A030665 -id:A030665 - cp's OEIS frontend

A018800 Smallest prime that begins with n.

11, 2, 3, 41, 5, 61, 7, 83, 97, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 2003, 211, 223, 23, 241, 251, 263, 271, 281, 29, 307, 31, 3203, 331, 347, 353, 367, 37, 383, 397, 401, 41, 421, 43, 443, 457, 461, 47, 487, 491, 503, 5101, 521, 53, 541, 557, 563, 571, 587, 59

Offset: 1

David W. Wilson

Conjecture: If a(n) = (n concatenated with k) then k < n. - Amarnath Murthy, May 01 2002

a(n) always exists. Proof. Suppose n is L digits long, and consider the numbers between n*10^B and n*10^B+10^C, where B > C are both large compared with L. All such numbers begin with the digits of n. Using the upper and lower bounds on pi(x) from Theorem 1 of Rosser and Schoenfeld, it follows that for sufficiently large B and C, at least one of these numbers is a prime. QED - N. J. A. Sloane, Nov 14 2014

T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Paolo P. Lava)
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), pp. 64-94.
Index entries for primes involving decimal expansion of n

Cf. A030665, A068164, A068695, A062584, A088781, A085608.
A164022 is the base-2 analog.
Cf. also A258337.
Row n=1 of A262369.

import Data.List (isPrefixOf, find); import Data.Maybe (fromJust)
a018800 n = read $ fromJust $
            find (show n `isPrefixOf`) $ map show a000040_list :: Int
-- Reinhard Zumkeller, Jul 01 2015

f:= proc(n) local x0, d,r,y;
   if isprime(n) then return(n) fi;
   for d from 1 do
     x0:= n*10^d;
     for r from 1 to 10^d-1 by 2 do
       if isprime(x0+r) then
          return(x0+r)
       fi
     od
   od
end proc:
seq(f(n),n=1..100); # Robert Israel, Dec 23 2014

Table[Function[d, FromDigits@ SelectFirst[ IntegerDigits@ Prime@ Range[10^4], Length@ # >= Length@ d && Take[#, Length@ d] == d &]][ IntegerDigits@ n], {n, 59}] (* Michael De Vlieger, May 24 2016, Version 10 *)

a(n{,base=10}) = for (l=0, oo, forprime (p=n*base^l, (n+1)*base^l-1, return (p))) \\ Rémy Sigrist, Jun 11 2017

from sympy import isprime
def a(n):
    if isprime(n): return n
    pow10 = 10
    while True:
        t, maxt = n * pow10 + 1, (n+1) * pow10
        while t < maxt:
            if isprime(t): return t
            t += 2
        pow10 *= 10
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Nov 02 2021

a(n) = prime(A085608(n)). - Michel Marcus, Oct 19 2013

A069691 Smallest prime with internal digits = n; or 0 if no such number exists.

Original entry on oeis.org

101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 1103, 1117, 1123, 2131, 2141, 1151, 1163, 1171, 1181, 1193, 1201, 1213, 1223, 1231, 1249, 1259, 2267, 1277, 1283, 1291, 1301, 1319, 1321, 2333, 2341, 2351, 1361, 1373, 1381, 1399, 1409, 2411, 1423, 1433, 1447

Offset: 0

Amarnath Murthy, Apr 06 2002

By placing one digit on both sides of n one get 36 different numbers that might be primes ( 1 to 9 on left and 1,3,7,9 on right). If none of these numbers is a prime then a(n) = 0.

The smallest value of n for which a(n) = 0 is 2437 = A032734(0). - Rick L. Shepherd and Reinhard Zumkeller, Jul 17 2002

			a(25) = 1259 is prime with internal digits =25.

Robert Price, Table of n, a(n) for n = 0..10000
Index entries for primes involving decimal expansion of n

Cf. A032734, A030665, A069691, A077501, A084413.

Corrected and extended by Larry Reeves, Rick L. Shepherd and Reinhard Zumkeller, Jul 17 2002

A077501 a(n) = smallest prime greater than a(n-1) and beginning with n.

Original entry on oeis.org

11, 23, 31, 41, 53, 61, 71, 83, 97, 101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 2003, 2111, 2203, 2309, 2411, 2503, 2609, 2707, 2801, 2903, 3001, 3109, 3203, 3301, 3407, 3511, 3607, 3701, 3803, 3907, 4001, 4111, 4201, 4327, 4409, 4507, 4603, 4703

Offset: 1

Amarnath Murthy, Nov 08 2002

First 20 terms are the same as that of A030665.

Robert Price, Table of n, a(n) for n = 1..5000
Index entries for primes involving decimal expansion of n

Cf. A030665.

a = {}; Do[AppendTo[a, Function[d, FromDigits@ SelectFirst[ IntegerDigits@ Prime@ Range[PrimePi@ Max@ a + 1, 10^4], And[Length@ # >= Length@ d, Take[#, Length@ d] == d] &]][IntegerDigits@ n]], {n, 47}]; a (* Michael De Vlieger, May 24 2016, Version 10 *)

Corrected and extended by Ray Chandler, Aug 11 2003

A256481 Smallest prime obtained by appending a number with identical digits to n or 0 if no such prime exists.

Original entry on oeis.org

2, 11, 23, 31, 41, 53, 61, 71, 83, 97, 101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 2011, 211, 223, 233, 241, 251, 263, 271, 281, 293, 307, 311, 3299, 331, 347, 353, 367, 373, 383, 397, 401, 419, 421, 431, 443, 457, 461, 479, 487, 491, 503, 511111, 521, 5333

Offset: 0

Chai Wah Wu, Mar 31 2015

For n <= 15392, a(n) = 0 if and only if n = 6930. Conjecture: if a(n) = 0, then n is divisible by 3. Conjecture verified for n <= 10^6. a(n) = 0 for n = 6930, 50358, 56574, 72975.

Chai Wah Wu, Table of n, a(n) for n = 0..6068
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.

Cf. A090287, A256480, A030665.

from gmpy2 import mpz, digits, is_prime
def A256481(n,limit=2000):
    if n in (6930,50358,56574,72975):
        return 0
    if n == 0:
        return 2
    sn = str(n)
    for i in range(1,limit+1):
        for j in range(1,10,2):
            si = digits(j,10)*i
            p = mpz(sn+si)
            if is_prime(p):
                return int(p)
    else:
        return 'search limit reached.'

A258337 Smallest prime starting with n that does not appear earlier.

Original entry on oeis.org

11, 2, 3, 41, 5, 61, 7, 83, 97, 101, 113, 127, 13, 149, 151, 163, 17, 181, 19, 2003, 211, 223, 23, 241, 251, 263, 271, 281, 29, 307, 31, 3203, 331, 347, 353, 367, 37, 383, 397, 401, 419, 421, 43, 443, 457, 461, 47, 487, 491, 503, 5101, 521, 53, 541, 557, 563, 571, 587, 59, 601, 613, 6203, 631, 641, 653, 661, 67, 683, 691, 701

Offset: 1

Vladimir Shevelev, May 27 2015

According to a comment in A018800, every term exists. So the sequence is a permutation of the primes. Indeed, every prime p appears in the sequence either as the p-th term or earlier.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000
Index entries for primes involving decimal expansion of n

Cf. A000040, A018800, A030665, A062584, A258190, A258339.

import Data.List (isPrefixOf, delete)
a258337 n = a258337_list !! (n-1)
a258337_list = f 1 $ map show a000040_list where
   f x pss = g pss where
     g (qs:qss) = if show x `isPrefixOf` qs
                     then (read qs :: Int) : f (x + 1) (delete qs pss)
                     else g qss
-- Reinhard Zumkeller, Jul 01 2015

N:= 100: # to get a(1) to a(N)
for n from 1 to N do
  p:= n;
  if n < 10 then
    p1:= 0
  else
    p1:= A[floor(n/10)];
  fi;
  if isprime(p) and p > p1 then
     A[n]:= p
  else
    for d from 1 while not assigned(A[n]) do
      for i from 1 to 10^d-1 do
        p:= 10^d*n+i;
        if p > p1 and isprime(p) then
           A[n]:= p;
           break
        fi;
      od
    od
  fi
od:
seq(A[i],i=1..N); # Robert Israel, Jun 29 2015

sw(m,n)=d1=digits(n);d2=digits(m);for(i=1,min(#d1,#d2),if(d1[i]!=d2[i],return(0)));1
v=[];n=1;while(n<70,k=1;while(k<10^3,p=prime(k);if(sw(p,n)&&!vecsearch(vecsort(v),p),v=concat(v,p);k=0;n++);k++));v \\ Derek Orr, Jun 13 2015

For n>=2, a(n) >= prime(n-1). The equality holds iff prime(n-1) did not already appear as a(k), k

A030675 Smallest nontrivial extension of n-th palindrome which is a prime.

Original entry on oeis.org

11, 23, 31, 41, 53, 61, 71, 83, 97, 113, 223, 331, 443, 557, 661, 773, 881, 991, 1013, 1117, 1213, 1319, 14107, 1511, 1613, 17107, 1811, 1913, 2027, 2129, 2221, 23201, 2423, 2521, 2621, 2729, 28201, 2927, 3037, 3137, 32303, 3331, 3433

Offset: 1

Patrick De Geest

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

Cf. A002113, A030665.

lpe:= proc(n) local b,d,x;
  for d from 1 do
    b:= 10^d*n;
    for x from b+1 to b+10^d-1 by 2 do
      if isprime(x) then return x fi
    od
  od
end proc:
digrev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
N:= 4: # to use all palindromes of up to N digits
Res:= seq(lpe(n),n=1..9):
for d from 2 to N do
  if d::even then
    m:= d/2;
    Res:= Res, seq(lpe(n*10^m + digrev(n)), n=10^(m-1)..10^m-1);
  else
    m:= (d-1)/2;
    Res:= Res, seq(seq(lpe(n*10^(m+1)+y*10^m+digrev(n)), y=0..9), n=10^(m-1)..10^m-1);
  fi
od:
Res; # Robert Israel, Sep 18 2018

d[n_]:=IntegerDigits[n]; Table[i=1; While[!PrimeQ[x=FromDigits[Flatten[{d[n],d[i]}]]],i=i+2]; x, {n,Select[Range[350],Reverse[x=d[#]]==x &]}]  (* Jayanta Basu, May 24 2013 *)

a(n) = A030665(A002113(n+1)). - Robert Israel, Sep 18 2018

Corrected by Robert Israel, Sep 18 2018

A064792 Append more digits to the n-th prime (leading zeros are permitted) until another prime is reached.

Original entry on oeis.org

23, 31, 53, 71, 113, 131, 173, 191, 233, 293, 311, 373, 419, 431, 479, 5303, 593, 613, 673, 719, 733, 797, 839, 8923, 971, 1013, 1031, 10709, 1091, 11311, 1277, 1319, 1373, 1399, 1493, 1511, 1571, 1637, 16703, 1733, 17903, 1811, 1913, 1931, 1973

Offset: 1

Reinhard Zumkeller, Oct 17 2001

Differs from A030670 in that here the appended digits may start with 0 which is not possible there (although zeros are allowed there, too, if they are not the first appended digit): The first difference appears at a(16) = 5303 where A030670(16) = 5323. - M. F. Hasler, Jan 15 2025

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A030665.

See A030670 for another version. Note that A030670 >= a(n). Cf. A065112.

apply( {A064792(n)=n=prime(n);for(L=1,oo, n*=10; forstep(s=1,10^L-1,2, isprime(n+s)&& return(n+s)))}, [1..44]) \\ M. F. Hasler, Jan 15 2025

a(n) = A030665(prime(n)). - Michel Marcus, Jan 17 2025

A065145 Smallest prime that begins with the n-th square in decimal notation.

Original entry on oeis.org

11, 41, 97, 163, 251, 367, 491, 641, 811, 1009, 1213, 1447, 1693, 19603, 2251, 25601, 2897, 32401, 3613, 4001, 44101, 48407, 5297, 57601, 6257, 6761, 7297, 7841, 8419, 9001, 9613, 10243, 10891, 115601, 12251, 12967, 13691, 14447, 15217, 16001

Offset: 1

Reinhard Zumkeller, Oct 17 2001

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

Cf. A030665, A062584, A000290, A000040, A065144.

f:= proc(n) local d,m,r;
  for d from 1 do
   for m from 1 to 10^d-1 by 2 do
    r:= 10^d*n^2 + m;
    if isprime(r) then return r fi
  od od
end proc:
map(f, [$1..100]); # Robert Israel, May 16 2018

a(n) = A030665(n^2). - Robert Israel, May 16 2018

A308438 a(n) is the smallest prime p whose decimal expansion begins with n and is such that the next prime is p+2n, or -1 if no such prime exists.

Original entry on oeis.org

11, 223, 31, 401, 547, 619, 773, 8581, 9109, 10223, 1129, 12073, 130553, 14563, 150011, 161471, 17257, 18803, 191189, 20809, 210557, 225383, 237091, 240209, 2509433, 2613397, 277429, 283211, 2901649, 308153, 313409, 3204139, 3300613, 3419063, 3507739, 360091, 3727313, 3806347, 3930061, 4045421, 41018911

Offset: 1

J. M. Bergot and Robert Israel, May 30 2019

			For n = 5, 547 is a prime starting with 5, and the next prime after 547 is 557 = 547 + 2*5.  Since this is the least number with these properties, a(5) = 547.

Chai Wah Wu, Table of n, a(n) for n = 1..100

Cf. A018800, A030665.

f:= proc(n) local d,p,q;
  for d from 0 do
    p:= nextprime(n*10^d-1);
    do
      q:= nextprime(p);
      if q - p = 2*n then return p fi;
      if q >= (n+1)*10^d then break fi;
      p:= q;
    od;
  od;
end proc:
map(f, [$1..50]);

from sympy import nextprime
def A308438(n):
    l, p = 1, nextprime(n)
    while True:
        q = nextprime(p)
        if q-p == 2*n:
            return p
        p = q
        if p >= (n+1)*l:
            l *= 10
            p = nextprime(n*l) # Chai Wah Wu, May 31 2019

A346979 Count of the prime decimal descendants of n.

Original entry on oeis.org

83, 63, 23, 22, 23, 11, 29, 23, 3, 4, 54, 1, 9, 14, 6, 7, 3, 4, 7, 40, 0, 4, 19, 15, 8, 7, 10, 14, 5, 6, 2, 7, 0, 16, 9, 11, 12, 13, 4, 1, 34, 1, 8, 14, 5, 1, 13, 5, 5, 16, 6, 0, 9, 0, 24, 4, 6, 19, 2, 9, 25, 16, 0, 7, 4, 4, 3, 11, 2, 7, 7, 4, 1, 15, 2, 8, 8

Offset: 0

Ya-Ping Lu, Aug 09 2021

The number of direct decimal descendants (i.e., decimal children) of n is A038800(n). The number of prime decimal descendants of the n-th prime is A214342(p_n). a(n) is the number of prime decimal descendants of n, which include the prime decimal children of n, the prime decimal children of the prime decimal children of n, and so on.
a(0) = Sum_{m=1..4} (A214342(m) + 1); a(1) = Sum_{m=5..8} (A214342(m) + 1).
a(A032352(m)) = 0; a(A119289(m)) = 0.
A214342 is a subset, as A214342(m) = a(prime(m)).
Conjecture 1: a(n) <= 83. Conjecture 2: lim_{n->oo} (n0/n) = 1, where n0 is the number of zero terms, a(k) = 0, for k <= n.

			a(4) = 23. The 23 prime decimal descendants of 4 are shown in the tree below.
       _____ 4__________________________
      /      |                          \
     41   ___43______________            47
    /    /   |               \             \
  419  431  433               439          479
            / \              /   \        /   \
        4337  4339         4391  4397   4793  4799
             /  |  \        |     |     /  \
        43391 43397 43399 43913 43973 47933 47939
                            |
                         439133
                            |
                        4391339

Cf. A030665, A032352, A038800, A119289, A122072, A214342, A218255, A256481.

Table[Length@Rest@Flatten[FixedPointList[(b=#;Select[Flatten[(a=#;FromDigits/@(Join[IntegerDigits@a,{#}]&/@If[b=={0},Range@9,{1,3,7,9}]))&/@b],PrimeQ])&,{n}]],{n,0,76}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

from sympy import isprime
def p_count(k):
    global ct; d = [2, 3, 5, 7] if k == 0 else [1, 3, 7, 9]
    for i in range(4):
        m = 10*k + d[i]
        if isprime(m): ct += 1; p_count(m)
    return ct
for n in range(100):
    ct = 0; print(p_count(n))

Showing 1-10 of 10 results.

cp's OEIS Frontend

A018800 Smallest prime that begins with n.

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A069691 Smallest prime with internal digits = n; or 0 if no such number exists.

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A077501 a(n) = smallest prime greater than a(n-1) and beginning with n.

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A256481 Smallest prime obtained by appending a number with identical digits to n or 0 if no such prime exists.

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A258337 Smallest prime starting with n that does not appear earlier.

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A030675 Smallest nontrivial extension of n-th palindrome which is a prime.

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A064792 Append more digits to the n-th prime (leading zeros are permitted) until another prime is reached.

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PARI

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A065145 Smallest prime that begins with the n-th square in decimal notation.

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