A124665 -id:A124665 - cp's OEIS frontend

A356272 a(n) is the least k such that exactly n consecutive integers starting from k belong to A124665.

20, 134, 1934, 9773, 19042, 138902, 104024, 512255, 1400180, 1558490, 1441174, 9363253, 20454244, 98854550, 57515874, 201139683, 49085531, 213492618, 475478220, 1152519092

Offset: 1

Michel Marcus, Aug 01 2022

			20 is in A124665, while 19 and 21 are not. This is the 1st such integer so a(1) = 20.
134 and 135 are in A124665, while 133 and 136 are not. This is the 1st such integer so a(2) = 134.

Cf. A124665.

is(n)=my(N=10*n, D=10^#Str(n)); forstep(k=n, n+9*D, D, if(isprime(k), return(0))); !(isprime(N+1)||isprime(N+3)||isprime(N+7)||isprime(N+9)); \\ A124665
isok(k, n) = if (!is(k-1), for (i=0, n-1, if (!is(k+i), return (0));); !is(k+n););
a(n) = my(k=2); while (!isok(k, n), k++); k;

from itertools import islice
from sympy import isprime
def ok(n):
    s = str(n)
    if any(isprime(int(s+c)) for c in "1379"): return False
    return not any(isprime(int(c+s)) for c in "0123456789")
def agen():
    adict = dict()
    n = k = 1
    while True:
        c = 0
        while ok(k): k += 1; c += 1
        if c not in adict: adict[c] = k-c
        while n in adict: yield adict[n]; n += 1
        k += 1
print(list(islice(agen(), 6))) # Michael S. Branicky, Aug 01 2022

a(14)-a(20) from Michael S. Branicky, Aug 02 2022

A032352 Numbers k such that there is no prime between 10k and 10k+9.

Original entry on oeis.org

20, 32, 51, 53, 62, 84, 89, 107, 113, 114, 126, 133, 134, 135, 141, 146, 150, 164, 167, 168, 171, 176, 179, 185, 189, 192, 196, 204, 207, 210, 218, 219, 232, 236, 240, 248, 249, 251, 256, 258, 282, 294, 298, 305, 309, 314, 315, 317, 323, 324, 326, 328, 342

Offset: 1

Jeff Burch

Numbers k with property that appending any single decimal digit to k does not produce a prime.

A007920(n*10) > 10.

			m=32: 321=3*107, 323=17*19, 325=5*5*13, 327=3*109, 329=7*47, therefore 32 is a term.

M. I. Wilczynski, Table of n, a(n) for n = 1..10000

Cf. A124665 (subsequence), A010051, A007811, A216292, A216293.

a032352 n = a032352_list !! (n-1)
a032352_list = filter
   (\x -> all (== 0) $ map (a010051 . (10*x +)) [1..9]) [1..]
-- Reinhard Zumkeller, Oct 22 2011

[n: n in [1..350] | IsZero(#PrimesInInterval(10*n, 10*n+9))]; // Bruno Berselli, Sep 04 2012

a:=proc(n) if map(isprime,{seq(10*n+j,j=1..9)})={false} then n else fi end: seq(a(n),n=1..350); # Emeric Deutsch, Aug 01 2005

f[n_] := PrimePi[10n + 10] - PrimePi[10n]; Select[ Range[342], f[ # ] == 0 &] (* Robert G. Wilson v, Sep 24 2004 *)
Select[Range[342], NextPrime[10 # ] > 10 # + 9 &] (* Maciej Ireneusz Wilczynski, Jul 18 2010 *)
Flatten@Position[10*#+{1,3,7,9}&/@Range@4000,{?CompositeQ ..}] (* _Hans Rudolf Widmer, Jul 06 2024 *)

is(n)=!isprime(10*n+1) && !isprime(10*n+3) && !isprime(10*n+7) && !isprime(10*n+9) \\ Charles R Greathouse IV, Mar 29 2013

from sympy import isprime
def aupto(limit):
  alst = []
  for d in range(2, limit+1):
    td = [10*d + j for j in [1, 3, 7, 9]]
    if not any(isprime(t) for t in td): alst.append(d)
  return alst
print(aupto(342)) # Michael S. Branicky, May 30 2021

a(n) ~ n. - Charles R Greathouse IV, Mar 29 2013

More terms from Miklos Kristof, Aug 27 2002

A065502 Positive numbers divisible by 2 or 5; 1/n not purely periodic after decimal point.

Original entry on oeis.org

2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 55, 56, 58, 60, 62, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106, 108, 110, 112, 114

Offset: 1

Len Smiley, Nov 25 2001

Complement of A045572. - Reinhard Zumkeller, Nov 15 2009
Numbers that cannot be prefixed by a single digit to form a prime in decimal representation: A124665 is a subsequence. - Reinhard Zumkeller, Oct 22 2011
Up to 198, this is almost identical to "a(n) = n such that 3^n-1 is not squarefree", with the only exceptions being 39 and 117, which are not in this sequence. Why is that? - Felix Fröhlich, Oct 19 2014
The asymptotic density of this sequence is 3/5. - Amiram Eldar, Mar 09 2021

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A000035, A001622, A045572, A051628, A079998, A124665, A047229 (numbers divisible by 2 or 3).

a065502 n = a065502_list !! (n-1)
a065502_list = filter ((> 1) . (gcd 10)) [1..]
-- Reinhard Zumkeller, Oct 22 2011

A065502 := proc(n)
     option remember;
     if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if (a mod 2) =0 or (a mod 5) =0 then
                return a;
            end if;
        end do:
    end if;
end proc; # R. J. Mathar, Jul 20 2012

Select[Range[114], Mod[#, 2] == 0 || Mod[#, 5] == 0 &] (* T. D. Noe, Jul 13 2012 *)
Select[ Range@ 114, MemberQ[{0, 2, 4, 5, 6, 8}, Mod[#, 10]] &] (* Robert G. Wilson v, May 22 2014 *)

isok(m) = ! ((m%2) && (m%5)); \\ Michel Marcus, Mar 09 2021

A000035(a(n))*(1-A079998(a(n)))=0. - Reinhard Zumkeller, Nov 15 2009
G.f.: x*(2*x^4+x^2+2) / ((x-1)^2*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jul 18 2013
a(n) = 10*floor(n/6)+s(n mod 6)-floor(((n-1)mod 6)/5), where s(n) = n+1+floor((n+1)/3). - Gary Detlefs, Oct 05 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/5 + log(phi)/sqrt(5), where phi is the golden ratio (A001622). - Amiram Eldar, Dec 28 2021

Offset changed from 0 to 1 by Harry J. Smith, Oct 20 2009

A124666 Numbers ending in 1, 3, 7 or 9 such that either prepending or following them by one digit doesn't produce a prime.

Original entry on oeis.org

891, 921, 1029, 1037, 1653, 1763, 1857, 2427, 2513, 2519, 2607, 3111, 3193, 3213, 3501, 3519, 3707, 3953, 4227, 4459, 4599, 4689, 4803, 4863, 5019, 5043, 5047, 5397, 5459, 5489, 5499, 6019, 6023, 6429, 6483, 6609, 6621, 7113

Offset: 1

Tanya Khovanova, Dec 23 2006

If the number doesn't end in 1, 3, 7 or 9, then the prepending requirement is automatically satisfied. Hence it becomes nonrestrictive and not very interesting.

			The definition means that 891, 1891, 2891, 3891, 4891, 5891, 6891, 7891, 8891, 9891, 8911, 8913, 8917 and 8919 are all composite numbers.

Harvey P. Dale, Table of n, a(n) for n = 1..4000

Cf. A124665.

dppQ[n_]:=AllTrue[Join[{n},Table[m*10^IntegerLength[n]+n,{m,9}],Table[ n*10+k,{k,{1,3,7,9}}]],CompositeQ]; Select[Range[8000],MemberQ[ {1,3,7,9},Mod[ #,10]]&&dppQ[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 19 2018 *)

from sympy import isprime
def ok(n):
    s = str(n)
    if s[-1] not in "1379": return False
    if any(isprime(int(s+c)) for c in "1379"): return False
    return not any(isprime(int(c+s)) for c in "0123456789")
print([k for k in range(7114) if ok(k)]) # Michael S. Branicky, Aug 02 2022

A125268 Numbers that end with decimal digit 1, 3, 7, or 9 and that produce only composite numbers when any of the digits 0,1,...,9 is inserted anywhere in them (including at the beginning or end).

Original entry on oeis.org

25011, 52647, 72753, 122313, 168699, 283251, 324021, 598041, 783441, 804131, 837207, 924807, 1247241, 1905759, 2514819, 3461101, 3514077, 3617389, 3905817, 4112913, 4142139, 4203151, 4229871, 4283679, 4531907, 4628827, 4828443, 5380413, 5478091, 5632671, 5714889, 5818569, 5989269, 5990961

Offset: 1

I. J. Kennedy, Jan 15 2007

Since digit 0 can be inserted at the beginning of a term, each term must be composite.

Max Alekseyev, Table of n, a(n) for n = 1..972 (all terms below 10^8)

Cf. A003459, A124665

filter:= proc(n) local x,y,d,t;
  x:= n; y:= 0;
  for d from 0 to ilog10(n)+1 do
     for t from 0 to 9 do
       if isprime(10^(d+1)*x+10^d*t + y) then return false fi;
     od;
     t:= x mod 10;
     y:= y + 10^d*t;
     x:= (x-t)/10;
  od;
  true
end proc:
select(filter, [seq(seq(10*i+j,j=[1,3,7,9]),i=0..10^6)]); # Robert Israel, Sep 12 2016

{ printA125268(U=8) = my(v,t); v=vector(10^U); forprime(p=11,10^(U+1), if(p<=U,v[p]=p); for(i=1,#Str(p), t=(p\10^i) * 10^(i-1) + (p%10^(i-1)); if(#Str(t)==#Str(p)-1,v[t]=p););); forstep(n=1,10^U,2, if(n%10==5||v[n],next); print1(n,", ");); } \\ prints terms below 10^U, by Max Alekseyev, Sep 12 2016

Corrected and extended by Robert G. Wilson v, Jan 26 2007

Removed incorrect terms and extended by Max Alekseyev, Sep 12 2016

A224953 Number of ways a digit can be appended or prepended to n and form a prime.

Original entry on oeis.org

4, 9, 3, 9, 3, 3, 2, 9, 2, 6, 4, 6, 1, 7, 1, 2, 2, 5, 1, 9, 0, 4, 3, 6, 1, 2, 2, 6, 2, 5, 1, 8, 0, 5, 2, 2, 1, 6, 2, 6, 2, 6, 1, 7, 2, 1, 3, 6, 1, 5, 2, 3, 2, 5, 2, 1, 2, 8, 1, 6, 2, 7, 0, 6, 3, 2, 1, 7, 1, 4, 2, 5, 1, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 7, 0, 3, 1

Offset: 0

Christian N. K. Anderson and Kevin L. Schwartz, Apr 20 2013

The prime number may be formed by adding a digit either before or after n, though only odd numbers can become prime by having digits added before n.
Appending a zero before n produces a prime if and only if n is prime. Conversely, for all prime numbers p, a(p) > 0.
In theory, a maximum of 7 digits could be added before any n, and 3 of the odd digits after n in cases where [10*n, 10*n+9] contains a number that is a factor of 3, 5 and 7 (the three single-digit odd primes). In practice, it appears that all 10 possibilities are never realized. There are 9 possibilities for n = {1, 3, 7, 19}.
The only example of a prime being formed two different ways is for n = 1, which can become 11 if a 1 is appended to either the front or the back. These are naively counted as two distinct alternatives. [This would also be true for n = A002275(A004023(k) - 1) for k > 1 as appending a 1 to either the front or the back forms the k-th repunit prime. - Michael S. Branicky, May 22 2024]
The term a(29587) is the first occurrence of 10. The primes are 29587, 129587, 329587, 429587, 729587, 929587, 295871, 295873, 295877, and 295879. This is the only occurrence of 10 for n < 10^8. - T. D. Noe, Apr 21 2013

			a(0) = 4 because there are 4 ways to concatenate a digit to 0 to produce a prime number: 02, 03, 05, and 07.
a(3) = 9 because a digit can be concatenated to 3 in 9 ways to produce a prime number: 03, 13, 23, 43, 53, 73, 83, 31, and 37.

Christian N. K. Anderson, Table of n, a(n) for n = 0..9999
Christian N. K. Anderson, Primes formed by appending digits to n for n = 0..9999

Cf. A087388, A087389, A087390.
Cf. A069686.
Cf. A075595.
Cr. A002275, A004023.
Index of zeros in this sequence: A124665.

Table[num = IntegerDigits[n]; cnt = 0; Do[If[PrimeQ[FromDigits[Prepend[num, k]]], cnt++], {k, 0, 9}]; Do[If[PrimeQ[FromDigits[Append[num, k]]], cnt++], {k, 0, 9}]; cnt, {n, 0, 86}] (* T. D. Noe, Apr 20 2013 *)

sapply(1:100, function(x) sum(sapply(as.numeric(c(paste0(0:9,x), paste0(x,c(1,3,7,9)))), is_prime  ))) # Christian N. K. Anderson, Apr 30 2024

A356273 a(n) is the position of the least prime in the ordered set of numbers obtained by inserting/placing any digit anywhere in the digits of n (except a zero before 1st digit), or 0 if there is no prime in that set.

Original entry on oeis.org

2, 5, 1, 5, 8, 7, 1, 11, 1, 2, 1, 10, 1, 14, 7, 10, 1, 10, 1, 0, 4, 7, 4, 7, 8, 11, 1, 11, 4, 10, 1, 0, 2, 14, 11, 16, 1, 14, 1, 5, 2, 7, 8, 11, 16, 11, 3, 19, 1, 8, 1, 8, 3, 10, 17, 14, 1, 20, 3, 7, 4, 0, 1, 11, 14, 13, 1, 17, 2, 8, 2, 16, 1, 14, 13, 14, 2, 22, 1, 17

Offset: 1

Michel Marcus, Aug 01 2022

It appears that a(n) = 0 for n in A124665.

891, a term of A124665 and with a(891) = 9, is the first counterexample. - Michael S. Branicky, Aug 01 2022

			For n=1, the resulting set is (10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91) where the least prime 11 is at position 2, so a(1) = 2.
For n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where the least prime 23 is at position 5, so a(2) = 5.

Michel Marcus, Table of n, a(n) for n = 1..10000

Related to the process in A068166, A068167, A068169, A068170, A068171, A068172, A068173, and A068174.

Cf. A124665.

Table[Function[w, If[IntegerQ[#], #, 0] &@ FirstPosition[Rest@ Union@ Flatten@ Table[FromDigits@ Insert[w, d, k], {k, Length[w] + 1, 1, -1}, {d, 0, 9}], ?PrimeQ][[1]]][IntegerDigits[n]], {n, 80}] (* _Michael De Vlieger, Aug 01 2022 *)

get(d, rd, n, k) = {if (n==0, return(fromdigits(concat(d, k)))); if (n==#d, return(fromdigits(concat(k, d)))); my(v = concat(Vec(d, #d-n), k)); v = concat(v, Vecrev(Vec(rd, n))); fromdigits(v);}
a(n) = {my(d=digits(n), rd = Vecrev(d), list = List(), p); for (n=0, #d, my(kstart = if (n==#d, 1, 0)); for (k=kstart, 9, listput(list, get(d, rd, n, k)););); my(svec = Set(Vec(list))); for (k=1, #svec, if (isprime(svec[k]), return(k)););}

from sympy import isprime
def a(n):
    s = str(n)
    out = set(s[:i]+c+s[i:] for i in range(len(s)+1) for c in "0123456789")
    out = sorted(int(k) for k in out if k[0] != "0")
    ptest = (i for i, k in enumerate(sorted(out), 1) if isprime(int(k)))
    return next(ptest, 0)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Aug 01 2022

cp's OEIS Frontend

A356272 a(n) is the least k such that exactly n consecutive integers starting from k belong to A124665.

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A032352 Numbers k such that there is no prime between 10*k and 10*k+9.

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A065502 Positive numbers divisible by 2 or 5; 1/n not purely periodic after decimal point.

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A124666 Numbers ending in 1, 3, 7 or 9 such that either prepending or following them by one digit doesn't produce a prime.

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A125268 Numbers that end with decimal digit 1, 3, 7, or 9 and that produce only composite numbers when any of the digits 0,1,...,9 is inserted anywhere in them (including at the beginning or end).

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A224953 Number of ways a digit can be appended or prepended to n and form a prime.

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A356273 a(n) is the position of the least prime in the ordered set of numbers obtained by inserting/placing any digit anywhere in the digits of n (except a zero before 1st digit), or 0 if there is no prime in that set.

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A032352 Numbers k such that there is no prime between 10k and 10k+9.