A370848 - OEIS

%I #24 Mar 26 2024 01:59:19

%S 13,17,31,71,113,1151,11131,112111,113111,131111,1111211,1111711,

%T 11111117,11111171,71111111,115111111,1111111121,1111115111,

%U 1115111111,1117111111,1151111111,1711111111,11111111113,11113111111,31111111111,111113111111,111511111111,1111171111111

%N Lesser of two consecutive primes such that the product of its digits is also prime and the sum of the digits of the other is composite.

%H Hugo Pfoertner, <a href="/A370848/b370848.txt">Table of n, a(n) for n = 1..3701</a>

%e 13 is a term because 13 is prime, the product of its digits is 3 which is also prime and the sum of the digits of 17, the next prime to 13, is 8 which is composite.

%e 23 is not a term because the product of its digits is 6 which is not prime.

%e 131 is not a term because although it is prime and the product of its digits is 3 which is also prime, the sum of the digits of 137, the next prime to 131, is 11 which is not composite.

%t Select[Prime[Range[5*10^6]],PrimeQ[Apply[Times,IntegerDigits[#]]]&&CompositeQ[Total[IntegerDigits[NextPrime[#]]]]&] (* _James C. McMahon_, Mar 03 2024 *)

%o (PARI) isok(p)=my(x=vecprod(digits(p)),y=sumdigits(nextprime(p+1)));isprime(x) && !isprime(y);

%o forprime(p=2,20000,if(isok(p),print1(p", ")))

%o (PARI) a370848(maxdigits=20) = {my (L=List()); for (n=2, maxdigits, my (r=(10^n-1)/9, d=digits(r)); foreach ([2,3,5,7], s, for (k=1, #d, my (dd=d); dd[k]=s; my(q=fromdigits(dd)); if (ispseudoprime(q) && ! isprime(sumdigits(nextprime(q+1))), listput(L,q))))); vecsort(Vec(L))};

%o a370848() \\ _Hugo Pfoertner_, Mar 03 2024

%o (Python)

%o from itertools import count, islice

%o from sympy import isprime, nextprime

%o def A370848_gen(): # generator of terms

%o     for l in count(1):

%o         k = (10**l-1)//9

%o         for m in range(l):

%o             a = 10**m

%o             for j in (1,2,4,6):

%o                 p = k+a*j

%o                 if isprime(p) and not isprime(sum(map(int,str(nextprime(p))))):

%o                     yield p

%o A370848_list = list(islice(A370848_gen(),20)) # _Chai Wah Wu_, Mar 25 2024

%Y Cf. A000040, A002808, A007605, A053666.

%Y Cf. A370850.

%Y Except for the first, all terms of this sequence are in A370851.

%K nonn,base

%O 1,1

%A _Claude H. R. Dequatre_, Mar 03 2024

%E a(17)-a(21) from _Michel Marcus_, Mar 03 2024

%E a(22)-a(28) from _Hugo Pfoertner_, Mar 03 2024