A136251 -id:A136251 - cp's OEIS frontend

A161973 Primes in A136251 in order of appearance.

3, 7, 3, 7, 3, 5, 3, 5, 2, 7, 3, 3, 3, 3, 7, 5, 3, 13, 11, 3, 7, 3, 3, 5, 5, 13, 7, 5, 2, 11, 13, 2, 5, 17, 13, 7, 3, 7, 3, 7, 19, 7, 5, 3, 3, 13, 3, 7, 7, 3, 11, 3, 19, 11, 17, 3, 3, 5, 5, 5, 7, 5, 17, 5, 17, 7, 11, 19, 5, 17, 13, 19, 13, 11, 7, 3, 7, 2, 11, 3, 3, 3, 7, 13, 5, 3, 13, 2, 3, 7, 3, 11, 7

Offset: 1

Juri-Stepan Gerasimov, Jun 23 2009

			The A136251(k) which are prime occur at k = 9, 10, 11, 12, 15, 16, 17, 18, ... and the subsequence of A136251 at these positions defines the sequence.

Cf. A000040, A136251.

read("transforms") ; A007605 := proc(n) digsum(ithprime(n)) ; end:
A136251 := proc(n) ithprime(n) mod A007605(n) ; end:
for n from 1 to 300 do p := A136251(n); if isprime(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Aug 02 2009

Edited, corrected and extended by R. J. Mathar, Aug 02 2009

A138792 Least prime, p, such that p mod (sum of the digits of p) = n.

Original entry on oeis.org

2, 11, 67, 23, 89, 53, 83, 29, 173, 19, 197, 193, 337, 167, 269, 79, 757, 397, 379, 479, 3677, 769, 997, 6967, 1699, 3889, 9857, 7867, 6959, 9949, 16987, 9887, 49697, 47599, 18899, 67979, 73999, 56999, 197699, 49999, 159899, 189989, 98899, 98999, 988877

Offset: 0

Robert G. Wilson v, Mar 28 2008

First occurrence of n in A136251.

			a(2) = 67 = 13*5+2 <--> 67 (mod 13) = 2.

David A. Corneth, Table of n, a(n) for n = 0..181 (first 74 terms from Robert G. Wilson v)

Cf. A136251, A138791.

V:= Array(0..50): count:= 0: p:= 1:
while count < 51 do
  p:= nextprime(p);
  s:= convert(convert(p,base,10),`+`);
  v:= p mod s;
  if v <= 50 and V[v] = 0 then V[v]:= p; count:= count+1;  fi
od:
convert(V,list); # Robert Israel, Mar 07 2023

f[n_] := Block[{p = Prime@ n}, Mod[p, Plus @@ IntegerDigits@ p]]; t = Table[0, {1000}]; Do[ a = f@n; If[a < 1000 && t[[a + 1]] == 0, t[[a + 1]] = Prime@ n; Print[{a, Prime@n}]], {n, 503200000}]
lp[n_]:=Module[{p=2},While[Mod[p,Total[IntegerDigits[p]]]!=n,p= NextPrime[ p]];p]; Array[lp,50,0] (* Harvey P. Dale, Jan 15 2019 *)

a(n) = my(p=2); while ((p % sumdigits(p)) != n, p=nextprime(p+1)); p; \\ Michel Marcus, Mar 07 2023

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    adict, n, p = dict(), 0, 2
    while True:
        v = p%sum(map(int, str(p)))
        if v not in adict: adict[v] = p
        while n in adict: yield adict[n]; n += 1
        p = nextprime(p)
print(list(islice(agen(), 45))) # Michael S. Branicky, Mar 07 2023

Name corrected by Robert Israel, Mar 07 2023

A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

Original entry on oeis.org

11, 13, 17, 41, 43, 97, 101, 131, 157, 181, 233, 239, 271, 311, 353, 401, 421, 491, 521, 541, 599, 617, 631, 647, 673, 743, 811, 859, 953, 1021, 1031, 1051, 1093, 1171, 1201, 1249, 1259, 1301, 1303, 1327, 1373, 1531, 1601, 1621, 1801, 1871, 2029, 2111, 2129, 2161

Offset: 1

Burak Muslu, Sep 10 2021

			97 is a term since its sum of digits is 9+7 = 16, and 97 mod 16 = 1.

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

Cf. A000040, A007605, A136251.
Subsequence of A209871.
A259866 \ {31}, and the primes associated with A056804 \ {1, 2} and A056797 are subsequences.

select(t -> isprime(t) and t mod convert(convert(t,base,10),`+`) = 1, [seq(i,i=3..10000,2)]); # Robert Israel, Mar 05 2024

Select[Range[2000], PrimeQ[#] && Mod[#, Plus @@ IntegerDigits[#]] == 1 &] (* Amiram Eldar, Sep 10 2021 *)

isok(p) = isprime(p) && ((p % sumdigits(p)) == 1); \\ Michel Marcus, Sep 10 2021

from sympy import primerange
def ok(p): return p%sum(map(int, str(p))) == 1
print(list(filter(ok, primerange(1, 2130)))) # Michael S. Branicky, Sep 10 2021

A344127 Primes p such that (p mod s) and (p mod t) are consecutive primes, where s is the sum of the digits of p and t is the product of the digits of p.

Original entry on oeis.org

23, 29, 313, 397, 431, 661, 941, 1129, 1193, 1223, 1277, 1613, 2621, 2791, 3461, 4111, 4159, 12641, 12911, 14419, 15271, 19211, 21611, 21773, 22613, 26731, 29819, 31181, 31511, 41381, 61211, 74611, 111191, 115811, 121181, 121727, 141161, 141221, 141269, 145513, 157523, 171713, 173141, 173891

Offset: 1

J. M. Bergot and Robert Israel, May 09 2021

Since p mod 0 is not defined, the digit 0 is not allowed.

			a(3) = 313 is a term because with s = 3+1+3 = 7 and t = 3*1*3 = 9, 313 mod 7 = 5 and 313 mod 9 = 7 are consecutive primes.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A007605, A053666, A136251.

filter:= proc(p) local L,s,t,q;
  L:= convert(p,base,10);
  s:= convert(L,`+`);
  t:= convert(L,`*`);
  if t = 0 then return false fi;
  q:= p mod s;
  isprime(q) and (p mod t) = nextprime(q)
end proc:
select(filter, [seq(ithprime(i),i=1..20000)]);

isok(p) = if (isprime(p), my(d=digits(p)); vecmin(d) && isprime(q=(p%vecsum(d))) && isprime(r=(p%vecprod(d))) && (nextprime(q+1)==r)); \\ Michel Marcus, May 10 2021

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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A161973 Primes in A136251 in order of appearance.

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A138792 Least prime, p, such that p mod (sum of the digits of p) = n.

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A347702 Prime numbers that give a remainder of 1 when divided by the sum of their digits.

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A344127 Primes p such that (p mod s) and (p mod t) are consecutive primes, where s is the sum of the digits of p and t is the product of the digits of p.

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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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Author

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Comments

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Programs

Maple

Mathematica

Extensions