A085563 -id:A085563 - cp's OEIS frontend

A342962 a(n) is the least prime that starts a string of exactly n distinct primes p_1, p_2, ..., p_n where p_{i+1} = p_i+A085563(p_i), but p_n+A085563(p_n) is either not prime or equal to p_n.

2, 29, 229, 5639, 35969, 54191353

Offset: 1

J. M. Bergot and Robert Israel, Mar 31 2021

No further terms up to 10^9.

			a(3) = 229 because 229, 229+A085563(229) = 233, and 233+A085563(233) = 241 are prime but 241+A085563(241) = 243 is not.

Cf. A085563, A342961.

f:= n -> n + convert(select(isprime,convert(n,base,10)),`+`):
F:= proc(n) option remember; local t,x;
    x:= f(n);
  if x = n or not isprime(x) then 1 else 1+procname(x) fi
end proc:
V:= Vector(6): count:= 0: p:= 1:
while count < 6 do
  p:= nextprime(p); v:= F(p);
  if v <= 6 and V[v] = 0 then V[v]:= p; count:= count+1 fi
od:
convert(V,list);

A342961 Primes p such that p + the sum of its prime digits is prime.

Original entry on oeis.org

11, 19, 29, 37, 41, 53, 61, 73, 89, 101, 109, 149, 181, 191, 199, 229, 233, 257, 269, 277, 281, 307, 331, 359, 379, 383, 401, 409, 419, 433, 449, 461, 491, 499, 563, 587, 593, 601, 619, 641, 653, 661, 673, 677, 691, 727, 797, 809, 811, 821, 881, 911, 919, 937, 941, 977, 991, 1009, 1019, 1033

Offset: 1

J. M. Bergot and Robert Israel, Mar 31 2021

			a(3) = 29 is a term because it is prime, the sum of its prime digits is 2, and 29+2 = 31 is also prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Includes A034844. Cf. A085563, A342962.

f:= p -> p + convert(select(isprime,convert(p,base,10)),`+`):
select(t -> isprime(t) and isprime(f(t)), [seq(i,i=3..2000,2)]);

Select[Prime@Range@200,PrimeQ@Total[Join[{#},Select[IntegerDigits@#,PrimeQ]]]&] (* Giorgos Kalogeropoulos, Apr 01 2021 *)

isok(p) = isprime(p) && isprime(p+sumdigits(p)); \\ Michel Marcus, Apr 01 2021

A341013 The cumulative sum of the prime digits so far in the sequence and the cumulative sum of the nonprime digits so far differ by n for all a(n) values.

Original entry on oeis.org

1, 3, 12, 16, 10, 34, 43, 56, 65, 78, 87, 100, 236, 258, 263, 285, 304, 326, 340, 359, 362, 395, 403, 430, 447, 474, 506, 528, 539, 560, 582, 593, 605, 623, 632, 650, 708, 744, 780, 807, 825, 852, 870, 935, 953, 1000, 1112, 1121, 1145, 1154, 1167, 1176

Offset: 1

Carole Dubois and Eric Angelini, Feb 02 2021

The prime digits are 2, 3, 5 and 7; the nonprime digits are 0, 1, 4, 6, 8 and 9.

This is the lexicographically earliest sequence of distinct integers > 0 having this property.

			Say that the current sequence is S, the cumulative sum at any moment of the prime digits of S is P, the cumulative sum at any moment of the nonprime digits of S is N and the absolute difference |P-N| is D. We would then have:
S =  1, 3, 12, 16, 10, 34, 43, 56, 65, 78, 87, 100,...
P =  0  3   5   5   5   8  11  16  21  28  35   35
N =  1  1   2   9  10  14  18  24  30  38  46   47
D =  1  2   3   4   5   6   7   8   9  10  11   12 <-- this is = n

Carole Dubois, Table of n, a(n) for n = 1..5000

Cf. A085562, A085563.

Cf. A341012 (the cumulative sums of even vs odd digits differ by n).

A341161 The sum of the prime digits of m and the sum of the nonprime digits of m differ by n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 119, 48, 49, 68, 69, 88, 89, 99, 199, 488, 489, 499, 689, 699, 889, 899, 999, 1999, 4889, 4899, 4999, 6899, 6999, 8899, 8999, 9999, 19999, 48899, 48999, 49999, 68999, 69999, 88999, 89999, 99999, 199999, 488999

Offset: 1

Carole Dubois and Eric Angelini, Feb 06 2021

This is the lexicographically earliest sequence of distinct terms > 0 with this property.

The nonprime digits are 0, 1, 4, 6, 8 and 9; the prime digits are 2, 3, 5 and 7. After a(7) = 7 no more terms will show a prime digit.

			The sequence:     1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 119, 48,...
Sum of prime dig. 0  2  3  0  5  0  7  0  0   0    0   0
Sum of nonprimes  1  0  0  4  0  6  0  8  9  10   11  12
Difference (= n)  1  2  3  4  5  6  7  8  9  10   11  12

Carole Dubois, Table of n, a(n) for n = 1..74

Cf. A085562, A085563, A341011.

cp's OEIS Frontend

A342962 a(n) is the least prime that starts a string of exactly n distinct primes p_1, p_2, ..., p_n where p_{i+1} = p_i+A085563(p_i), but p_n+A085563(p_n) is either not prime or equal to p_n.

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Maple

A342961 Primes p such that p + the sum of its prime digits is prime.

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Maple

Mathematica

PARI

A341013 The cumulative sum of the prime digits so far in the sequence and the cumulative sum of the nonprime digits so far differ by n for all a(n) values.

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A341161 The sum of the prime digits of m and the sum of the nonprime digits of m differ by n.

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