A065512 -id:A065512 - cp's OEIS frontend

A088580 a(n) = 1 + sigma(n).

2, 4, 5, 8, 7, 13, 9, 16, 14, 19, 13, 29, 15, 25, 25, 32, 19, 40, 21, 43, 33, 37, 25, 61, 32, 43, 41, 57, 31, 73, 33, 64, 49, 55, 49, 92, 39, 61, 57, 91, 43, 97, 45, 85, 79, 73, 49, 125, 58, 94, 73, 99, 55, 121, 73, 121, 81, 91, 61, 169, 63, 97, 105, 128, 85, 145, 69, 127, 97

Offset: 1

James East, Nov 20 2003

Number of reflection subgroups of the (dihedral) Coxeter group of type I_2(n).

			a(2)=4. If W=<s, t|s^2=t^2=1, st=ts> then the reflection subgroups are {1}, <s>, <t>, <s, t>.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A000203 (sum of divisors of n).

Cf. A065512 (indices of primes in this sequence), A258430 (corresponding primes).

a088580 = (+ 1) . a000203  -- Reinhard Zumkeller, Dec 20 2014

[1+SumOfDivisors(n): n in [1..100]]; // Vincenzo Librandi, May 30 2015

map(1+numtheory:-sigma, [$1..1000]); # Robert Israel, May 29 2015

Table[1 + DivisorSigma[1, n], {n, 100}] (* Robert Price, May 29 2015 *)

a(n) = 1 + A000203(n).

G.f.: x/(1 - x) + Sum_{k>=1} x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Mar 17 2017

A039698 Numbers k such that phi(k) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 46, 47, 48, 49, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 76, 77, 79, 82, 83, 86, 88, 89, 91, 93, 94, 95, 97, 98, 99, 100, 101, 103

Offset: 1

Olivier Gérard

Positive integers k for which values of A039649(k) are primes. - Vladimir Shevelev, May 10 2008
For every prime p, the numbers p and 2p are terms of this sequence. - Vladimir Shevelev, May 10 2008
Union of A000040 and A066071. - Ray Chandler, May 26 2008

			phi(10)+1 = 4+1 = 5, a prime number, so 10 is a term.

Antti Karttunen, Table of n, a(n) for n = 1..26197 (first 1000 terms from Vincenzo Librandi)

Cf. A000010, A000040, A006093, A039649, A066071, A007614.
Cf. A039689 (complement), A296079 (characteristic function).
Cf. also A065512, A078892, A263028, A248792.

[n: n in [1..200] | IsPrime(EulerPhi(n)+1)]; // Vincenzo Librandi, Aug 13 2013

Select[Range[300], PrimeQ[EulerPhi[#] + 1]&] (* Vincenzo Librandi, Aug 13 2013 *)

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

A248792 Numbers n such that sigma(n) - 1 is a prime p.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 26, 29, 30, 31, 33, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 63, 65, 67, 71, 73, 74, 76, 78, 79, 83, 84, 85, 86, 88, 89, 90, 92, 93, 96, 97, 101, 103, 105, 107, 109

Offset: 1

Jaroslav Krizek, Nov 01 2014

Union of primes (A000040) and terms of A066073 (composites).
Numbers n such that A039653(n) is prime.
Corresponding values of primes p are in A248793.

			6 is in sequence because sigma(6) - 1 = 12 - 1 = 11 (prime).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A000203 (sum of divisors), A000040 (primes).
Cf. A039653 (sigma(n)-1), A066073 (subsequence of composites), A248793.
Cf. A065512 (numbers n such that sigma(n) + 1 is prime).

[n: n in[1..1000] | IsPrime(SumOfDivisors(n) - 1)];

with(numtheory): A248792:=n->`if`(isprime(sigma(n)-1), n, NULL): seq(A248792(n), n=1..200); # Wesley Ivan Hurt, Jul 09 2015

Select[Range[110], PrimeQ[DivisorSigma[1, #] - 1] &] (* Vincenzo Librandi, Nov 02 2014 *)

for(n=1,10^3,if(isprime(sigma(n)-1),print1(n,", "))) \\ Derek Orr, Nov 01 2014

A258430 Primes in A088580.

Original entry on oeis.org

2, 5, 7, 13, 19, 13, 29, 19, 43, 37, 61, 43, 41, 31, 73, 61, 43, 97, 79, 73, 73, 73, 61, 97, 127, 97, 73, 97, 127, 109, 181, 113, 157, 103, 211, 193, 163, 109, 281, 241, 211, 181, 157, 313, 337, 241, 271, 139, 337, 193, 181, 223, 229, 151, 373, 193, 241, 379

Offset: 1

Robert Price, May 29 2015

nonn

These primes are neither sorted nor uniqued. They are listed in the order found in A088580.

Robert Price, Table of n, a(n) for n = 1..2967
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)

Cf. A088580, A065512.

[a: n in [1..300] | IsPrime(a) where a is 1 + SumOfDivisors(n)]; // Vincenzo Librandi, May 30 2015

with(numtheory): A258430:=n->`if`(isprime(1+sigma(n)), 1+sigma(n), NULL): seq(A258430(n), n=1..300); # Wesley Ivan Hurt, Jul 09 2015

Select[Table[1 + DivisorSigma[1, n], {n, 10000}], PrimeQ]

select(x->isprime(x), vector(200, n, 1+sigma(n))) \\ Michel Marcus, Jun 04 2015

a(n) = A088580(A065512(n)). - Michel Marcus, Jun 04 2015

A296212 a(n) = 1 if sigma(n) + 1 is prime, 0 otherwise.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

Characteristic function of A065512, numbers n such that sigma(n) + 1 is prime.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for characteristic functions

Cf. A000203, A010051, A065512, A088580, A296092.

Cf. also A296077, A296079, A296211.

(define (A296212 n) (A010051 (+ 1 (A000203 n))))

a(n) = A010051(1+A000203(n)) = A010051(A088580(n)).

A259973 Numbers n such that sigma(n) + product of divisors of n is prime.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 23, 27, 29, 32, 41, 50, 53, 57, 83, 85, 89, 111, 113, 128, 131, 161, 173, 179, 191, 215, 233, 237, 239, 245, 251, 265, 275, 281, 293, 319, 355, 359, 365, 391, 413, 419, 431, 437, 443, 453, 481, 485, 491, 493, 505, 509, 511, 535, 589, 593, 603

Offset: 1

K. D. Bajpai, Jul 15 2015

nonn

If p is prime, then (sigma(p) + product of divisors of p) = 2*p+1. So the subsequence of primes gives the Sophie Germain primes: A005384. - Michel Marcus, Jul 16 2015

			a(5) = 8; divisors(8) = {1,2,4,8}; sum = 1+2+4+8 = 15; product = 1*2*4*8 = 64; 15 + 64 = 79, which is prime.
a(8) = 27; divisors(27) = {1,3,9,27}; sum = 1+3+9+27 = 40; product = 1*3*9*27 = 729; 40+729 = 769, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000203, A007955, A005384, A065512, A118369.

[n: n in[1..1000] | IsPrime(&*Divisors(n) + SumOfDivisors(n))];

Select[Range[2000], PrimeQ[DivisorSigma[1, #] + Times@@Divisors[#]] &]

for(n=1, 1000, d=divisors(n); k=sigma(n) + prod(i=1,#d,d[i]); if(isprime(k),print1(n,", ")));

A260108 Primes of the form sigma(k) + product of divisors of k.

Original entry on oeis.org

2, 5, 7, 11, 79, 23, 47, 769, 59, 32831, 83, 125093, 107, 3329, 167, 7333, 179, 12473, 227, 268435711, 263, 26113, 347, 359, 383, 46489, 467, 56489, 479, 14706467, 503, 70549, 20797247, 563, 587, 102121, 126457, 719, 133669, 153313, 171049, 839, 863, 191449, 887

Offset: 1

K. D. Bajpai, Jul 16 2015

nonn

Alternatively: Primes arising in A259973.

			a(5) = 79; divisors(8) = {1,2,4,8}; sum = 1+2+4+8 = 15; product = 1*2*4*8 = 64; 15 + 64 = 79 which is prime.
a(8) = 769; divisors(27) = {1,3,9,27}; sum = 1+3+9+27 = 40; product = 1*3*9*27 = 729; 40+729 = 769 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000203, A007955, A065512, A118369, A259973.

[k: n in[1..1000] | IsPrime(k) where k is (&*Divisors(n) + SumOfDivisors(n))];

with(numtheory): A260108:= n-> (sigma(n) + convert( divisors(n), `*`)): select(isprime, [seq((A260108 (n), n=1..800))]);

Select[Table[DivisorSigma[1, n] + Times @@ Divisors[n], {n, 1, 1000}], PrimeQ]

for(n=1, 1000, d=divisors(n); k=sigma(n) + prod(i=1, #d, d[i]); if( isprime(k) , print1(k, ", ")));

A007955(n)=if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2))
list(lim)=v=List([2]); forprime(p=2,(lim-1)\2, if(isprime(2*p+1), listput(v,2*p+1))); forprime(p=2,sqrtnint(lim\1,3), my(t=p^3+p^2+p+1); if(t>lim,break); if(isprime(t), listput(v,t))); forcomposite(n=4,sqrtint(lim\1), my(t=A007955(n)+sigma(n)); if(t<=lim && isprime(t), listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jul 17 2015

cp's OEIS Frontend

A088580 a(n) = 1 + sigma(n).

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A039698 Numbers k such that phi(k) + 1 is prime.

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A248792 Numbers n such that sigma(n) - 1 is a prime p.

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A258430 Primes in A088580.

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A296212 a(n) = 1 if sigma(n) + 1 is prime, 0 otherwise.

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A259973 Numbers n such that sigma(n) + product of divisors of n is prime.

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Magma

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A260108 Primes of the form sigma(k) + product of divisors of k.

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