A370848 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.
13, 17, 31, 71, 113, 1151, 11131, 112111, 113111, 131111, 1111211, 1111711, 11111117, 11111171, 71111111, 115111111, 1111111121, 1111115111, 1115111111, 1117111111, 1151111111, 1711111111, 11111111113, 11113111111, 31111111111, 111113111111, 111511111111, 1111171111111
Offset: 1
Examples
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. 23 is not a term because the product of its digits is 6 which is not prime. 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.
Links
- Hugo Pfoertner, Table of n, a(n) for n = 1..3701
Crossrefs
Programs
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Mathematica
Select[Prime[Range[5*10^6]],PrimeQ[Apply[Times,IntegerDigits[#]]]&&CompositeQ[Total[IntegerDigits[NextPrime[#]]]]&] (* James C. McMahon, Mar 03 2024 *)
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PARI
isok(p)=my(x=vecprod(digits(p)),y=sumdigits(nextprime(p+1)));isprime(x) && !isprime(y); forprime(p=2,20000,if(isok(p),print1(p", ")))
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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))}; a370848() \\ Hugo Pfoertner, Mar 03 2024 -
Python
from itertools import count, islice from sympy import isprime, nextprime def A370848_gen(): # generator of terms for l in count(1): k = (10**l-1)//9 for m in range(l): a = 10**m for j in (1,2,4,6): p = k+a*j if isprime(p) and not isprime(sum(map(int,str(nextprime(p))))): yield p A370848_list = list(islice(A370848_gen(),20)) # Chai Wah Wu, Mar 25 2024
Extensions
a(17)-a(21) from Michel Marcus, Mar 03 2024
a(22)-a(28) from Hugo Pfoertner, Mar 03 2024