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Iterated sum-of-digits of A078403(n).
+20
3
2, 3, 5, 7, 2, 5, 2, 5, 7, 2, 5, 7, 7, 2, 7, 2, 5, 5, 2, 5, 7, 5, 2, 2, 7, 2, 5, 7, 5, 2, 7, 2, 5, 5, 7, 2, 7, 5, 7, 2, 7, 5, 2, 5, 5, 7, 7, 2, 7, 2, 2, 5, 5, 7, 5, 2, 2, 5, 7, 5, 7, 2, 5, 2, 7, 2, 7, 7, 7, 5, 5, 5, 2, 2, 7, 2, 5, 7, 2, 2, 5, 2, 5, 2, 7, 5, 2, 5, 7, 5, 7, 7, 7, 2, 5, 2, 7, 2, 2, 5, 7, 2, 5, 7, 7
PROG
(PARI) drp(n) = { forprime(x=2, n, r = x%9; if(isprime(r), print1(r" "); ); ); }
Primes congruent to {2, 3, 5, 7} mod 11.
+10
7
2, 3, 5, 7, 13, 29, 47, 71, 73, 79, 101, 113, 137, 139, 157, 167, 179, 181, 211, 223, 227, 233, 269, 271, 277, 293, 311, 313, 337, 359, 379, 401, 409, 421, 431, 443, 467, 487, 491, 509, 541, 557, 563, 577, 599, 601, 607, 619, 641, 643, 673, 709, 733, 739, 751
COMMENTS
Primes p such that p mod 11 is prime.
Primes of the form 11*n+r where n >= 0 and r is in {2, 3, 5, 7}.
MATHEMATICA
Select[Prime[Range[600]], MemberQ[{2, 3, 5, 7}, Mod[#, 11]]&] (* Vincenzo Librandi, Aug 05 2012 *)
PROG
(Magma) [ p: p in PrimesUpTo(760) | p mod 11 in {2, 3, 5, 7} ];
[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 11] | exists(u){ r: r in {2, 3, 5, 7} | p eq (11*n+r) } } ];
CROSSREFS
Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.
Primes congruent to {2, 3, 5, 7, 11} mod 12.
+10
7
2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331, 347, 353, 359
COMMENTS
Primes p such that p mod 12 is prime.
Primes of the form 12*n+r where n >= 0 and r is in {2, 3, 5, 7, 11}.
Except for the prime 2, these are the primes that are encountered in the set of numbers {x, f(f(x))} where x is of the form 4k+3 with k>=0, and where f(x) is the 3x+1-problem function, and f(f(x)) the second iteration value. Indeed this sequence is the set union of 2 and A002145 (4k+3 primes) and A007528 (6k+5 primes), since f(f(4k+3))=6k+5. Equivalently one does not get any prime from A068228 (the complement of the present sequence). - Michel Marcus and Bill McEachen, May 07 2016
MAPLE
isA167135 := n -> isprime(n) and not modp(n, 12) != 1:
MATHEMATICA
Select[Prime[Range[400]], MemberQ[{2, 3, 5, 7, 11}, Mod[#, 12]]&] (* Vincenzo Librandi, Aug 05 2012 *) Select[Prime[Range[72]], Mod[#, 12] != 1 &] (* Peter Luschny, Mar 28 2018 *)
PROG
(Magma) [ p: p in PrimesUpTo(760) | p mod 12 in {2, 3, 5, 7, 11} ];
(Magma) [ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 12] | exists(u){ r: r in {2, 3, 5, 7, 11} | p eq (12*n+r) } } ];
CROSSREFS
Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.
Primes congruent to 2, 3, 5, 7 or 11 (mod 13).
+10
5
2, 3, 5, 7, 11, 29, 31, 37, 41, 59, 67, 83, 89, 107, 109, 137, 163, 167, 193, 197, 211, 223, 239, 241, 263, 271, 293, 317, 349, 353, 367, 379, 397, 401, 419, 421, 431, 449, 457, 479, 499, 509, 523, 557, 577, 587, 601, 613, 631, 653, 661, 683, 691, 709, 733, 739, 743, 757
COMMENTS
Primes which have a remainder mod 13 that is prime.
EXAMPLE
11 mod 13 = 11, 29 mod 13 = 3, 31 mod 13 = 5, hence 11, 29 and 31 are in the sequence.
MATHEMATICA
f[n_]:=PrimeQ[Mod[n, 13]]; lst={}; Do[p=Prime[n]; If[f[p], AppendTo[lst, p]], {n, 6, 6!}]; lst
Select[Prime[Range[4000]], MemberQ[{2, 3, 5, 7, 11}, Mod[#, 13]]&] (* Vincenzo Librandi, Aug 05 2012 *)
PROG
(PARI) {forprime(p=2, 740, if(isprime(p%13), print1(p, ", ")))} \\ Klaus Brockhaus, Oct 28 2009 (Magma) [ p: p in PrimesUpTo(740) | p mod 13 in {2, 3, 5, 7, 11} ]; // Klaus Brockhaus, Oct 28 2009
CROSSREFS
Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135: primes p such that p mod k is prime, for k = 3..12 resp.
Primes such that iterated sum-of-digits ( A038194) is a square.
+10
4
13, 19, 31, 37, 67, 73, 103, 109, 127, 139, 157, 163, 181, 193, 199, 211, 229, 271, 283, 307, 337, 373, 379, 397, 409, 433, 463, 487, 499, 523, 541, 571, 577, 607, 613, 631, 643, 661, 733, 739, 751, 757, 769, 787, 811, 823, 829, 859, 877, 883, 919, 937, 967
MAPLE
select(isprime, map(t -> (9*t+1, 9*t+4), [$1..1000])); # Robert Israel, Sep 04 2014
MATHEMATICA
sQ[n_]:=MemberQ[{1, 4, 9}, NestWhile[Total[IntegerDigits[#]]&, n, #>9&]]; Select[Prime[Range[300]], sQ] (* Harvey P. Dale, Dec 06 2012 *)
PROG
(PARI) forprime(p=2, 1000, if(issquare(p%9), print1(p, ", ")))
Primes such that iterated sum-of-digits ( A038194) is odd.
+10
3
3, 5, 7, 19, 23, 37, 41, 43, 59, 61, 73, 79, 97, 109, 113, 127, 131, 149, 151, 163, 167, 181, 199, 223, 239, 241, 257, 271, 277, 293, 307, 311, 313, 331, 347, 349, 367, 379, 383, 397, 401, 419, 421, 433, 439, 457, 487, 491, 509, 523, 541, 547, 563, 577, 599
COMMENTS
Primes congruent to 1, 3, 5, 7 mod 18. - Robert Israel, Jun 08 2015
MAPLE
select(isprime, [3, seq(seq(i*18+j, j=[1, 5, 7]), i=0..100)]); # Robert Israel, Jun 08 2015
MATHEMATICA
Select[Prime[Range[120]], OddQ[Mod[#, 9]] &] (* Bruno Berselli, Aug 31 2012 *)
PROG
(PARI) forprime(p=2, 600, if((p%9)%2==1, print1(p, ", ")))
(Magma) [p: p in PrimesUpTo(600) | p mod 18 in [1, 3, 5, 7]]; // Vincenzo Librandi, Jun 07 2015 (Magma) [a: n in [0..1000] | IsPrime(a) where a is Floor(9*n/5)]; // Vincenzo Librandi, Jun 08 2015
Primes such that iterated sum-of-digits ( A038194) is even.
+10
2
2, 11, 13, 17, 29, 31, 47, 53, 67, 71, 83, 89, 101, 103, 107, 137, 139, 157, 173, 179, 191, 193, 197, 211, 227, 229, 233, 251, 263, 269, 281, 283, 317, 337, 353, 359, 373, 389, 409, 431, 443, 449, 461, 463, 467, 479, 499, 503, 521, 557, 569, 571, 587, 593, 607
MATHEMATICA
Select[Prime[Range[120]], EvenQ[Mod[#, 9]] &] (* Bruno Berselli, Aug 31 2012 *)
PROG
(PARI) forprime(p=2, 600, if((p%9)%2==0, print1(p, ", ")))
Palindromic primes with prime digital roots.
+10
1
2, 3, 5, 7, 11, 101, 131, 151, 191, 313, 353, 383, 727, 797, 929, 10301, 10501, 11311, 12821, 13331, 13831, 15451, 16061, 16661, 17471, 17971, 19391, 19991, 30103, 30803, 32423, 35053, 35753, 36263, 36563, 37573, 38183, 38783, 70207, 70607
The number of primes less than 10^n whose digital root ( A038194) is also prime.
+10
0
4, 15, 85, 619, 4800, 39266, 332276, 2880818, 25423985, 227527467
EXAMPLE
a(2) = 15 because the only primes less than 100 whose have digital roots are also prime are {2,3,5,7,11,23,29,41,43,47,59,61,79,83,97}.
MATHEMATICA
c = 0; k = 1; Do[ While[ k < 10^n, If[ PrimeQ[k] && PrimeQ[ Mod[k, 9]], c++ ]; k++ ]; Print[c], {n, 1, 8}]
PROG
(Python)
# use primerange (slower) vs. sieve.primerange (>> memory) for larger terms
from sympy import isprime, sieve
def afind(terms):
s = 0
for n in range(1, terms+1):
s += sum(isprime(p%9) for p in sieve.primerange(10**(n-1), 10**n))
print(s, end=", ")
Primes with prime subscripts whose digits are primes, whose digital root is prime, whose sum of digits is prime and whose reversal is also prime.
+10
0
3, 5, 353, 32732237, 35225327, 75527537, 75535277, 75557723, 75737723, 75755257, 77322233, 77752733, 322375577, 322775737, 325725577, 325773727, 337735553, 352272233, 355322777, 357333377, 357735773, 372577727, 372753727, 375577733, 375722377, 375727237, 377725723, 377752723
EXAMPLE
32732237 is in the sequence because 32732237 is the 2016197-th prime number, 2016197 is prime, digits 2, 3 and 7 are primes, 32732237 -> 3 + 2 + 7 + 3 + 2 + 2 + 3 + 7 = 29 (is prime) -> 2 + 9 = 11 -> 1 + 1 = 2, 2 is prime and 73223723 is also prime.
MATHEMATICA
Select[Table[Prime[Prime[n]], {n, 1500000}], Complement[IntegerDigits[#1], {2, 3, 5, 7}] == {} && PrimeQ[#1 - 9 Floor[(#1 - 1)/9]] && PrimeQ[Total[IntegerDigits[#1]]] && PrimeQ[FromDigits[Reverse[IntegerDigits[#1]]]] & ]
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