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Remainder when 2nd order prime pp(n) = A006450(n) is divided by n-th prime = A000040(n).
+20
6
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
EXAMPLE
a(4) = 3 since prime(prime(4)) (mod prime(4)) = prime(7) (mod 7) = 17 (mod 7) = 3. - Michael De Vlieger, Mar 25 2017
MAPLE
a:= n-> (p-> irem(ithprime(p), p))(ithprime(n)):
MATHEMATICA
Table[Mod @@ Map[Nest[Prime, n, #] &, {2, 1}], {n, 65}] (* Michael De Vlieger, Mar 25 2017 *)
PROG
(PARI) a(n) = prime(prime(n)) % prime(n); \\ Michel Marcus, Mar 25 2017
Remainder when 2nd order prime pp(n)= A006450(n) is divided by n.
+20
3
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
PROG
(Magma) [NthPrime(NthPrime(n)) mod(n): n in [1..100]]; // Vincenzo Librandi, Jul 10 2017
3, 5, 11, 17, 41, 59, 179, 191, 431, 461, 599, 617, 1031, 1787, 2027, 2081, 2381, 2549, 3299, 4091, 4217, 4421, 4517, 4787, 5021, 5441, 5651, 8999, 9041, 9461, 10457, 13217, 13709, 13757, 14591, 14867, 15641, 16061, 16451, 16901, 17189, 17291
EXAMPLE
a(10) = 461 since prime(10) = 89 and prime(89 + 1) - prime(89) = 463 - 461 = 2.
MATHEMATICA
Prime[Prime[Flatten[Position[Table[Prime[Prime[n]+1] -Prime[Prime[n]], {n, 1, 1000}], 2]]]]
a(n) = pip(n)^pip(n) where pip(n) is the n-th prime-indexed prime (see A006450).
+20
2
27, 3125, 285311670611, 827240261886336764177, 17069174130723235958610643029059314756044734431, 1330877630632711998713399240963346255985889330161650994325137953641
PROG
(PARI) piptopip(n) = { local(x, y); for(x=1, n, y=pip(x)^pip(x); print1(y", "); ) } pip() = { return(prime(prime(n))) }
Squares of A006450: a(n) = prime(prime(n))^2.
+20
1
9, 25, 121, 289, 961, 1681, 3481, 4489, 6889, 11881, 16129, 24649, 32041, 36481, 44521, 58081, 76729, 80089, 109561, 124609, 134689, 160801, 185761, 212521, 259081, 299209, 316969, 344569, 358801, 380689, 502681, 546121, 597529, 635209
Cubes of A006450: a(n) = prime(prime(n))^3.
+20
1
27, 125, 1331, 4913, 29791, 68921, 205379, 300763, 571787, 1295029, 2048383, 3869893, 5735339, 6967871, 9393931, 13997521, 21253933, 22665187, 36264691, 43986977, 49430863, 64481201, 80062991, 97972181, 131872229, 163667323
4, 0, 8, -4, 8, -10, 8, 10, -8, 12, -8, -10, 8, 10, 6, -30, 42, -26, -8, 20, -4, 0, 18, -10, -22, 8, -12, 6, 74, -62, 4, -10, 38, -44, 24, 6, -24, 16, -8, -8, 42, -48, 12, -14, 64, 32, -88, -10, 10
PROG
(PARI) f(n) = prime(prime(n))
a(n) = f(n+2)-2*f(n+1)+f(n)
Number of ways to write n as the sum of distinct super-primes ( A006450).
+20
1
0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 0, 1, 3, 0, 1, 2, 0, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 0, 3, 2, 0, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 0, 2, 3, 1, 4
COMMENTS
a(n) > 0 for n > 96 (cf. Dressler, Parker, 1975).
EXAMPLE
There are two ways to write 31 as the sum of distinct super-primes: 31 (a single summand, as 31 is itself a super-prime) and 17 + 11 + 3 (three summands), so a(31) = 2.
PROG
(PARI) isokp(pt) = {for (k=1, #pt, if (! isprime(pt[k]) || !isprime(primepi(pt[k])), return (0)); ); #pt == #Set(pt); }
a(n) = {if (n < 3, return (0)); nb = 0; forpart(pt = n, if (isokp(pt), nb++), [3, n]); nb; } \\ Michel Marcus, Apr 06 2016
1, 2, 3, 4, 5, 6, 7, 12, 45, 64, 121, 144, 238, 261, 415, 2296, 2847
COMMENTS
Corresponding primes are 5, 11, 41, 227, 2341 and 30071.
1, 4, 64, 121, 144 are squares for initial terms of sequence. Are there any other squares in this sequence?
EXAMPLE
a(1) = 1 because 2 + 3 = 5 is prime.
a(2) = 2 because 2*3 + 5 = 11 is prime.
a(3) = 3 because 2*3*5 + 11 = 41 is prime.
MATHEMATICA
Select[Range@ 500, PrimeQ[Product[Prime@ k, {k, #}] + Prime@ Prime@ #] &] (* Michael De Vlieger, Dec 02 2015 *)
PROG
(PARI) lista(nn) = {s = 1; for(k=1, nn, s *= prime(k); if(ispseudoprime(s + prime(prime(k))), print1(k, ", ")); ); }
Primeth recurrence: a(n+1) = a(n)-th prime.
(Formerly M0734)
+10
285
1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, 88362852307, 2428095424619, 75063692618249, 2586559730396077, 98552043847093519, 4123221751654370051, 188272405179937051081, 9332039515881088707361, 499720579610303128776791, 28785866289100396890228041
COMMENTS
A007097(n) = Min {k : A109301(k) = n} = the first k whose rote height is n, the level set leader or minimum inverse function corresponding to A109301. - Jon Awbrey, Jun 26 2005
a(n) is the Matula-Goebel number of the rooted path tree on n+1 vertices. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, Feb 18 2012 Conjecture: log(a(1))*log(a(2))*...*log(a(n)) ~ a(n). - Thomas Ordowski, Mar 26 2015
REFERENCES
Lubomir Alexandrov, unpublished notes, circa 1960.
L. Longeri, Towards understanding nature and the aesthetics of prime numbers, https://www.longeri.org/prime/nature.html [Broken link, but leave the URL here for historical reasons]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Lubomir Alexandrov, "The Eratosthenes Progression p(k+1)=π^{-1}(p(k)), k=0,1,2,..., p(0)=1,4,6,... Determines an Inner Prime Number Distribution Law", Second Int. Conf. "Modern Trends in Computational Physics", Jul 24-29, 2000, Dubna, Russia, Book of Abstracts, p. 19. Available at arXiv:math/0105154 [math.NT], 2001.
FORMULA
a(n) = prime^{[n]}(1), with the prime function prime(k) = A000040(k), with a(0) = 1. See the name and the programs. - Wolfdieter Lang, Apr 03 2018
PROG
(PARI) print1(p=1); until(, print1(", "p=prime(p))) \\ M. F. Hasler, Oct 09 2011 (Haskell)
a007097 n = a007097_list !! n
(GAP) P:=Filtered([1..60000], IsPrime);;
a:=[1];; for n in [2..10] do a[n]:=P[a[n-1]]; od; a; # Muniru A Asiru, Dec 22 2018
EXTENSIONS
a(20)-a(21) found by Andrey V. Kulsha using a program by Xavier Gourdon, Oct 02 2011 a(23) from David Baugh using Kim Walisch's primecount, May 16 2016
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