577, 1457, 3361, 4801, 6961, 8191, 10081, 15841, 20401, 31249, 34321, 55777, 57121, 59857, 131071, 167041, 171697, 293377, 524287, 559681, 916657, 982801, 1062881, 1104097, 1158241, 1195057, 1367857, 1407841, 1414561, 1468897, 1659841

Offset: 1

Jaroslav Krizek, Sep 16 2017

Corresponding values of primes q are in A062700.
Prime terms are in A292447.
Mersenne primes p = 2^k - 1 (A000668) are terms: sigma((p + 1) / 2) = sigma((2^k - 1 + 1) / 2) = sigma(2^(k - 1)) = 2^k - 1.
This sequence also has terms of the form p^(q-1) where p and q are odd primes, i.e., A002315(1)^2 = 7^2 and A002315(3)^2 = 239^2. Terms that are not squarefree are 49, 55777, 57121, 167041, 2789521, 50060017, ... - Altug Alkan, Oct 02 2017

			49 is a term because sigma((49 + 1) / 2) = sigma(25) = 31 (prime).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A000668, A023194, A023195, A062700, A292447, A292448.

[n: n in [1..10^8] | IsOdd(n) and IsPrime(SumOfDivisors((n+1) div 2))];

Select[Range[1,166*10^4,2],PrimeQ[DivisorSigma[1,(#+1)/2]]&] (* Harvey P. Dale, Jun 22 2022 *)

isok(n) = (n%2) && isprime(sigma((n+1)/2)); \\ Michel Marcus, Sep 16 2017

a(n) = 2*A023194(n) - 1.

A377786 Numbers k such that 5^sigma(k) - k is a prime.

Original entry on oeis.org

4, 524, 7206, 11156

Offset: 1

J.W.L. (Jan) Eerland, Nov 11 2024

			4 is in the sequence because 5^sigma(4) - 4 = 5^7 - 4 = 78121 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651, A367460, A377927.

[n: n in[1..10000] | IsPrime((5^SumOfDivisors(n)) - n)];

a[n_] := Select[Range@ n, PrimeQ[5^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[5^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

a(4) from Michael S. Branicky, Nov 11 2024

A378512 Numbers k such that 6^sigma(k) - k is a prime.

Original entry on oeis.org

1, 7, 13, 77, 395, 2867, 3959, 5023

Offset: 1

J.W.L. (Jan) Eerland, Nov 29 2024

a(9) > 10^5. - Michael S. Branicky, Dec 01 2024

			7 is in the sequence because 6^sigma(7) - 7 = 6^8 - 7 = 1679609 is prime.

Cf. A000043, A000203, A000668, A023194, A023195, A253850, A253851, A368651, A367460, A377927, A377786.

[n: n in[1..10000] | IsPrime((6^SumOfDivisors(n)) - n)];

a[n_] := Select[Range@ n, PrimeQ[6^DivisorSigma[1, #] - #] &]; a[20000]
DeleteCases[ParallelTable[If[PrimeQ[6^DivisorSigma[1,k]-k],k,n],{k,1,10^4}],n]

isok(k) = ispseudoprime(6^sigma(k) - k); \\ Michel Marcus, Dec 09 2024

cp's OEIS Frontend

A187823 Primes of the form (p^x - 1)/(p^y - 1), where p is prime, y > 1, and y is the largest proper divisor of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A248963 Prime powers p^m for which sigma(p^2m) is not prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A331036 Odd values of the sum-of-divisors function sigma (A000203), listed by increasing size and with multiplicity.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A377927 Numbers k such that 4^sigma(k) - k is a prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

Extensions

A243765 Numbers that have all their divisors in A002191 (possible values for sigma(n), A000203).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A292446 Numbers k such that sigma((k + 1) / 2) is a prime q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A377786 Numbers k such that 5^sigma(k) - k is a prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

Extensions

A378512 Numbers k such that 6^sigma(k) - k is a prime.

Views

Author

Keywords

Comments