A062584 -id:A062584 - cp's OEIS frontend

A174926 Smallest prime which has a decimal representation which shows n^2 embedded in otherwise only decimal square digits 0, 1, 4 and 9.

101, 11, 41, 19, 1601, 251, 1361, 149, 641, 811, 1009, 12101, 14401, 1699, 11969, 2251, 12569, 1289, 13241, 1361, 4001, 4441, 48409, 10529, 15761, 62501, 946769, 4729, 7841, 8419, 9001, 9619, 102409, 10891, 115601, 12251, 129641, 11369, 14449

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 02 2010

There are four decimal square digits: 0 = 0^2 = 0, 1 = 1^2, 4 = 2^2, 9 = 3^2.
It is conjectured that sequence is infinite.
Some primes of the form n^2//1 = 10 * n^2 + 1 are in this sequence: for n = 1, 2, 5, ...
Note this curiosity of a double appearance of 1361 as  1//6^2//1 = p(6^2) = 1361 = p(19^2) = 1//19^2 or of 13691 = prime(1618) = 37^2//1 > 11369 = prime(1373) = 1//37^2 = p(37^2), 38th term of sequence

			Let // denote concatenation of digits. Then:
101 = prime(26) = 1//0^2//1.
11 = prime(5) = 1^2//1.
41 = prime(13) = 2^2//1.
19 = prime(8) = 1//3^2.
1601 = prime(252) = 4^2//0//1.
251 = prime(54) = 5^2//1.
1361 = prime(218) = 1//6^2//1.
149 = prime(35) = 1//7^2.
641 = prime(116) = 8^2//1.
811 = prime(141) = 9^2//1.
1009 = prime(169) = 10^2//9.
12101 = prime(1448) = 11^2//0//1.

Cf. A174884, A062584, A113616.

A197816 Smallest composite number m such that m and the greatest prime divisor of m begin with n.

Original entry on oeis.org

102, 203, 36, 410, 50, 603, 70, 801, 970, 1010, 110, 1270, 130, 1490, 1510, 1630, 170, 1810, 190, 20030, 2110, 2230, 230, 2410, 2510, 2630, 2710, 2810, 290, 3070, 310, 32030, 3310, 3470, 3530, 3670, 370, 3830, 3970, 4010, 410, 4210, 430, 4430, 4570, 4610, 470

Offset: 1

Michel Lagneau, Oct 18 2011

A majority of numbers are divisible by 10.

The case m prime gives A062584 (First occurrence of n in the decimal representation of primes).

			a(6) = 603 = 3^2*67 => 603 and 67 start with 6.

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

Cf. A018800, A062584.

with(numtheory): for n from 1 to 47 do: l1:=length(n):i:=0:for m from 2 to 100000 while(i=0) do: x:=factorset(m):k:=nops(x):y:=x[k]: l2:=length(m):x1:=floor(m/(10^(l2-l1))): l3:=length(y):x2:=floor(y/(10^(l3-l1))):if x1=n and x2=n and l2>=l1 and l3 >=l1 and type(m,prime)=false then i:=1: printf(`%d, `,m):else fi :od:od:
# Alternative:
f:= proc(n) local d,k,p;
  for d from 1 do
    for k from 10^d*n to 10^d*(n+1)-1 do
       if not isprime(k) then
         p:= max(numtheory:-factorset(k));
         if p >= n and floor(p/10^(length(p)-length(n))) = n then return k fi
       fi od od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 04 2018

a(n) = 10*A018800(n) for n >= 9. - Robert Israel, Jun 04 2018

A377483 Smallest index k such that prime(k) in base-2 contains n in base-2 as a contiguous substring.

Original entry on oeis.org

1, 1, 2, 7, 3, 6, 4, 7, 8, 13, 5, 24, 6, 10, 11, 19, 7, 12, 8, 13, 14, 24, 9, 25, 24, 16, 17, 30, 10, 18, 11, 32, 19, 33, 20, 21, 12, 63, 22, 38, 13, 68, 14, 24, 29, 70, 15, 25, 30, 26, 27, 47, 16, 29, 48, 30, 50, 51, 17, 53, 18, 54, 31, 55, 32, 77, 19, 33, 34, 60, 20, 79

Offset: 1

Charles Marsden, Oct 29 2024

The intersections between this sequence and similar sequences in base-B occur at values of n that are the sequence of prime numbers, and values of a(n) that are the sequence of positive integers.

			For n=1 -> 1 in base-2. The first prime containing 1 in its base-2 form is prime(1)=2 -> 10. Therefore, a(1)=1.
For n=3 -> 11 in base-2. The first prime containing 11 in its base-2 form is prime(2)=3 -> 11. Therefore, a(3)=2.
For n=5 -> 101 in base-2. The first prime containing 101 in its base-2 form is prime(3)=5 -> 101. Therefore, a(5)=3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Charles Marsden, Python program

Cf. A000040, A004676, A062584, A082058.

s={}; Do[k=0;Until[SequenceCount[IntegerDigits[Prime[k],2],IntegerDigits[n,2]]>0,k++]; AppendTo[s,k],{n,72}];s (* James C. McMahon, Nov 20 2024 *)

a(n) = { my (w = 2^#binary(n), k = 0, r); forprime (p = 2, oo, k++; r = p; while (r >= n, if (r % w == n, return (k), r \= 2;););); } \\ Rémy Sigrist, Nov 20 2024

Python
```
# See links.
```

from sympy import nextprime, primepi
def A377483(n):
    p, k, a = nextprime(n-1), primepi(n-1)+1, bin(n)[2:]
    while True:
        if a in bin(p)[2:]:
            return k
        p = nextprime(p)
        k += 1 # Chai Wah Wu, Nov 20 2024

A381606 a(n) is the smallest prime number greater than n that contains n as a substring of its digits.

Original entry on oeis.org

101, 11, 23, 13, 41, 53, 61, 17, 83, 19, 101, 113, 127, 113, 149, 151, 163, 173, 181, 191, 1201, 211, 223, 223, 241, 251, 263, 127, 281, 229, 307, 131, 1321, 233, 347, 353, 367, 137, 383, 139, 401, 241, 421, 431, 443, 457, 461, 347, 487, 149, 503, 151, 521, 353, 541

Offset: 0

Joost de Winter, Mar 01 2025

			The first prime number greater than 0 that contains "0" is 101, so a(0) = 101.
The first prime number greater than 1 that contains "1" is 11, so a(1) = 11.
The first prime number greater than 2 that contains "2" is 23, so a(2) = 23.

Joost de Winter, Table of n, a(n) for n = 0..9000
David A. Corneth, PARI program
Joost de Winter, MATLAB function

Cf. A062584, A354114, A381099, A064735.

MATLAB
```
\\ See De Winter link
```

f:= proc(n) local m,d,d1,x,y,L;
  m:= length(n);
  for d from 1 do
    L:= sort([seq(10^d * n + x, x = 1 .. 10^d-1, 2),
             seq(n+10^m*x, x=10^(d-1) .. 10^d-1),
             seq(seq(seq(10^d1*n + x + 10^(m+d1)*y, x=1 .. 10^d1-1,2),y=10^(d-d1-1) .. 10^(d-d1)-1),d1=1..d-1)]);
    for x in L do if isprime(x) then return x fi od
  od
end proc:
f(0):= 101:
map(f, [$0..100]); # Robert Israel, Mar 02 2025

a[n_] := Module[{p = NextPrime[n + 1], s = ToString[n]}, While[! StringContainsQ[ToString[p], s], p = NextPrime[p]]; p]; Array[a, 100, 0] (* Amiram Eldar, Mar 03 2025 *)

a(n) = my(p=nextprime(n+1), s=Str(n)); while (#strsplit(Str(p), s) < 2, p = nextprime(p+1)); p; \\ Michel Marcus, Mar 01 2025

PARI
```
\\ See Corneth link
```

a(n) > 2n. For large enough n, a(n) < n^5 by the strongest known version of Linnik's theorem. - Charles R Greathouse IV, Mar 01 2025

A065144 Smallest prime containing the n-th square in decimal notation.

Original entry on oeis.org

11, 41, 19, 163, 251, 367, 149, 641, 181, 1009, 1213, 1447, 1693, 11969, 2251, 12569, 1289, 13241, 1361, 4001, 2441, 14843, 3529, 15761, 6257, 6761, 2729, 7841, 6841, 9001, 6961, 10243, 10891, 115601, 12251, 12967, 11369, 14447, 15217, 16001

Offset: 1

Reinhard Zumkeller, Oct 17 2001

Essentially the same as A029948. [From R. J. Mathar, Oct 02 2008]

A062584, A000290, A000040, A065145.

A089754 Smallest prime ending (least significant side) in n if possible else beginning in n.

Original entry on oeis.org

11, 2, 3, 41, 5, 61, 7, 83, 19, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 2003, 421, 223, 23, 241, 251, 263, 127, 281, 29, 307, 31, 3203, 233, 347, 353, 367, 37, 383, 139, 401, 41, 421, 43, 443, 457, 461, 47, 487, 149, 503, 151, 521, 53, 541, 557, 563, 157, 587, 59, 601, 61, 6203, 163

Offset: 1

Amarnath Murthy, Nov 22 2003

Cf. A018800, A062584, A068164.

read("transforms)"):
nleadzero := proc(n,dgs)
    local ndgs ;
    if n = 0 then
        ndgs := 0 ;
    else
        ndgs := ilog10(n)+1 ;
    end if;
    [op(convert(n,base,10)),seq(0,i=1..dgs-ndgs)]
end proc:
digcatL2 := proc(L1,L2)
    digcatL([op(L1),op(L2)]) ;
end proc:
A089754 := proc(n)
    local prep,a,dgs ;
    if isprime(n) then
        n;
    elif modp(n,2) = 0 or modp(n,5) = 0 then
        ndgs := convert(n,base,10) ;
        for dgs from 1 do
            for prep from 0 to 10^dgs-1 do
                a := digcatL2(ListTools[Reverse](ndgs),ListTools[Reverse](nleadzero(prep,dgs))) ;
                if isprime(a) then
                    return a;
                end if;
            end do:
        end do:
    else
        for prep from 1 do
            a := digcat2(prep,n) ;
            if isprime(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 18 2015

Corrected by R. J. Mathar, Sep 18 2015

A103611 Smallest prime p with at least two non-overlapping occurrences of n in decimal representation of p.

Original entry on oeis.org

11, 223, 233, 443, 557, 661, 277, 881, 199, 10103, 11113, 112121, 13313, 14143, 115151, 116167, 11717, 18181, 19219, 20201, 21121, 22229, 12323, 24247, 25253, 26261, 22727, 28283, 29129, 30307, 23131, 32321, 23333, 134341, 35353, 136361

Offset: 1

Reinhard Zumkeller, Mar 24 2005

A062584(n) <= a(n).

Harvey P. Dale, Table of n, a(n) for n = 1..100

With[{prs=Prime[Range[15000]]},Table[SelectFirst[prs, SequenceCount[ IntegerDigits[ #], IntegerDigits[n]]>1&],{n,40}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 30 2016 *)

A178546 Emirps in which n first occurs as a substring.

Original entry on oeis.org

107, 13, 1021, 13, 149, 157, 167, 17, 389, 79, 107, 113, 1201, 13, 149, 157, 167, 17, 1181, 199, 1201, 1021, 1223, 1223, 1249, 1259, 12269, 1279, 1283, 1229, 1301, 31, 1321, 337, 347, 359, 3613, 37, 389, 739, 1409, 941, 1429, 743, 11447, 1453, 3463, 347, 1487, 149, 7507

Offset: 0

Lekraj Beedassy, May 29 2010

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A006567, A062584.

emrpQ[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn]; idn!=ridn && PrimeQ[ FromDigits[ ridn]]]; nss[n_]:= Module[{idn=IntegerDigits[n]}, Select[e,MemberQ[Partition[ IntegerDigits[#],Length[idn],1],idn]&,1]]; With[{e=Select[Prime[Range[50000]],emrpQ]},Flatten[ Join[{107}, Table[ nss[i], {i,50}]]]] (* Harvey P. Dale, Oct 08 2012 *)

Corrected and extended by Harvey P. Dale, Oct 08 2012

A246398 Smallest number of digits that have to be inserted into or prepended to decimal representation of n, to get a prime.

Original entry on oeis.org

2, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1

Offset: 0

Reinhard Zumkeller, Nov 13 2014

a(n) = 0 iff n is a prime number, a(A000040(n)) = 0;

a(n) = A055642(A062584(n)) - A055642(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000040, A055642, A062584, A068164.

a246398 n = a246398_list !! n
a246398_list = f 0 $ map show a000040_list where
   f x pss = (length ps - length xs) :
             f (x + 1) (dropWhile (== xs) pss)
     where ps = head [qs | qs <- pss, isin xs qs]; xs = show x
   isin [] _  = True
   isin _  [] = False
   isin (u:us) vs = not (null ws) && isin us ws
                    where ws = dropWhile (/= u) vs

A263450 Smallest integer k>0 such that there is at least one zero in the decimal representation of prime(n)^k.

Original entry on oeis.org

10, 10, 8, 4, 5, 6, 7, 4, 6, 4, 6, 3, 5, 3, 2, 2, 3, 5, 3, 2, 3, 3, 5, 3, 2, 1, 1, 1, 1, 4, 3, 3, 6, 4, 2, 2, 4, 3, 5, 4, 2, 4, 4, 3, 2, 2, 5, 3, 3, 3, 6, 4, 2, 2, 2, 4, 3, 3, 5, 3, 2, 4, 1, 3, 3, 2, 2, 6, 2, 2, 2, 4, 3, 5, 4, 6, 4, 2, 1, 1, 3, 4, 3, 5, 3, 3, 2, 2, 5

Offset: 1

Zak Seidov, Oct 18 2015

Conjecture: there are an infinite number of ones in the sequence.
Corresponding values of prime(n)^k: 1024, 59049, 390625, 2401, 161051, 4826809, 410338673, 130321 (not yet in OEIS).
From Robert Israel, Oct 19 2015: (Start)
By Dirichlet's theorem there are infinitely many n for which prime(n) == 1 (mod 100), and these all have a(n) = 1.
All a(n) <= 20, since every x coprime to 10 has x^20 == 1 (mod 100). (End)

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

Cf. A062584, A071531, A103662, A103663.

f:= proc(m) local k;
for k from 1 do
     if has(convert(m^k,base,10),0) then return k fi
   od
end proc:
seq(f(ithprime(i)), i=1..1000); # Robert Israel, Oct 19 2015

Reap[Do[p=Prime[n];k=1;While[Min[IntegerDigits[p^k]]>0,k++];Sow[k],{n,1,200}]][[2,1]]

a(n) = {p = prime(n); k = 1; while (vecmin(digits(p^k)), k++); k;} \\ Michel Marcus, Oct 21 2015

a(n) = A071531(prime(n)). - Michel Marcus, Oct 21 2015

cp's OEIS Frontend

A174926 Smallest prime which has a decimal representation which shows n^2 embedded in otherwise only decimal square digits 0, 1, 4 and 9.

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A197816 Smallest composite number m such that m and the greatest prime divisor of m begin with n.

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A377483 Smallest index k such that prime(k) in base-2 contains n in base-2 as a contiguous substring.

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PARI

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A381606 a(n) is the smallest prime number greater than n that contains n as a substring of its digits.

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MATLAB

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A065144 Smallest prime containing the n-th square in decimal notation.

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A089754 Smallest prime ending (least significant side) in n if possible else beginning in n.

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Maple

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A103611 Smallest prime p with at least two non-overlapping occurrences of n in decimal representation of p.

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Mathematica

A178546 Emirps in which n first occurs as a substring.

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Mathematica

Extensions

A246398 Smallest number of digits that have to be inserted into or prepended to decimal representation of n, to get a prime.

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