A141887 -id:A141887 - cp's OEIS frontend

A244765 Prime numbers ending in the prime number 19.

19, 419, 619, 719, 919, 1019, 1319, 1619, 2719, 2819, 3019, 3119, 3319, 3719, 3919, 4019, 4219, 4519, 4919, 5119, 5419, 5519, 6619, 6719, 7019, 7219, 7919, 8219, 8419, 8719, 8819, 9319, 9419, 9619, 9719, 11119, 11519, 11719, 12119, 12619, 12919, 13219

Offset: 1

Vincenzo Librandi, Jul 06 2014

Also primes of the form 100*n+19. Subsequence of A141887, A141942.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A141887, A141942.

Cf. similar sequences listed in A244763.

[n: n in PrimesUpTo(16000) | n mod 100 eq 19];

Select[Prime[Range[5, 6000]], Take[IntegerDigits[#], -2]=={1, 9} &]
Select[Prime[Range[1600]],Mod[#,100]==19&] (* Harvey P. Dale, Jul 29 2018 *)

select(x->(x % 100)==19, primes(2000)) \\ Michel Marcus, Jul 06 2014

A244772 Prime numbers ending in the prime number 59.

Original entry on oeis.org

59, 359, 659, 859, 1259, 1459, 1559, 1759, 2459, 2659, 3259, 3359, 3559, 3659, 4159, 4259, 4759, 5059, 5659, 6359, 6659, 6959, 7159, 7459, 7559, 7759, 8059, 9059, 9859, 10159, 10259, 10459, 10559, 10859, 11059, 11159, 11959, 12659, 12959, 13159, 13259

Offset: 1

Vincenzo Librandi, Jul 07 2014

Also primes of the form 100*n+59. Subsequence of A141887, A141934.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. similar sequences listed in A244763.

Cf. A141887, A141934.

[n: n in PrimesUpTo(14000) | n mod 100 eq 59];

Select[Prime[Range[5, 6000]], Take[IntegerDigits[#], -2]=={5, 9} &]

select(x->(x % 100)==59, primes(2000)) \\ Michel Marcus, Jul 07 2014

A244775 Prime numbers ending in the prime number 79.

Original entry on oeis.org

79, 179, 379, 479, 1279, 1579, 1879, 1979, 2179, 2579, 2879, 3079, 3779, 4079, 4679, 5179, 5279, 5479, 5779, 5879, 6079, 6379, 6679, 6779, 7079, 7879, 8179, 8779, 9479, 9679, 10079, 10979, 11279, 11579, 11779, 12379, 12479, 12979, 13679, 13879

Offset: 1

Vincenzo Librandi, Jul 07 2014

Also primes of the form 100*n+79. Subsequence of A141887, A141930.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. similar sequences listed in A244763.

Cf. A141887, A141930.

[n: n in PrimesUpTo(14000) | n mod 100 eq 79];

Select[Prime[Range[5, 6000]], Take[IntegerDigits[#], -2]=={7, 9} &]
Select[Prime[Range[2000]],Mod[#,100]==79&] (* Harvey P. Dale, Nov 29 2017 *)

select(x->(x % 100)==79, primes(2000)) \\ Michel Marcus, Jul 07 2014

A216257 a(n) = 840n^2 - 23100n + 86861.

Original entry on oeis.org

86861, 64601, 44021, 25121, 7901, -7639, -21499, -33679, -44179, -52999, -60139, -65599, -69379, -71479, -71899, -70639, -67699, -63079, -56779, -48799, -39139, -27799, -14779, -79, 16301, 34361, 54101, 75521, 98621, 123401, 149861, 178001, 207821, 239321, 272501

Offset: 0

Arkadiusz Wesolowski, Mar 15 2013

|a(n)| are distinct primes for 0 <= n <= 32.
The values of this polynomial are never divisible by a prime less than 79.
All terms are congruent to 1 (mod 20).

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A141881, A141887, A215145.

[ 840*n^2-23100*n+86861 : n in [0..34]];

Maple
```
seq(840*n^2-23100*n+86861, n=0..34);
```

Table[840*n^2 - 23100*n + 86861, {n, 0, 34}]

for(n=0, 34, print1(840*n^2-23100*n+86861, ", "))

G.f.: (86861 - 195982*x + 110801*x^2)/(1-x)^3.
From Elmo R. Oliveira, Feb 10 2025: (Start)
E.g.f.: exp(x)*(86861 - 22260*x + 840*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A306431 Least number x > 1 such that n*x divides 1 + Sum_{k=1..x-1} k^(x-1).

Original entry on oeis.org

2, 3, 13, 7, 19, 31, 41, 31, 13, 19, 43, 31, 23, 83, 139, 31, 61, 67, 113, 79, 251, 43, 19, 31, 199, 23, 13, 167, 53, 139, 83, 127, 157, 67, 293, 431, 443, 151, 103, 79, 61, 251, 113, 47, 337, 19, 179, 31, 41, 199, 67, 23, 19, 499, 181, 367, 607, 139, 257, 359

Offset: 1

Paolo P. Lava, Apr 05 2019

If n = 1, all the solutions of x | 1 + Sum_{k=1..x-1} k^(x-1) should be prime numbers, according to Giuga's conjecture.
If n*x | 1 + Sum_{k=1..x-1} k^(x-1), then certainly x does, so Giuga's conjecture would say x must be prime. Similarly if x^n divides it, so does x, so again Giuga would say x is prime. - Robert Israel, Apr 26 2019
E.g., the first solution for x^2 | 1 + Sum_{k=1..x-1} k^(x-1) is x = 1277, that is prime.

			a(4) = 7 because (1 + 1^6 + 2^6 + 3^6 + 4^6 + 5^6 + 6^6) / (4*7) = 67172 / 28 = 2399 and it is the least prime to have this property.

Robert Israel, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Giuga's Conjecture

Cf. A191677. All the solutions for n = m: A000040 (m=1), A002145 (m=2), A007522 (m=4), A127576 (m=8), A141887 (m=10), A127578 (m=16), A142198 (m=20), A127579 (m=32), A095995 (m=50).

P:=proc(j) local k,n; for n from 2 to 10^6 do
if frac((add(k^(n-1),k=1..n-1)+1)/(j*n))=0
then RETURN(n); break; fi; od; end: seq(P(i),i=1..60);

a[n_] := For[x = 2, True, x++, If[Divisible[1+Sum[k^(x-1), {k, x-1}], n x], Return[x]]];
Array[a, 60] (* Jean-François Alcover, Oct 16 2020 *)

a(n) = my(x=2); while (((1 + sum(k=1, x-1, k^(x-1))) % (n*x)), x++); x; \\ Michel Marcus, Apr 27 2019

Least solution of n*x | 1 + Sum_{k=1..x-1} k^(x-1), for n = 1, 2, 3, ...

cp's OEIS Frontend

A244765 Prime numbers ending in the prime number 19.

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A244772 Prime numbers ending in the prime number 59.

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A244775 Prime numbers ending in the prime number 79.

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A216257 a(n) = 840*n^2 - 23100*n + 86861.

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A306431 Least number x > 1 such that n*x divides 1 + Sum_{k=1..x-1} k^(x-1).

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Maple

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Formula

A216257 a(n) = 840n^2 - 23100n + 86861.