A174414 -id:A174414 - cp's OEIS frontend

A376220 Record values in A174414.

3, 9, 19, 23, 27, 37, 39, 107, 1007, 1041, 1047, 1051, 1073, 10000011, 10000047, 10000109, 1000000000000017, 1000000000000053, 1000000000000071, 1000000000000291, 1000000000000449, 10000000000000000000000000000113, 10000000000000000000000000000193, 10000000000000000000000000000249

Offset: 1

Robert Israel, Sep 16 2024

Numbers m such that for some x, the concatenation (m+x)||m is prime, and for every j < x there is some k < m such that (k+j)||k is prime.

			a(3) = 19 because A376219(3) = 11 and A174414(11) = 19.  Thus 19 is the least k such that the concatenation (k+11)||k is prime, and for all j < 11 we have (k+j)||k prime for some k < 19.

Chai Wah Wu, Table of n, a(n) for n = 1..28

Cf. A174414, A376219.

tcat:= proc(a, b) a*10^(1+ilog10(b))+b end proc:
f:= proc(n) local k, d;
    for d from 1 do
      if igcd(n, 10^d+1) > 1 then next fi;
      for k from 10^(d-1)+`if`(d=1, 0, 1) to 10^d by 2 do
        if isprime(tcat(n+k, k)) then return k fi
    od od
end proc:
R:= NULL: m:= 0:
for n from 1 to 10^6 do
   v:= f(n);
   if v > m then m:= v; R:= R, m fi
od:
R;

from itertools import count, islice
from math import gcd
from sympy import isprime
def A376220_gen(): # generator of terms
    c = 0
    for n in count(1):
        for l in count(1):
            if gcd(n,(m:=10**l)+1)==1:
                r = m//10
                a = m*(n+r)+r
                for k in range(r,m):
                    if isprime(a):
                        if k>c:
                            yield k
                            c = k
                        break
                    a += m+1
                else:
                    continue
                break
A376220_list = list(islice(A376220_gen(),22)) # Chai Wah Wu, Sep 19 2024

a(n) = A174414(A376219(n)).

A376219 Positions of records in A174414.

Original entry on oeis.org

1, 10, 11, 33, 88, 132, 341, 505, 1111, 2828, 10504, 31512, 34138, 81103, 152207, 304414, 1378751, 2587519, 2757502, 5175038, 126845092, 486699103, 883779391, 973398206, 1857177597, 1942660159, 3095295995, 7307153656

Offset: 1

Robert Israel, Sep 16 2024

Positive integers k such that the least m for which the concatenation (m+k)||m is prime is greater than it is for all previous k.
If k and 10^d+1 are not coprime, then A174414(k) can't have d digits.
Therefore, assuming A174414(n) always exists, it is unbounded and this sequence is infinite.

			a(3) = 11 because A174414(11) = 19 is greater than A174414(1),..., A174414(10).

Cf. A174414, A376220.

tcat:= proc(a,b) a*10^(1+ilog10(b))+b end proc:
f:= proc(n) local k,d;
    for d from 1 do
      if igcd(n, 10^d+1) > 1 then next fi;
      for k from 10^(d-1)+`if`(d=1,0,1) to 10^d by 2 do
        if isprime(tcat(n+k,k)) then return k fi
    od od
end proc:
J:= NULL: m:= 0:
for n from 1 to 10^6 do
   v:= f(n);
   if v > m then m:= v; J:= J,n fi
od:
J;

from itertools import count, islice
from math import gcd
from sympy import isprime
def A376219_gen(): # generator of terms
    c = 0
    for n in count(1):
        for l in count(1):
            if gcd(n,(m:=10**l)+1)==1:
                r = m//10
                a = m*(n+r)+r
                for k in range(r,m):
                    if isprime(a):
                        if k>c:
                            yield n
                            c = k
                        break
                    a += m+1
                else:
                    continue
                break
A376219_list = list(islice(A376219_gen(), 30)) # Chai Wah Wu, Sep 18 2024

a(23)-a(28) from Chai Wah Wu, Sep 20 2024

A375552 Yet unseen terms in the enumeration of A375553, prepended by [2, 5].

Original entry on oeis.org

2, 5, 3, 7, 19, 31, 11, 17, 13, 29, 23, 41, 47, 139, 61, 37, 53, 67, 59, 43, 89, 103, 109, 83, 73, 97, 79, 101, 71, 131, 167, 137, 107, 199, 163, 151, 191, 233, 113, 127, 227, 211, 179, 173, 239, 157, 193, 149, 181, 277, 313, 197, 223, 307, 251, 271, 331, 263, 241, 229, 349

Offset: 1

Peter Luschny, Sep 17 2024

nonn

Conjecture: This is a permutation of the prime numbers.

Chai Wah Wu, Table of n, a(n) for n = 1..242

Cf. A000040, A375553, A174414, A055642.

aList := proc(upto) local P, p, q, Y; Y := 2,5;
   P := select(isprime, [seq(2..upto)]):
   for p in P do for q in P do
      if isprime(q+(p+q)*10^(1+ilog10(q))) then break fi od:
   if not member(q, [Y]) then Y := Y,q fi od;
Y end: aList(100000);

spq[p_] := Module[{k = 2}, While[!PrimeQ[(p + k)*10^IntegerLength[k] + k], k = NextPrime[k]]; k];
Join[{2, 5}, DeleteDuplicates @ Table[spq[p], {p, Prime[Range[30000]]}]]
(* Jean-François Alcover, Oct 01 2024, after Harvey P. Dale in A375553 *)

f(n) = my(k=2); while (!isprime(eval(concat(Str(prime(n)+k), Str(k)))), k = nextprime(k+1)); k; \\ A375553
lista(nn) = my(list=List()); listput(list, 2); listput(list, 5); for (n=1, nn, my(k=f(n)); if (#select(x->(x==k), Vec(list)) == 0, listput(list, k));); Vec(list); \\ Michel Marcus, Sep 17 2024

from itertools import count, islice
from sympy import isprime, nextprime
def A375552_gen(): # generator of terms
    p, a = 2, set()
    yield from (2,5)
    while True:
        q, m = 2, 10
        for l in count(1):
            while qA375552_list = print(list(islice(A375552_gen(),61))) # Chai Wah Wu, Sep 18 2024

from more_itertools import unique_everseen
def f(p):
    for q in Primes():
        if is_prime(q + (p + q)*10^(1 + int(log(q, 10)))): return q
a = lambda n: unique_everseen((f(p) for p in prime_range(n)))
print([2, 5] + list(a(999)))

A375553 a(n) is the smallest prime q such that the concatenation (p + q)"q is a prime number, where p = prime(n).

Original entry on oeis.org

3, 7, 3, 3, 19, 3, 31, 3, 3, 7, 11, 17, 3, 3, 3, 3, 13, 3, 29, 3, 23, 3, 3, 7, 41, 7, 3, 3, 3, 3, 3, 31, 7, 3, 3, 3, 11, 3, 7, 19, 3, 11, 7, 11, 3, 11, 3, 23, 7, 47, 19, 3, 23, 3, 7, 3, 7, 11, 3, 3, 11, 3, 23, 7, 3, 3, 3, 29, 7, 11, 7, 3, 11, 23, 3, 3, 3, 3, 13

Offset: 1

Peter Luschny, Sep 17 2024

Conjecture: The image of this sequence joined with {2, 5} are the prime numbers, {2, 5} union imag(a) = P.

Sam Sweet, Table of n, a(n) for n = 1..500

Cf. A000040, A375552, A174414.

P := select(isprime, [seq(2..405)]):
g := p -> local q;
   for q in P do
       if isprime(q + (p + q)*10^(1 + ilog10(q))) then return q fi
   od:
map(g, P);

spq[p_]:=Module[{k=2},While[!PrimeQ[(p+k)*10^IntegerLength[k]+k],k=NextPrime[k]];k]; Table[spq[p],{p,Prime[Range[80]]}] (* Harvey P. Dale, Sep 24 2024 *)

a(n) = my(k=2); while (!isprime(eval(concat(Str(prime(n)+k), Str(k)))), k = nextprime(k+1)); k; \\ Michel Marcus, Sep 17 2024

from itertools import count
from sympy import prime, isprime, nextprime
def A375553(n):
    p, q, m = prime(n), 2, 10
    for l in count(1):
        while qChai Wah Wu, Sep 18 2024

def f(p):
    for q in Primes():
        if is_prime(q + (p + q)*10^(1 + int(log(q, 10)))): return q
print([f(p) for p in prime_range(405)])

A174583 a(k) is the least n such that the concatenation (n - k)"n is a prime number, for k >= 0.

Original entry on oeis.org

1, 3, 3, 7, 7, 17, 7, 9, 9, 11, 13, 17, 13, 19, 17, 19, 21, 21, 23, 27, 27, 23, 43, 33, 41, 27, 27, 29, 31, 33, 31, 33, 39, 47, 37, 39, 37, 39, 39, 41, 51, 47, 47, 61, 47, 49, 49, 53, 49, 51, 51, 59, 57, 57, 61, 57, 57, 61, 63, 63, 71, 63, 63, 67, 67, 77, 67, 69, 77, 71, 73, 77

Offset: 0

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 23 2010

See comments and references for A174414.
10^d*(n - k) + n has to be prime for the least d-digit n > k (k >= 0).
For (n - k)"n to be a prime, n must end in the digit 1, 3, 7, or 9.
Conjecture: a(k) = a(k+1) for an infinite number of k's.
As n > k, the number of a(k) is finite, and can be easily bounded from above.
1, 11, ... appear only once in the sequence; 3, 9, 13, 19, 21, 23, ... appear twice; 7, 17, ... three times; and so on.
Does each n that ends in the digit 1, 3, 7, or 9 appear in this sequence?
Note this interesting observation that first occurs for k = 84: 9291013 = prime(620602) = (1013 - 84)"1013, a(84) = 1013. A second example is: 9381037 = prime(626219) = (1037 - 99)"1037.
Let k be a multiple of 7, 11, or 13, then no 3-digit n exists such that (n - k)"n is prime. Proof: 10^3*(n - k) + n = n * (10^3+1) - k * 10^3 = 7 * 11 * 13 * n - k * 10^3 is not prime, as k is a multiple of 7, 11, or 13.
Similar for k-digit n with given divisors and k > 3: 10^4 + 1 = 73 * 137, 10^5 + 1 = 11 * 9091.

			11 = prime(5) = (1 - 0)"1, thus a(0) = 1.
23 = prime(9) = (3 - 1)"3, thus a(1) = 3.
13 = prime(6) = (3 - 2)"3, thus a(2) = 3.
139 = prime(34) = (39 - 38)"39, thus a(38) = 39.
9109 = prime(1130) = (109 - 100)"109, thus a(100) = 109.

Cf. A174414, A089712, A171154, A173976, A174031.

mycat := (k, n) -> parse(cat(convert(n - k, string), convert(n, string))):
sol := (k, n) -> isprime(mycat(k, n)):
a := proc(k) local n; for n from k + 1 while not sol(k, n) do od; n end:
seq(a(k), k = 0..71);  # Peter Luschny, Sep 20 2024

Edited, offset set to 0 and a(71) corrected by Peter Luschny, Sep 20 2024

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A376220 Record values in A174414.

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A376219 Positions of records in A174414.

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A375552 Yet unseen terms in the enumeration of A375553, prepended by [2, 5].

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A375553 a(n) is the smallest prime q such that the concatenation (p + q)"q is a prime number, where p = prime(n).

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Programs

Maple

Mathematica

PARI

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A174583 a(k) is the least n such that the concatenation (n - k)"n is a prime number, for k >= 0.

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