A049203 -id:A049203 - cp's OEIS frontend

A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

1, 2, 1, 3, 9, 2, 8, 10, 14, 22, 3, 9, 15, 19, 23, 29, 41, 39, 63, 69, 2, 6, 16, 16, 24, 42, 48, 52, 54, 52, 74, 84, 88, 102, 114, 122, 134, 152, 156, 166, 168, 1, 7, 13, 19, 23, 31, 71, 71, 73, 73, 65, 77, 91, 79, 91, 109, 115, 125, 137, 149, 155, 185, 197, 203, 197, 235

Offset: 1

Labos Elemer, Oct 08 2002

			a(4) = 3 since prime(prime(4)) (mod prime(4)) = prime(7) (mod 7) = 17 (mod 7) = 3. - _Michael De Vlieger_, Mar 25 2017

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076241, A076242, A076243.

a:= n-> (p-> irem(ithprime(p), p))(ithprime(n)):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 09 2015

Table[Mod @@ Map[Nest[Prime, n, #] &, {2, 1}], {n, 65}] (* Michael De Vlieger, Mar 25 2017 *)

a(n) = prime(prime(n)) % prime(n); \\ Michel Marcus, Mar 25 2017

a(n) = prime^2(n) mod prime(n) = A006450(n) mod A000040(n).

A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 4, 3, 8, 9, 5, 7, 10, 5, 7, 3, 2, 11, 17, 1, 20, 21, 11, 19, 12, 17, 14, 17, 18, 19, 18, 23, 28, 27, 11, 19, 15, 7, 2, 21, 40, 25, 31, 1, 19, 15, 9, 31, 46, 47, 10, 15, 43, 23, 14, 9, 17, 19, 18, 41, 24, 27, 50, 3, 14, 29, 13, 3, 4, 39, 21, 1, 47, 19, 31, 13, 6, 17

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240, A076241, A076242.

MapIndexed[Mod[#1, First@ #2] &, Nest[Prime, Range@ 79, 3]] (* Michael De Vlieger, Jul 22 2017 *)

a(n) = ppp(n) mod n = A038580(n) mod n.

A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 5, 3, 3, 2, 9, 6, 1, 10, 9, 1, 1, 5, 13, 8, 13, 10, 5, 17, 5, 9, 1, 23, 27, 19, 17, 27, 3, 14, 15, 19, 13, 31, 17, 16, 31, 38, 37, 35, 27, 31, 21, 28, 17, 12, 47, 43, 43, 39, 31, 26, 45, 13, 1, 17, 23, 17, 53, 11, 15, 1, 53, 10, 25, 64, 41, 38, 41, 68, 33, 59, 63, 65

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

[NthPrime(NthPrime(n)) mod(n): n in [1..100]]; // Vincenzo Librandi, Jul 10 2017

Table[Mod[Prime[Prime[n]], n], {n, 100}] (* Vincenzo Librandi, Jul 10 2017 *)

a(n) = prime(prime(n)) % n; \\ Michel Marcus, Jul 09 2017

a(n) = A006450(n) mod n.

A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

Original entry on oeis.org

1, 2, 1, 3, 6, 10, 5, 8, 17, 19, 27, 31, 38, 35, 28, 39, 17, 17, 10, 38, 68, 63, 13, 55, 48, 4, 74, 100, 37, 29, 47, 121, 115, 136, 105, 28, 128, 109, 159, 90, 114, 31, 151, 4, 86, 108, 81, 147, 149, 189, 185, 119, 231, 166, 88, 238, 197, 233, 64, 186, 258, 111, 128, 260

Offset: 1

Labos Elemer, Oct 08 2002

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

Cf. A006450, A038580, A049090, A049203, A049202, A057809, A076240-A076243.

Table[Mod[Prime[Prime[Prime[n]]],Prime[n]],{n,70}] (* Harvey P. Dale, Sep 28 2013 *)

a(n) = Mod[A038580(n), A000040(n)]

A088973 Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation.

Original entry on oeis.org

5, 20, 25, 76, 51, 93, 61, 100, 176, 122, 207, 156, 89, 152, 249, 280, 44, 412, 178, 90, 293, 270, 282, 374, 340, 157, 186, 121, 169, 913, 263, 235, 255, 597, 162, 406, 457, 263, 418, 339, 221, 645, 161, 300, 133, 855, 1235, 236, 162, 240, 256, 243, 786, 261, 514, 590, 156, 481, 374, 211

Offset: 1

Cino Hilliard, Oct 30 2003

nonn

Conjecture: The interval [PIPS4(n), PIPS4(n+1)] always contains at least one twin prime pair. (This implies the Twin Prime Conjecture.)

			a(1) = 5, since there are five pairs of twin primes at least PIPS4(1) = 31 and at most PIPS4(2) = 127: (41,43), (59,61), (71,73), (101,103), and (107,109).

Cf. A006450, A049203, A077800, A088971.

piptwins4(m,n) = { for(x=m,n, f=1; c=0; p1 = prime(prime(prime(prime(prime(x))))); p2 = prime(prime(prime(prime(prime(x+1))))); forprime(j=p1,p2-2, if(isprime(j+2),f=0; c++) ); print1(c","); ) }

def PIP(n,i): # Returns the n-th prime-indexed prime of order i
    if i==0:
        return primes_first_n(n)[n-1]
    else:
        return PIP(PIP(n,i-1),0)
def A088973(n):
    return len([i for i in range(PIP(n,4),PIP(n+1,4),2) if (is_prime(i) and is_prime(i+2))])
A088973(60) # Danny Rorabaugh, Mar 30 2015

PIPS4(x) = A049203(x) = the x-th prime-indexed primes of order 4 = prime(prime(prime(prime(prime(x))))) where prime(x) = A000040(x) is the x-th prime. a(n) = number of twin prime pairs in [PIPS4(n), PIPS(n+1)].

Edited to count twin pairs entirely within [PIPS4(n), PIPS4(n+1)], rather than pairs with the first prime in that interval. - Danny Rorabaugh, Apr 01 2015

A094722 Primes p whose order of primeness A078442(p) is at least 13.

Original entry on oeis.org

3657500101, 88362852307, 2428095424619, 12055296811267, 75063692618249, 156740126985437

Offset: 1

Robert G. Wilson v, May 22 2004

N. Fernandez, An order of primeness, F(p).
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A078442, A000040, A006450, A038580, A049090, A049203, A049202, A057849, A057850, A057851, A057847, A058332, A093047.

Mathematica
```
Nest[ Prime, Range[35], 13]
```

a(n) = prime(A093047(n)). - Andrew Howroyd, Nov 17 2024

a(6) from Robert G. Wilson v, May 22 2004

Name clarified by Andrew Howroyd, Nov 17 2024

A162253 Smallest value of the n-fold nesting prime(prime(...(k)...)) with a prime digital sum.

Original entry on oeis.org

2, 3, 5, 11, 1787, 5381, 5381, 5381, 648391, 648391, 414507281407, 414507281407

Offset: 1

Cino Hilliard, Jun 29 2009

n-deep nestings prime(prime(...(prime(k))...)) = prime^n(k) can be arranged in a table T(n,k),
    2   3    5     7    11    13 : A000040, n=0
    3   5   11    17    31    41 : A006450, n=1
    5  11   31    59   127   179 : A038580, n=2
   11  31  127   277   709  1063 : A049090
   31 127  709  1787  5381  8527 : A049203
  127 709 5381 15299 52711 87803 : A049202
a(n) is the leftmost value in the n-th row (the one with the smallest k) with a digit sum which is prime.
In order to generate the entries a(11) and a(12), prime2() was used which reads a large 880 gigabyte file of all primes < 10^12.

			1st nesting is prime(1) = 2 which has a prime digit sum: a(0). The second nesting is prime(prime(1)) = 3, which has a prime digits sum: a(1)=3. The 3rd and 4th nesting also succeed for k=1 while the fifth nesting prime(prime(prime(prime(prime(4))))) = 1787 is the first occurrence of sum of digits is prime. Here nesting for k = 1,2,3 does not sum to a prime number.

for(j=1,12,print(j","sodip2(100,j)","));
sodip2(n,m) = \\multiple nesting of prime(prime(prime..(n)
{
local(s=0,a,x,y,j,p);
for(x=1,n,
for(i=1,m,p=prime2(p));
a=eval(Vec(Str(p)));
y=sum(j=1,length(a),a[j]);
if(isprime(y),return(p));
)
}

{min A000040^n(k): A000040^n(k) in A028834}. - R. J. Mathar, Jul 16 2009

Definition rephrased by R. J. Mathar, Jul 16 2009

cp's OEIS Frontend

A076240 Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).

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A076243 Remainder when 3rd-order prime ppp(n) = A038580(n) is divided by n.

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A076241 Remainder when 2nd order prime pp(n)=A006450(n) is divided by n.

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A076242 Remainder when 3rd order prime A038580(n) is divided by n-th prime=A000040(n).

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A088973 Number of twin prime pairs between consecutive prime-indexed primes of order 4. The bounds are included in the calculation.

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A094722 Primes p whose order of primeness A078442(p) is at least 13.

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A162253 Smallest value of the n-fold nesting prime(prime(...(k)...)) with a prime digital sum.

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PARI

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