A209871 -id:A209871 - cp's OEIS frontend

A341189 Numbers that when divided by the sum of their digits leave 19 as remainder.

299, 479, 659, 679, 839, 877, 1199, 1379, 1559, 1669, 1739, 1859, 1867, 1919, 1969, 2099, 2279, 2459, 2639, 2659, 2687, 2819, 2857, 2869, 2879, 2894, 3179, 3359, 3539, 3649, 3719, 3769, 3799, 3847, 3929, 3994, 4079, 4259, 4439, 4619, 4639, 4669, 4757

Offset: 1

Eric Angelini and Carole Dubois, Feb 06 2021

			a(1) = 299 and 299 is 20*14 with remainder 19;
a(2) = 479 and 479 is 20*23 with remainder 19; etc.

Carole Dubois, Table of n, a(n) for n = 1..5001

Cf. A005349 (Niven numbers: remainder = 0), A209871 (Quasi-Niven numbers: remainder = 1), A341169 to A341182 (remainders = 2 to 15).

A341190 Numbers that when divided by the sum of their digits leave 20 as remainder.

Original entry on oeis.org

779, 986, 2849, 2995, 3298, 3496, 3677, 3694, 3884, 3892, 3895, 4288, 4298, 4486, 4684, 4778, 4795, 4882, 4919, 4979, 5278, 5476, 5674, 5695, 5747, 5788, 5872, 5948, 5954, 6268, 6368, 6466, 6575, 6595, 6664, 6782, 6796, 6862, 6884, 7196, 7258, 7456

Offset: 1

Eric Angelini and Carole Dubois, Feb 06 2021

			a(1) = 779 and 779 is 23*33 with remainder 20;
a(2) = 986 and 986 is 23*42 with remainder 20; etc.

Carole Dubois, Table of n, a(n) for n = 1..5001

Cf. A005349 (Niven numbers: remainder = 0), A209871 (Quasi-Niven numbers: remainder = 1), A341169 to A341182 (remainders = 2 to 15).

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

Original entry on oeis.org

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

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A341189 Numbers that when divided by the sum of their digits leave 19 as remainder.

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A341190 Numbers that when divided by the sum of their digits leave 20 as remainder.

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

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