A341715 -id:A341715 - cp's OEIS frontend

A341717 a(n) = smallest m >= n such that the decimal concatenation n||n+1||n+2||...||m is prime, or -1 if no such prime exists.

2, 3, 7, 5, 7, 7, 9, 187

Offset: 2

N. J. A. Sloane, Feb 22 2021

These are the values of n+k from A341715. See that entry for more information.
See A084559 (the same for nonprime indices) for further information. - M. F. Hasler, Feb 22 2021
Suppose n > 3. If n == 2 (mod 3), then a(n) == 3 or 5 (mod 6). If n = 0 or 1 (mod 3), then a(n) == 1 (mod 6) (see A341716). - Chai Wah Wu, Feb 22 2021

Cf. A084559, A140793, A341715, A341716.

Array[Block[{k = #, s = #}, While[! PrimeQ[s], k++; s = FromDigits[IntegerDigits[s]~Join~IntegerDigits[k]]]; k] &, 8, 2] (* Michael De Vlieger, Feb 22 2021 *)

A341717(n)=if(isprime(n), n, A084559(n)) \\ M. F. Hasler, Feb 22 2021

a(n) = n iff n is prime, otherwise a(n) = A084559(n). - M. F. Hasler, Feb 22 2021

A341716 a(n) = smallest k such that the decimal concatenation n||n+1||n+2||...||n+k is prime, or -1 if no such prime exists.

Original entry on oeis.org

0, 0, 3, 0, 1, 0, 1, 178

Offset: 2

N. J. A. Sloane, Feb 22 2021

These are the values of k from A341715. See that entry for more information.
From Chai Wah Wu, Feb 22 2021: (Start)
Theorem: For n > 1, a(n)+n is odd. If a(n) > 0, then a(n) !== 2 (mod 3) and a(n)+2n !== 0 (mod 3).
Proof: For n > 1, n||n+1||...||n+k needs to be odd to be prime, i.e., n+k is odd. n||n+1||...||n+k can be written as x+y where x = n||n||...||n||n and y = 1||0...2||...||0...k. The only exception is when n+i has more decimal digits than n for some i <= k in which case there are some extra zeros inserted into x and y.
In any case x is divisible by n and by 10..10..1 whose digit sum is k+1, i.e., x == n*(k+1) (mod 3).
Similarly the sum of digits of y mod 3 == Sum_{i=1..k} (i mod 3) == (Sum_{i=1..k} i) == k(k+1)/2 (mod 3). This means that x+y == (k+1)(k+2n)/2 (mod 3). If either k+1 or k+2n is a multiple of 3, then x+y is a multiple of 3. We know x+y > 3 since a(n) > 0 and thus x+y is not a prime.
Corollary: Suppose n > 3. If n == 0 (mod 6), then a(n) == 1 (mod 6). If n == 1 (mod 6), then a(n) == 0 (mod 6). If n == 2 (mod 6), then a(n) == 1 or 3 (mod 6). If n == 3 (mod 6), then a(n) == 4 (mod 6). If n == 4 (mod 6), then a(n) == 3 (mod 6). If n == 5 (mod 6), then a(n) == 0 or 4 (mod 6).
(End)

Cf. A084559, A140793, A341715, A341717.

a(n) = 0 iff n is prime, otherwise a(n) = A084559(n)-n. - M. F. Hasler, Feb 22 2021

A084559 Smallest a(n) > n such that concatenation of numbers from n to a(n) is a prime or 0 if no such number exists.

Original entry on oeis.org

3, 19, 7, 17, 7, 13, 9, 187

Offset: 2

Zak Seidov, Jun 27 2003

Terms a(1) and a(10) (and many other terms) are currently unknown.
a(11) = 309, a(12) = 13.
a(1) > 344869 (see A007908). - Sean A. Irvine, Jun 17 2019
More terms: a(14..17) = (17, 19, 43, 39), a(20) = 23, a(23) = 41, a(25) = 49, a(26) = 147, a(28) = 73, a(33..39) = (103, 37, 603, 37, 43, 57, 43), a(42) = 43, a(44) = 51, a(49) = 241, a(50) = 51, a(n) > 1000 for 12 < n < 50 not mentioned here. - M. F. Hasler, Feb 22 2021
a(10) > 10010, a(18) = 3607, a(66) = 1003, a(275) = 1089. If n == 2 (mod 3), then a(n) == 3 or 5 (mod 6). If n == 0 or 1 (mod 3), then a(n) == 1 (mod 6) (see A341716). - Chai Wah Wu, Feb 22 2021
a(10) > 50000. - Michael S. Branicky, Feb 25 2025

			a(4) = 7 because 4567 is a prime.

Eric Weisstein's World of Mathematics, Consecutive Number Sequences.
Eric Weisstein's World of Mathematics, Smarandache Prime

Cf. A007908, A030458, A030471, A052087, A052088, A052089.

Cf. also A341715, A341716, A341717 (similar but a(n) = n when n is prime).

A084559(n,N=n,T=10^logint(n,10))=while(T*=10,for(m=n+1,n=T-1,ispseudoprime(N=N*T+m)&&return(m))) \\ M. F. Hasler, Feb 22 2021

Edited by Max Alekseyev, Jan 28 2012
a(4) corrected by Daniel Suteu, Jun 16 2019
Escape clause added to definition by Chai Wah Wu, Feb 22 2021

A140793 Smallest prime which is the concatenation of 2 or more consecutive numbers starting with n, or -1 if no such prime exists.

Original entry on oeis.org

23, 345678910111213141516171819, 4567, 567891011121314151617, 67, 78910111213, 89

Offset: 2

Lekraj Beedassy, Jul 15 2008

Comment from Ray G. Opao, Sep 30 2008: a(9) is the following number:
91011121314151617181920212223242526272829303132333435363738394041424
34445464748495051525354555657585960616263646566676869707172737475767
77879808182838485868788899091929394959697989910010110210310410510610
71081091101111121131141151161171181191201211221231241251261271281291
30131132133134135136137138139140141142143144145146147148149150151152
15315415515615715815916016116216316416516616716816917017117217317417
5176177178179180181182183184185186187.

Cf. A341715, A341716, A341717.

f[n_] := Block[{c = n, k = n + 1}, While[c = 10^IntegerLength[k] c + k; p = c; !PrimeQ[c], k++]; c]; Table[f[n], {n, 2, 8}] (* Robert G. Wilson v, Jun 24 2012 *)
pr[n_]:=Module[{k=1},While[!PrimeQ[FromDigits[Flatten[IntegerDigits/@Range[n,n+k]]]],k++];FromDigits[Flatten[IntegerDigits/@Range[n,n+k]]]]; Array[pr,8,2] (* Harvey P. Dale, Mar 14 2014 *)

A140793(n,N=n,T=10^logint(n,10))=while(T*=10,for(m=n+1,n=T-1,ispseudoprime(N=N*T+m)&&return(N))) \\ M. F. Hasler, Feb 22 2021

a(n) = concatenate( n, ..., A084559(n)). - M. F. Hasler, Feb 22 2021

Added escape clause to definition. - N. J. A. Sloane, Feb 22 2021

A341701 a(n) = largest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 2, 3, 3, 5, -1, 7, -1, -1, 9, 11, -1, 13, -1, -1, -1, 17, -1, 19, -1, -1, 21, 23, 23, 13, -1, 23, -1, 29, -1, 31, -1, -1, 33, -1, -1, 37, -1, -1, -1, 41, 41, 43, -1, -1, 39, 47, 41, -1, -1, 47, 37, 53, -1, 43, 47, -1, 57, 59, 47, 61, -1, -1, -1, -1, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

A variant of A341717. a(82) = 1. Are there other n such that a(n) = 1?

Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then a(n) is odd, n-a(n) !== 2 (mod 3) and n+a(n) !== 0 (mod 3).

			a(4) = 3 since 43 is prime, a(25) = 13 since 25242322212019181716151413 is prime.

Cf. A052088, A052089, A341702, A341715, A341716, A341717.

from sympy import isprime
def A341701(n):
    k, m = n, n-1
    while not isprime(k) and m > 0:
        k = int(str(k)+str(m))
        m -= 1
    return m+1 if isprime(k) else -1

a(p) = p if and only if p is prime.

A341702 a(n) is the smallest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 0, 0, 1, 0, -1, 0, -1, -1, 1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, 1, 0, 1, 12, -1, 4, -1, 0, -1, 0, -1, -1, 1, -1, -1, 0, -1, -1, -1, 0, 1, 0, -1, -1, 7, 0, 7, -1, -1, 4, 15, 0, -1, 12, 9, -1, 1, 0, 13, 0, -1, -1, -1, -1, -1, 0, 57, -1, 1, 0, -1, 0

Offset: 0

Chai Wah Wu, Feb 23 2021

A variation of A341716. a(n) = n-1 for n = 82. Are there other n such that a(n) = n-1?

Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then n-a(n) is odd, a(n) !== 2 (mod 3) and 2n-a(n) !== 0 (mod 3).

			a(10) = 1 since 109 is prime. a(22) = 1 since 2221 is prime.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A052088, A052089, A054211, A341701, A341715, A341716, A341717.

tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
f:= proc(n) local x,k;
  x:= n;
  for k from 0 to n-1 do
    if isprime(x) then return k fi;
    x:= tcat(x,n-k-1)
  od;
  -1
end proc:
map(f, [$0..100]); # Robert Israel, Mar 02 2022

a(n) = my(k=0, s=Str(n)); while (!isprime(eval(s)), k++; n--; if (k>=n, return(-1)); s = concat(s, Str(n-k))); return(k); \\ Michel Marcus, Mar 02 2022

from sympy import isprime
def A341702(n):
    k, m = n, n-1
    while not isprime(k) and m > 0:
        k = int(str(k)+str(m))
        m -= 1
    return n-m-1 if isprime(k) else -1

a(n) = n-A341701(n).

a(p) = 0 if and only if p is prime.

A341931 a(n) = smallest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 2, 3, 3, 5, -1, 3, -1, -1, 7, 11, -1, 13, -1, -1, -1, 17, -1, 19, -1, -1, 19, 23, 23, 13, -1, 23, -1, 29, -1, 31, -1, -1, 33, -1, -1, 37, -1, -1, -1, 41, 41, 43, -1, -1, 3, 47, 17, -1, -1, 47, 37, 41, -1, 27, 47, -1, 57, 59, 47, 61, -1, -1, -1, -1, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

a(n) <= A341701(n). a(82) = 1, are there any other n such that a(n) = 1?
Primes p such that a(p) < p: 7, 53, 73, 79, 89, 103, ...
n such that a(n) < A341701(n): 7, 10, 22, 46, 48, 53, 55, 73, ...
Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then a(n) is odd, n-a(n) !== 2 (mod 3) and n+a(n) !== 0 (mod 3).

			a(7) = 3 since 76543 is prime and 765432, 7654321 are not. a(10) = 7 since 10987 is prime.

Cf. A052088, A052089, A341701, A341702, A341715, A341716, A341717, A341932.

from sympy import isprime
def A341931(n):
    k, m, r = n, n-1, n if isprime(n) else -1
    while m > 0:
        k = int(str(k)+str(m))
        if isprime(k):
            r = m
        m -= 1
    return r

A341932 a(n) = largest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 0, 0, 1, 0, -1, 4, -1, -1, 3, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, 3, 0, 1, 12, -1, 4, -1, 0, -1, 0, -1, -1, 1, -1, -1, 0, -1, -1, -1, 0, 1, 0, -1, -1, 43, 0, 31, -1, -1, 4, 15, 12, -1, 28, 9, -1, 1, 0, 13, 0, -1, -1, -1, -1, -1, 0, 57, -1, 1, 0, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

a(82) = 81, are there any other n such that a(n) = n-1?
Primes p such that a(p) > 0: 7, 53, 73, 79, 89, 103, ...
n such that a(n) > A341702(n): 7, 10, 22, 46, 48, 53, 55, 73, ...
Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then n-a(n) is odd, a(n) !== 2 (mod 3) and 2n-a(n) !== 0 (mod 3).

			a(22) = 3 since 22212019 is prime.

Cf. A052088, A052089, A341701, A341702, A341715, A341716, A341717, A341931.

from sympy import isprime
def A341932(n):
    k, m, r = n, n-1, 0 if isprime(n) else -1
    while m > 0:
        k = int(str(k)+str(m))
        if isprime(k):
            r = n-m
        m -= 1
    return r

a(n) = n-A341931(n) >= A341702(n).

A341720 a(n) = smallest k > 0 such that the decimal concatenation n||n+1||n+2||...||n+k is prime, or -1 if no such prime exists.

Original entry on oeis.org

1, 16, 3, 12, 1, 6, 1, 178

Offset: 2

N. J. A. Sloane, Feb 24 2021

If they exist, a(10) > 22496; a(21)-a(22) > 17510; a(24) > 15696; a(27) > 13641; a(29)-a(32), a(40), a(41), a(43), a(45), a(46)-a(48) > 7000; a(52) > 16359; a(54) > 16458; a(55) > 14261; a(60) > 19920; a(61) > 9029; a(11) = 298, a(12) = 1, a(13) = 7284, a(14) = 3, a(15) = 4, a(16) = 27, a(17) = 22, a(18) = 3589, a(19) = 1452, a(20) = 3, a(23) = 18, a(25) = 24, a(26) = 121, a(28) = 45, a(33) = 70, a(34) = 3, a(35) = 568, a(36) = 1, a(37) = 6, a(38) = 19, a(39) = 4, a(42) = 1, a(44) = 7, a(49) = 192, a(50) = 1, a(51) = 7228, a(53) = 430, a(56) = 1, a(57) = 4, a(58) = 3, a(59) = 18. - J.W.L. (Jan) Eerland, Feb 24 2023

If it exists, a(10) > 50000. - Michael S. Branicky, Aug 29 2024

If k=0 is allowed we get the sequences [A341715, A341716, A341717], but if k must be at least 1 we get [A140793, the present sequence, A084559].

Table[k=1;Monitor[Parallelize[While[True,If[PrimeQ[ToExpression[StringJoin[ToString/@Table[n+a,{a,0,k}]]]],Break[]];k++];k],k],{n,2,9}] (* J.W.L. (Jan) Eerland, Dec 22 2022 *)

f(n,k) = my(s=Str(n)); for(i=1, k, s=concat(s, n+i)); eval(s);
a(n) = my(k=1); while(!ispseudoprime(f(n,k)), k++); k; \\ Michel Marcus, Dec 22 2022

cp's OEIS Frontend

A341717 a(n) = smallest m >= n such that the decimal concatenation n||n+1||n+2||...||m is prime, or -1 if no such prime exists.

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A341716 a(n) = smallest k such that the decimal concatenation n||n+1||n+2||...||n+k is prime, or -1 if no such prime exists.

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A084559 Smallest a(n) > n such that concatenation of numbers from n to a(n) is a prime or 0 if no such number exists.

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A140793 Smallest prime which is the concatenation of 2 or more consecutive numbers starting with n, or -1 if no such prime exists.

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A341701 a(n) = largest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

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A341702 a(n) is the smallest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

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A341931 a(n) = smallest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

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A341932 a(n) = largest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

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