A053183 -id:A053183 - cp's OEIS frontend

A193574 Smallest divisor of sigma(n) that does not divide n.

3, 2, 7, 2, 4, 2, 3, 13, 3, 2, 7, 2, 3, 2, 31, 2, 13, 2, 3, 2, 3, 2, 5, 31, 3, 2, 8, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 31, 3, 3, 2, 7, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 127, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3

Offset: 2

Franklin T. Adams-Watters, Aug 27 2011

nonn

a(n) = 2 iff n is an odd number that is not a perfect square.
From Hartmut F. W. Hoft, May 05 2017: (Start)
(1)  Every a(n) > n is a prime: Because of the minimality of a(n), a(n) = u*v with gcd(u,v)=1 leads to the contradiction (u*v)|n. Similarly, a(n)=p^k with p prime an k>1 leads to the contradiction (p^k-1)/(p-1) | n.
(2)  n=p^(2*k), k>=1 and 2*k+1 prime, when a(n) = sigma(n) for n>2: Because n having two distinct prime factors implies sigma(n) composite, and if n is an odd power of a prime then 2|sigma(n). Finally, if 2*k+1=u*v  with  u,v > 1 then sigma(p^(u-1)) divides sigma(p^(2*k)), but not p^(2k), for any prime p, contradicting minimality of a(n). For example, no number sigma(p^8) for any prime p is in the sequence.
(3)  The converse of (2) is false since, e.g. sigma(7^2) = 3*19 so that a(7^2) = 3, and sigma(2^10) = 23*89 so that a(2^10) = 23.
(4)  Conjecture: a(n) > n implies a(n) = sigma(n); tested through n = 20000000.
(5)  Subsequences are: A053183 (sigma(p^2) is prime for prime p), A190527 (sigma(p^4) is prime for prime p), A194257 (sigma(p^6) is prime for prime p), A286301 (sigma(p^10) is prime for prime p)
(6)  Subsequences are: A000668 (primes of form 2^p-1), A076481 (primes of form (3^p-1)/2), A086122 (primes of form (5^p-1)/4), A102170 (primes of form (7^p-1)/6), all when p is prime.
(End)
Up to n = 10^6, there are 89 distinct elements. For those n, a(n) is prime. If it's not, it's a power of 2, a power of 3 or a perfect square <= 121. - David A. Corneth, May 10 2017

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A000203, A007978, A027750, A135718.

Subsequences: A000668, A053183, A076481, A086122, A102170, A190527, A194257, A286301.

import Data.List ((\\))
a193574 n = head [d | d <- [1..sigma] \\ nDivisors, mod sigma d == 0]
   where nDivisors = a027750_row n
         sigma = sum nDivisors
-- Reinhard Zumkeller, May 20 2015, Aug 28 2011

a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]
Map[a193574, Range[2, 80]] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)

a(n)=local(ds);ds=divisors(sigma(n));for(k=2,#ds,if(n%ds[k],return(ds[k])))

A136242 Numbers k among A008864 such that k^2 - k + 1 is prime.

Original entry on oeis.org

3, 4, 6, 18, 42, 60, 72, 90, 102, 132, 168, 174, 294, 384, 678, 702, 744, 762, 774, 828, 840, 858, 912, 1092, 1098, 1164, 1182, 1194, 1218, 1374, 1428, 1488, 1560, 1584, 1710, 1812, 1848, 1932, 1974, 2130, 2274, 2310, 2340, 2412, 2664, 2730, 2790, 2958

Offset: 1

Lekraj Beedassy, Dec 24 2007

See A053183 for the primes associated with a(n).

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

Cf. A008864, A053182, A053183, A136243.

Select[Prime[Range[500]] + 1, PrimeQ[#^2 - # + 1] &] (* Amiram Eldar, Apr 19 2024 *)

lista(pmax) = forprime(p=1, pmax, if(isprime(p^2+p+1), print1(p+1, ", "))); \\ Amiram Eldar, Apr 19 2024

a(n) = A053182(n) + 1.

A342691 Primes of the form (p^k)^2 + p^k + 1 with prime p and positive integer k.

Original entry on oeis.org

7, 13, 31, 73, 307, 757, 1723, 3541, 5113, 8011, 10303, 17293, 28057, 30103, 86143, 147073, 262657, 459007, 492103, 552793, 579883, 598303, 684757, 704761, 735307, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307, 1772893, 1886503, 2037757, 2212657

Offset: 1

Martin Becker, May 18 2021

nonn

Also, primes of the form (p^3^m)^2 + p^3^m + 1 with prime p and nonnegative integer m, since k must be a power of 3, from the theory of cyclotomic polynomials.

			31 = (5^1)^2 + 5^1 + 1 is in the sequence as 31 is prime and 5 is prime and 1 is a positive integer.
73 = (2^3)^2 + 2^3 + 1 is in the sequence as it is prime and 2 is prime and 3 is a positive integer.

Martin Becker, Table of n, a(n) for n = 1..20000

Contains A053183 and A063784.
Intersection of A335865 and A000040 minus {3}.
Cf. A342690, A344448.

Select[Table[q^2 + q + 1, {q, Select[Range[1500], PrimePowerQ[#] &]}], PrimeQ] (* Amiram Eldar, Aug 16 2024 *)

for(q=2,2048,if(isprimepower(q),m=q^2+q+1;if(isprime(m),print1(m, ", "))))

A243471 Primes p such that p^6 - p^5 + 1 is prime.

Original entry on oeis.org

3, 31, 73, 181, 367, 373, 523, 631, 733, 1021, 1039, 1171, 1489, 1723, 1777, 2203, 2557, 2683, 3121, 3187, 3319, 4441, 4591, 4621, 4801, 4957, 5113, 5167, 5323, 5431, 5659, 5839, 5851, 5857, 6883, 7057, 7129, 7297, 7309, 7477, 7993, 8017, 8209, 8221, 8689, 8821

Offset: 1

K. D. Bajpai, Jun 05 2014

nonn

			31 appears in the sequence because it is prime and 31^6 - 31^5  + 1 = 858874531 is also prime.
73 appears in the sequence because it is prime and 73^6 - 73^5  + 1 = 149261154697 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5457

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

A243471 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5+1; if isprime (b) then RETURN (a); fi; end: seq(A243471 (), n=1..2000);

c=0; Do[k=Prime[n]; If[PrimeQ[k^6-k^5+1], c++; Print[c," ",k]], {n,1,200000}];

A243472 Primes p such that p^6 - p^5 - 1 is prime.

Original entry on oeis.org

2, 31, 101, 151, 181, 199, 229, 277, 307, 317, 379, 439, 479, 491, 647, 691, 797, 911, 997, 1039, 1051, 1181, 1291, 1367, 1381, 1471, 1511, 1549, 1657, 1709, 1847, 1867, 1987, 2081, 2099, 2111, 2207, 2467, 2621, 2707, 3041, 3221, 3259, 3541, 3571, 3581, 3769

Offset: 1

K. D. Bajpai, Jun 05 2014

nonn

			31 appears in the sequence because it is prime and 31^6 - 31^5 - 1 = 858874529 is also prime.
101 appears in the sequence because it is prime and 101^6 - 101^5  - 1 = 1051010050099 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..4699

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

A243472 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5-1; if isprime (b) then RETURN (a); fi; end: seq(A243472 (), n=1..2000);

c = 0;  Do[k=Prime[n]; If[PrimeQ[k^6-k^5-1], c++; Print[c," ",k]], {n,1,200000}];
Select[Prime[Range[600]],PrimeQ[#^6-#^5-1]&] (* Harvey P. Dale, Jan 21 2015 *)

s=[]; forprime(p=2, 4000, if(isprime(p^6-p^5-1), s=concat(s, p))); s \\ Colin Barker, Jun 06 2014

A238400 Primes in A238399.

Original entry on oeis.org

2, 3, 7, 1237, 66067, 525593, 974167, 1412473, 2675759, 4471237, 5264333, 8107961, 8308271, 12615151, 20145407, 34926433, 43167569, 94772749, 104612297, 115103327, 144450221, 153124973, 165108557, 196634723, 211696049, 213507241, 255963131, 263979101

Offset: 1

Torlach Rush, Feb 26 2014

nonn

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A238399, A053182, A053183.

Select[(PrimePi[#^2 + #] - PrimePi[#]) & /@ Select[Prime@Range[3000], PrimeQ[#^2 + # + 1] &], PrimeQ] (* Giovanni Resta, Feb 27 2014 *)

Corrected by Torlach Rush, Feb 26 2014

a(16)-a(28) from Giovanni Resta, Feb 27 2014

A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

Original entry on oeis.org

18, 50, 578, 1458, 3362, 4802, 6962, 10082, 15842, 20402, 31250, 34322, 55778, 57122, 59858, 167042, 171698, 293378, 559682, 916658, 982802, 1062882, 1104098, 1158242, 1195058, 1367858, 1407842, 1414562, 1468898, 1659842, 2380562, 2406818, 2705138, 2789522

Offset: 1

Michel Lagneau, May 30 2016

nonn

a(n) is of the form 2q^2 where q is an odd numbers for which sigma(q^2) is prime (A193070).
The corresponding primes p are 13, 31, 307, 1093, 1723, 2801, 3541, 5113, 8011, 10303, 19531, 17293, 28057, 30941, 30103, 88741, 86143, 147073, 292561, 459007, 492103, 797161, 552793, 579883, 598303, 684757, 704761, 732541, 735307, 830833, 1191373, 1204507, ...
We observe an interesting property: each prime p is element of A053183 (primes of the form m^2 + m + 1 when m is prime) or element of A247837 (primes of the form sigma(2m-1) for a number m) or element of both A053183 and A247837.
Examples:
The numbers 13, 31, 307, 1723, 3541, 5113,... are in A053183;
The numbers 13, 31, 307, 1093, 1723, 2801, 3541,...are in A247837;
The numbers 13, 31, 307, 1723, 3541,... are in A053183 and A247837.

			18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18}. The sum of the odd divisors is 1 + 3 + 9 = 13 and the sum of the even divisors is 2 + 6 + 18 = 26 = 2*13.

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

Cf. A002384, A053183, A193070, A247837, A278911.

with(numtheory):
for n from 2 by 2  to 500000 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     ii:=0:
        if isprime(s1) and s0=2*s1
        then
        printf(`%d, `, n):
         else fi:
     od:

Select[Range[2, 3000000, 2], And[PrimeQ[Total@ Select[#, EvenQ]/2], PrimeQ@ Total@ Select[#, OddQ]] &@ Divisors@ # &] (* Michael De Vlieger, May 30 2016 *)
sodpQ[n_]:=Module[{d=Divisors[n],s},s=Total[Select[d,OddQ]];PrimeQ[ s] && Total[ Select[d,EvenQ]]==2s]; Select[Range[2,279*10^4,2],sodpQ] (* Harvey P. Dale, Dec 01 2020 *)
2 * Select[Range[1, 1200, 2]^2, PrimeQ@DivisorSigma[1, #] &] (* Amiram Eldar, Jul 19 2022 *)

is(n)=my(t); n%4==2 && issquare(n/2,&t) && isprime(n/2+t+1) \\ Charles R Greathouse IV, Jun 08 2016

a(n) >> n^2. - Charles R Greathouse IV, Jun 08 2016

a(n) = 2 * A278911(n) = 2 * A193070(n)^2. - Amiram Eldar, Jul 19 2022

A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 29, 31, 41, 43, 109, 131, 157, 181, 271, 307, 379, 811, 929, 991, 1721, 1723, 2161, 2861, 3539, 3541, 3659, 4421, 4423, 4969, 5113, 6163, 6971, 8009, 8011, 9311, 10099, 10301, 10303, 10711, 16001, 17029, 17291, 17293, 19181, 19183, 22051, 22349, 22651

Offset: 1

Ralf Steiner, Aug 11 2017

nonn

This sequence contains prime chains and prime trees using an appropriate mapping form p^2 +- p +- 1 in each step, such as the chain: 3 -> 5 -> 19 -> 379 -> 143263 -> 20524143907 and the tree: 41 -> {1721, 1723}.

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

Cf. A000040.

Union of A053183, A053185, A074268, A091568.

{p^2+(-1)^k*p+(-1)^s:p in PrimesUpTo(150), s,k in [1..2]|IsPrime(p^2+(-1)^k*p+(-1)^s)}; //  Marius A. Burtea, Nov 28 2019

select(isprime, [3,seq(op([p^2-p-1,p^2-p+1,p^2+p-1,p^2+p+1]),p=select(isprime,[seq(i,i=3..1000,2)]))]); # Robert Israel, Nov 27 2019

Select[Union[Flatten[{(#^2 + # + 1 ), (#^2 + # - 1 ), (#^2 - # + 1 ), (#^2 - # - 1 )}] &[Prime[Range[100]]]], (PrimeQ[#]) &]

cp's OEIS Frontend

A193574 Smallest divisor of sigma(n) that does not divide n.

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A136242 Numbers k among A008864 such that k^2 - k + 1 is prime.

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A342691 Primes of the form (p^k)^2 + p^k + 1 with prime p and positive integer k.

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A243471 Primes p such that p^6 - p^5 + 1 is prime.

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A243472 Primes p such that p^6 - p^5 - 1 is prime.

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A238400 Primes in A238399.

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A273459 Even numbers such that the sum of the odd divisors is a prime p and the sum of the even divisors is 2p.

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A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

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