A347702 -id:A347702 - cp's OEIS frontend

A356374 a(n) is the first prime that starts a string of exactly n consecutive primes that are in A347702.

131, 41, 11, 178909, 304290583, 8345111009

Offset: 1

J. M. Bergot and Robert Israel, Aug 04 2022

a(n) is the first prime that starts a string of exactly n consecutive primes that are quasi-Niven numbers, i.e., have remainder 1 when divided by the sum of their digits.

a(7) > 3*10^11, if it exists. - Amiram Eldar, Aug 04 2022

			a(3) = 11 because [11, 13, 17] is the first string of exactly 3 consecutive primes that are quasi-Niven numbers: 11 mod (1+1) = 1, 13 mod (1+3) = 1 and 17 mod (1+7) = 1, while the preceding prime 7 and the next prime 23 are not quasi-Niven.

Cf. A007605, A136251, A209871, A347702.

filter:= proc(n) n mod convert(convert(n,base,10),`+`) = 1 end proc:
V:= Vector(5): count:= 0:
s:= 0: p:= 1:
while count < 5 do
p:= nextprime(p);
  if filter(p) then
    s:= s+1;
    if s = 1 then p0:= p fi
  elif s > 0 then
  if s <= 5 and V[s] = 0 then V[s]:= p0; count:= count+1 fi;
    s:= 0;
fi
od:
convert(V,list);

seq[len_, pmax_] := Module[{s = Table[0, {len}], v = {}, p = 2, c = 0, pfirst = 2, i}, While[c < len && p < pmax, If[Divisible[p - 1, Plus @@ IntegerDigits[p]], AppendTo[v, p]; If[pfirst == 0, pfirst = p], i = Length[v]; v = {}; If[0 < i <= len && s[[i]] == 0, s[[i]] = pfirst]; pfirst = 0]; p = NextPrime[p]]; s]; seq[4, 10^6] (* Amiram Eldar, Aug 04 2022 *)

a(5)-a(6) from Amiram Eldar, Aug 04 2022

A136251 a(n) = n-th prime reduced modulo the sum of its digits.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 9, 3, 7, 3, 7, 1, 1, 3, 5, 3, 5, 2, 7, 3, 15, 6, 4, 1, 1, 3, 3, 9, 3, 7, 1, 5, 9, 9, 4, 1, 3, 13, 8, 9, 1, 4, 11, 10, 9, 3, 6, 7, 8, 1, 1, 3, 3, 5, 10, 14, 1, 5, 6, 10, 13, 7, 1, 5, 9, 2, 12, 11, 13, 1, 2, 15, 9, 18, 5, 9, 17, 1, 6, 13, 1, 7, 3, 7, 3, 7, 9, 10, 8, 8, 19, 12, 1, 15, 7, 5

Offset: 1

Odimar Fabeny, Mar 17 2008

First occurrence of k: A138792. - Robert G. Wilson v, Mar 27 2008

			2 = 2*1 + 0
3 = 3*1 + 0
5 = 5*1 + 0
7 = 7*1 + 0
11 = 2*5 + 1 (the sum of the digits of 11 is equal to 2)
13 = 4*3 + 1
17 = 8*2 + 1
19 = 10*1 + 9

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

Cf. A000040, A070635, A138791, A138792, A347702.

P := select(isprime, [2,seq(i,i=3..10^3,2)]):
map(p -> p mod convert(convert(p,base,10),`+`), P); # Robert Israel, Mar 05 2024

f[n_] := Block[{p = Prime@n}, Mod[p, Plus @@ IntegerDigits@p]]; Array[f, 97] (* Robert G. Wilson v, Mar 27 2008 *)

a(n) = my(p=prime(n)); p % sumdigits(p); \\ Michel Marcus, Mar 07 2023

a(n) = A070635(A000040(n)). - Michel Marcus, Mar 07 2023

More terms from Robert G. Wilson v, Mar 27 2008

cp's OEIS Frontend

A356374 a(n) is the first prime that starts a string of exactly n consecutive primes that are in A347702.

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