A038700 -id:A038700 - cp's OEIS frontend

A093870 Triangle read by rows: T(n,k) is the k-th prime = -1 (mod n), 1 <= k <= n.

2, 3, 5, 2, 5, 11, 3, 7, 11, 19, 19, 29, 59, 79, 89, 5, 11, 17, 23, 29, 41, 13, 41, 83, 97, 139, 167, 181, 7, 23, 31, 47, 71, 79, 103, 127, 17, 53, 71, 89, 107, 179, 197, 233, 251, 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 43, 109, 131, 197, 241, 263, 307, 373, 439, 461, 571

Offset: 1

Amarnath Murthy, Apr 20 2004

			2;
3,5;
2,5,11;
3,7,11,19;
...

Cf. A077316, A038700 (first column), A093871 (main diagonal).

row[n_] := Reap[Module[{k, p}, For[k = 0; p = n - 1, k < n, p += n, If[PrimeQ[p], k++; Sow[p]]]]][[2, 1]];
Array[row, 12] // Flatten (* Jean-François Alcover, Jun 08 2020 *)

Edited and extended by Franklin T. Adams-Watters, Aug 29 2006
Definition clarified by Zak Seidov, Apr 16 2013
Duplicate term 139 removed by Jean-François Alcover, Jun 08 2020

A275391 Least k such that n divides sigma(k^k) (k > 0).

Original entry on oeis.org

1, 3, 5, 3, 3, 5, 2, 3, 5, 3, 19, 11, 11, 5, 15, 7, 15, 5, 11, 3, 5, 19, 10, 11, 7, 11, 17, 11, 13, 15, 5, 7, 29, 15, 23, 11, 11, 11, 11, 3, 15, 5, 35, 19, 23, 21, 22, 15, 13, 7, 15, 11, 23, 17, 19, 11, 11, 13, 28, 15, 11, 5, 5, 15, 15, 29, 21, 15, 65, 23, 34, 11, 4, 11, 29, 11, 39, 11, 23, 7, 17

Offset: 1

Altug Alkan, Aug 07 2016

nonn

From Robert Israel, Aug 09 2016: (Start)
a(n) <= A038700(n) if n >= 4, since sigma(k^k) == 0 (mod n) if k is an odd prime == -1 (mod n).
If n is prime and n-2 is squarefree, then a(n) <= n-2 since sigma((n-2)^(n-2)) == 0 (mod n).
Conjecture: a(n) <= n-2 for all n > 15, but a(n) = n-2 for infinitely many n. (End)

			a(11) = 19 because sigma(19^19) is divisible by 11.

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

Cf. A000203, A038700, A062727, A275800.

N:= 200: # to get a(1)..a(N)
S:= {$1..N}:
for k from 1 while S <> {} do
  v:= numtheory:-sigma(k^k);
  F:= select(t -> v mod t = 0, S);
  for n in F do A[n]:= k od:
  S:= S minus F;
od:
seq(A[n],n=1..N); # Robert Israel, Aug 09 2016

a(n) = {my(k=1); while(sigma(k^k) % n != 0, k++); k; }

A086508 If n is even, a(n) = smallest prime == -1 (mod n), If n is odd, a(n) = smallest prime == 1 (mod n).

Original entry on oeis.org

2, 3, 7, 3, 11, 5, 29, 7, 19, 19, 23, 11, 53, 13, 31, 31, 103, 17, 191, 19, 43, 43, 47, 23, 101, 103, 109, 83, 59, 29, 311, 31, 67, 67, 71, 71, 149, 37, 79, 79, 83, 41, 173, 43, 181, 137, 283, 47, 197, 149, 103, 103, 107, 53, 331, 167, 229, 173, 709, 59, 367, 61, 127

Offset: 1

Amarnath Murthy, Jul 29 2003

nonn

Cf. A034694, A038700, A086507.

Join[{2},With[{prs=Prime[Range[500]]},Flatten[Table[If[OddQ[n],Select[ prs,Mod[#,n] == 1&,1],Select[prs,Mod[#,n]==n-1&,1]],{n,70}]]]] (* Harvey P. Dale, Nov 16 2013 *)

More terms from David Wasserman, Mar 09 2005

A225192 Number of primes p such that p is -1 mod n where p < n-th prime.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 3, 0, 0, 1, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 3, 0, 1, 1, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 1, 0, 1, 1, 3, 1, 3, 1, 2, 0, 2, 1, 4, 0, 1, 2

Offset: 1

Irina Gerasimova, May 01 2013

nonn

Primes p(n) such that a(n) = a(n + 1): 2, 5, 17, 19, 23, 73, 97, 103, 173, 193, 233, 239, 263, 293, 347, 349, 353, 373, 449, 467,...
Primes p(n) such that p is not -1 mod n and mod n+1 for all prime p < p(n+1): 2, 97, 829, 1597, 2251,...
Smallest k such that a(k) = n:, 1, 3, 6, 24, 84, 90,...
Numbers n such that a(n) is equal to number of primes p such that n is -1 mod p where p < n-th prime: 1, 2, 3, 4, 7, 8, 10, 14, 15, 20, 22, 28, 31, 32, 34, 40, 44, 45, 46, 50, 52, 55, 57, 63, 65, 70, 72, 87,...

			Prime 11 == - 1 (mod 12), prime 23 == -1 (mod 12) and 11, 23 < prime(12) = 37, so a(12) = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A038700, A219109.

Table[s = Prime[Range[n - 1]]; Length[Select[s, Mod[#, n] == n - 1 &]], {n, 93}] (* T. D. Noe, May 13 2013 *)

a(n)=my(s); forstep(p=n-1,prime(n)-1,n,s+=isprime(p)); s \\ Charles R Greathouse IV, Mar 18 2014

A282049 Least prime p such that n divides sigma(p*n).

Original entry on oeis.org

2, 3, 2, 3, 19, 5, 13, 7, 17, 19, 43, 2, 103, 2, 19, 31, 67, 5, 37, 19, 41, 43, 137, 5, 149, 103, 53, 3, 173, 19, 61, 31, 43, 67, 139, 71, 73, 37, 233, 3, 163, 13, 257, 43, 29, 137, 281, 11, 97, 149, 67, 103, 211, 17, 109, 13, 113, 173, 353, 2, 487, 61, 251, 127, 389, 43, 401, 67

Offset: 1

Altug Alkan, Feb 05 2017

nonn

a(n) <= A038700(n). Numbers n such that a(n) is not equal to A038700(A017666(n)) are 6, 14, 24, 28, 60, 78, 84, 90, 120, 234, 248, 270, 336, 496, 532, 546, ...

			a(5) = 19 because 5 divides sigma(19*5) = 120 and 19 is the least prime number with this property.

Altug Alkan, Table of n, a(n) for n = 1..30000

Cf. A000203, A227470

a(n) = {my(p=2); while (sigma(p*n) % n != 0, p = nextprime(p+1)); p; } \\ David A. Corneth, Feb 05 2017

A225318 Numbers n such that either prime(n-1) == -1 (mod n) or prime(n+1) == -1 (mod n) but not both.

Original entry on oeis.org

2, 4, 7, 8, 14, 16, 26, 27, 32, 33, 35, 76, 78, 169, 170, 172, 175, 177, 183, 184, 185, 434, 446, 1054, 1056, 2638, 2702, 6468, 15930, 40069, 40070, 40080, 40112, 40115, 40157, 251721, 251758, 251767, 251770, 251788, 637286, 4124464, 4124704

Offset: 1

Irina Gerasimova, May 05 2013

nonn

			2nd prime is 3 and 2 is a member because 1st prime, 2, is congruent to 0 mod 2 and 3rd prime, 5, is congruent to -1 mod 2;
6th prime is 11 and 6 is not a member because 5th prime, 11, is congruent to -1 mod 6 and 7th prime, 17, is congruent to -1 mod 6;
7th prime is 17 and 7 is a member because 6th prime, 13, is congruent to -1 mod 7 and 8th prime, 19, is congruent to 1 mod 6;
14th prime is 43 and 14 is a member because 13th prime, 41, is congruent to -1 mod 14 and 15th prime, 47, is congruent to 5 mod 14.

Cf. A038700, A045924, A078931, A219109.

for n from 2 to 100000 do
    if modp(ithprime(n-1),n) = modp(-1,n) then
        pn := true ;
    else
        pn := false ;
    end if;
    if modp(ithprime(n+1),n) = modp(-1,n) then
        pm := true ;
    else
        pm := false ;
    end if;
    if pn <> pm then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 09 2013

is(n)=my(p=prime(n-1),q=nextprime(nextprime(p+1)+1),v=[p+1,q+1]%n); !vecmin(v) && vecmax(v) \\ Charles R Greathouse IV, Mar 18 2014

Corrected by R. J. Mathar, May 09 2013

a(36)-a(43) from Alois P. Heinz, May 18 2013

cp's OEIS Frontend

A093870 Triangle read by rows: T(n,k) is the k-th prime = -1 (mod n), 1 <= k <= n.

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A275391 Least k such that n divides sigma(k^k) (k > 0).

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A086508 If n is even, a(n) = smallest prime == -1 (mod n), If n is odd, a(n) = smallest prime == 1 (mod n).

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A225192 Number of primes p such that p is -1 mod n where p < n-th prime.

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Mathematica

PARI

A282049 Least prime p such that n divides sigma(p*n).

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PARI

A225318 Numbers n such that either prime(n-1) == -1 (mod n) or prime(n+1) == -1 (mod n) but not both.

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