A209252 -id:A209252 - cp's OEIS frontend

A276694 Where records occur in A209252.

0, 11, 13, 101, 107, 177, 357, 1001, 1011, 10759, 11487, 42189, 113183, 344253, 1851759, 4787769, 15848679, 139367847, 240889077, 1167555543, 111738007953

Offset: 1

Paolo P. Lava, Sep 14 2016

Except for the first term, all terms are coprime to 10. - Chai Wah Wu, Sep 19 2016

Cf. A000040, A209252.

with(numtheory): P:= proc(q) local d,j,k,n,t,x; x:=4; print(0);
for n from 1 to q do d:=ilog10(n)+1; t:=0;
for k from 0 to d-1 do for j from 0 to 9 do
if isprime(trunc(n/10^(k+1))*10^(k+1)+j*10^k+(n mod 10^k))
then t:=t+1; fi; od; od;
if isprime(n) then if t-d>x then x:=t-d; print(n); fi; else
if t>x then x:=t; print(n); fi; fi; od; end: P(10^9);

a(18)-a(19) from Chai Wah Wu, Sep 19 2016
a(20) from Michael S. Branicky, Apr 21 2025
a(21) from Michael S. Branicky, May 21 2025

A265733 Number of n-digit primes whose digits include at least n-1 digits "1".

Original entry on oeis.org

4, 8, 10, 9, 10, 13, 7, 16, 10, 11, 11, 13, 9, 11, 6, 7, 8, 16, 7, 11, 8, 9, 9, 10, 8, 12, 6, 13, 13, 21, 7, 12, 8, 7, 15, 16, 8, 9, 17, 9, 5, 22, 9, 15, 6, 12, 9, 20, 11, 13, 14, 11, 13, 12, 9, 15, 7, 4, 8, 12, 4, 11, 8, 15, 10, 17, 12, 12, 4, 9, 9, 14, 8, 14, 10, 7, 10, 14, 5, 14

Offset: 1

Michael De Vlieger and Robert G. Wilson v, Dec 14 2015

Inspired by A241100 and its conjecture by Chai Wah Wu of Dec 10 2015.
The average value of a(n) is 10.6, median is 10, and mode is 11 for the first 1000 terms.
The first occurrence of k>0: 433, 361, 229, 1, 41, 15, 7, 2, 4, 3, 10, 26, 6, 51, 35, 8, 39, 180, 84, 48, 30, 42, 306, 138, 948, 642, ..., .
0 < a(n) < 27 for n < 1551.
It appears that a(n)/(9*n + 0^(n-1)) has its maximum at n = 2. - Altug Alkan, Dec 16 2015

			a(1) = 4 because {2, 3, 5, 7} are primes;
a(2) = 8 because {11, 13, 17, 19, 31, 41, 61, 71} are two digit primes with at least one digit being 1;
a(3) = 10 because {101, 113, 131, 151, 181, 191, 211, 311, 811, 911} are three digit primes with at least two digit being 1, etc.

Michael De Vlieger and Robert G. Wilson v, Table of n, a(n) for n = 1..1550

Cf. A209252, A241100.

A:= proc(n) local x,X,i,y;
  x:= (10^n-1)/9;
  y:= [x,seq(seq(x+10^i*y,y=1..8),i=0..n-1),seq(x - 10^i,i=0..n-2)];
  nops(select(isprime,y))
end proc:
map(A, [$1..100]); # Robert Israel, Dec 31 2015

f1[n_] := Block[{cnt = k = 0, r = (10^n - 1)/9, s = {-1, 1, 2, 3, 4, 5, 6, 7, 8}}, If[ PrimeQ@ r, cnt++]; If[ PrimeQ[(10^(n - 1) - 1)/9], cnt--]; While[k < n, p = Select[r + 10^k*s, PrimeQ]; cnt += Length@ p; k++]; cnt]; Array[f1, 70]

Number of n-digit primes of the form 111...d...111, where the number of ones is at least n-1 and d is any other decimal digit.

A118118 Composite numbers that always remain composite when a single decimal digit of the number is changed.

Original entry on oeis.org

200, 204, 206, 208, 320, 322, 324, 325, 326, 328, 510, 512, 514, 515, 516, 518, 530, 532, 534, 535, 536, 538, 620, 622, 624, 625, 626, 628, 840, 842, 844, 845, 846, 848, 890, 892, 894, 895, 896, 898, 1070, 1072, 1074, 1075, 1076, 1078, 1130

Offset: 1

Adam Panagos (adam.panagos(AT)gmail.com), May 12 2006

The term "prime-proof" for this property is found on projecteuler.net (cf. link). The nontrivial subsequence A143641 is that of odd elements not ending in 5 (i.e. not ending in 0,2,4,5,6 or 8); it starts 212159,595631,872897,... - M. F. Hasler, Sep 04 2008

Also indices n such that A209252(n) is zero. - Ray G. Opao, Aug 01 2020

			a(1) = 200 is in the sequence because changing any digit of 200 (for example 300, 220, or 209) is still composite. The integer 100 is not in the sequence because it can be changed to 107 which is prime.

Paolo P. Lava, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 200
Project Euler, Problem 200. Find the 200th prime-proof sqube containing the contiguous sub-string "200" (2008)

Cf. A143641, A050249, A209252.

IsA118118:=function(n); D:=Intseq(n); return forall{ : k in [1..#D], j in [0..9] | j eq D[k] or not IsPrime(Seqint(S)) where S:=Insert(D, k, k, [j]) }; end function; [ n: n in [1..1200] | IsA118118(n) ]; // Klaus Brockhaus, Feb 28 2011

unprimeableQ[n_] := Block[{d = IntegerDigits@ n, t = {}}, Do[AppendTo[t, FromDigits@ ReplacePart[d, i -> #] & /@ DeleteCases[Range[0, 9], x_ /; x == d[[i]]]], {i, Length@ d}]; ! AnyTrue[Flatten@ t, PrimeQ]]; Select[Range@ 1200, unprimeableQ] (* Michael De Vlieger, Nov 09 2015, Version 10 *)

/* return 1 if no digit can be changed to make it prime; if d=1, print a prime if n is not prime-proof */ isA118118(n,d=0)={ forstep( k=n\10*10+1, n\10*10+9,2, isprime(k) || next; d && print("prime:",k); return); if( n%2==0 || n%5==0, /* even or ending in 5: no other digit can make it prime, except for the case where the last digit is prime and the first digit is the only other nonzero one */ return( !isprime(n%10) || 9 < n % 10^( log(n+.5)\log(10) ) || (d && print("prime:",n%10)) )); o=10; until( n < o*=10, k=n-o*(n\o%10); for( i=0,9, isprime(k) && return(d && print("prime:",k)); k+=o));1} \\ M. F. Hasler, Sep 04 2008

from sympy import isprime
def selfplusneighs(n):
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L))
def ok(n): return all(not isprime(k) for k in selfplusneighs(n))
print([k for k in range(1131) if ok(k)]) # Michael S. Branicky, Jun 19 2022

Edited by Charles R Greathouse IV, Aug 05 2010

A346064 Number of primes that may be generated by changing any two digits of n simultaneously.

Original entry on oeis.org

21, 17, 20, 15, 21, 20, 21, 16, 21, 17, 23, 18, 22, 17, 23, 22, 23, 17, 23, 19, 23, 19, 22, 16, 23, 22, 23, 18, 23, 18, 22, 18, 21, 16, 22, 21, 22, 17, 22, 17, 23, 18, 22, 17, 23, 22, 23, 17, 23, 19, 23, 19, 22, 16, 23, 22, 23, 18, 23, 18, 22, 18, 21, 16, 22

Offset: 10

Franz Vrabec, Jul 03 2021

Indices of low records are given by A345289. By heuristic considerations it is conjectured that a(n) > 0 for all n >= 10.

			Changing two digits of the number 17 simultaneously yields the primes 02,03,05,23,29,31,41,43,53,59,61,71,73,79,83,89, so a(17) = 16.

M. Filaseta, M. Kozek, Ch. Nicol and J. Selfridge, Composites that Remain Composite After Changing a Digit, J. Comb. Number Theory, Vol. 2, No. 1 (2010), pp. 25-36.

Cf. A345289 (indices of record lows).

Cf. A209252 (changing one digit).

A346064 := proc(n)
local a, d, e, r, s, l, N, NN, nn, i;
a := 0;
N := convert(n, base, 10);
l := nops(N);
for d to l - 1 do
   for e from d + 1 to l do
      for r from 0 to 9 do
         for s from 0 to 9 do
        if r <> op(d, N) and s <> op(e, N) then
           NN := subsop(d = r, e = s, N);
           nn := add(op(i, NN)*10^(i - 1), i = 1 .. l);
           if isprime(nn) then a := a + 1; end if;
        end if;
        end do;
      end do;
   end do;
end do;
a;
end proc:

Table[Count[Flatten[FromDigits/@Tuples[ReplacePart[t=List/@IntegerDigits[n],{#->Complement[Range[0,9],t[[#]]],#2->Complement[Range[0,9],t[[#2]]]}]&@@#]&/@Subsets[Range@IntegerLength@n,{2}]],?PrimeQ],{n,10,100}] (* _Giorgos Kalogeropoulos, Jul 23 2021 *)

from sympy import isprime
from itertools import combinations, product
def change2(s):
    for i, j in combinations(range(len(s)), 2):
        for c, d in product("0123456789", repeat=2):
            if c != s[i] and d != s[j]:
                yield s[:i] + c + s[i+1:j] + d + s[j+1:]
def a(n): return sum(isprime(int(t)) for t in change2(str(n)))
print([a(n) for n in range(10, 101)]) # Michael S. Branicky, Jul 23 2021

A383271 Number of primes (excluding n) that may be generated by replacing any binary digit of n with a digit from 0 to 1.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 0, 2, 2, 1, 1, 1, 0, 3, 1, 1, 2, 3, 0, 4, 1, 3, 0, 2, 0, 3, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 3, 1, 1, 1, 4, 0, 5, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 0, 3, 1, 1, 1, 2, 0, 4, 0, 3, 2, 3, 0, 3, 1, 4, 1, 1, 0, 5, 0, 4, 1, 1, 0, 4, 1, 2, 0, 0, 0, 3, 1, 1

Offset: 0

Michael S. Branicky, Apr 21 2025

Also, the number of prime neighbors of n in H(A070939(n), 2), where H(k, b) is the Hamming graph whose vertices are the sequences of length k over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1 (see A145667).

Prepending 1 is not allowed.

			a(3) = 1 since 3 = 11_2 can be changed to 10_2 = 2, which is prime.
a(5) = 1 since 5 = 101_2 can be changed to 001_2 = 1, 111_2 = 7 (prime), or 100_2 = 4.
a(6) = 2 since 6 = 110_2 can be changed to 010_2 = 2 (prime), 100_2 = 4, or 111_2 = 7 (prime).
a(7) = 2 since 7 = 111_2 can be changed to 011_2 = 3 (prime), 101_2 = 5 (prime), or 110_2 = 6.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Base-2 analog of A209252.

Cf. A070939, A145667, A352942.

a:= n-> nops(select(isprime, [seq(Bits[Xor](2^i, n), i=0..ilog2(n))])):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 23 2025

A383271[n_] := Count[BitXor[n, 2^Range[0, BitLength[n] - 1]], _?PrimeQ];
Array[A383271, 100, 0] (* Paolo Xausa, Apr 23 2025 *)

from gmpy2 import is_prime
def a(n):
    if n == 0:
        return 0
    if n&1 == 0:
        return int(is_prime(n + 1)) + int(1<<(n.bit_length()-1)^n == 2)
    mask, c = 1, 0
    for i in range(n.bit_length()):
        if is_prime(mask^n):
            c += 1
        mask <<= 1
    return c
print([a(n) for n in range(90)])

cp's OEIS Frontend

A276694 Where records occur in A209252.

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A265733 Number of n-digit primes whose digits include at least n-1 digits "1".

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A118118 Composite numbers that always remain composite when a single decimal digit of the number is changed.

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A346064 Number of primes that may be generated by changing any two digits of n simultaneously.

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A383271 Number of primes (excluding n) that may be generated by replacing any binary digit of n with a digit from 0 to 1.

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