A055782 -id:A055782 - cp's OEIS frontend

A232210 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation prime(n)b_k is prime, or a(n)=0 if there is no such prime.

1, 0, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 3, 1, 2, 6, 2, 2, 1, 1, 2, 1, 4, 4, 23, 1, 2, 1, 6, 2, 2, 5, 1, 10, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 4, 2, 1, 1, 1, 2, 4, 1, 2, 5, 4, 2, 3, 1, 1, 5, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 4, 2, 14, 2, 4, 1, 3

Offset: 1

Vladimir Shevelev, Sep 13 2014

Conjecture: for n>=3, a(n)>0.
Records are 1,14,23,50,252,4752,...
The corresponding primes are 2,13,131,653,883,1279,...
These primes beginning with the second one we call "stubborn primes".
Counter-conjecture: a(2889)=0. - Hans Havermann, Oct 15 2014
If a(n)=1, then the resulting primes are in A092993 and form A055782; if a(n)=2, then they form sequence 4133,4733,5333,7933,..., etc. - Vladimir Shevelev, Oct 16 2014
If a prime p divides Pb_k, then it also divides Pb_{k+m(p-1)} for all m>=0. This follows from Fermat's little theorem applied to b_x=(10^x-1)/3 with x=p-1. - M. F. Hasler, Oct 20 2014

			For n=1, start with prime(1)=2 and get already at the first step the prime 23. So a(1)=1.
For n=2, starting with prime(2)=3, one never gets a prime by appending further digits "3", therefore a(2)=0.
For n=3, n=4, n=5, one gets after the first step the primes 53, 73, 113, and therefore a(n)=1.
For n=6, start with prime(6)=13; one has to append 14 "3"s in order to get a new prime, so a(6)=14.
For n=2889, start with prime(2889) = 26293. (Do not mix up with prime(2899) = 26393...!) Appending 2k-1 or 6k-4 or 6k-2 or 18k-6 or 36k-18 or 180k-144 digits "3" yields a number divisible by 11 resp. 7 resp. 13 resp. 19 resp. 101 resp. 31. For 18k-12 and 36k (with k <> 1 (mod 5)) digits "3" there is no simple pattern and both yield sometimes large primes in the factorization, but (so far) always composite numbers 26293...3 (up to several thousand digits). - _M. F. Hasler_, Oct 16 2014

Hans Havermann, Table of n, a(n) for n = 1..2888 (first 200 terms from Michel Marcus)
Hans Havermann, "Google+ discussion relating to this sequence"
Vladimir Shevelev, "Stubborn primes"
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found

Cf. A055782, A092993, A242775, A247341, A247342, A248919.

f[n_] := Block[{k = 1, p = Prime@ n}, While[ !PrimeQ[p*10^k + (10^k - 1)/3], k++]; k]; f[2] = 0; Array[f, 100] (* Robert G. Wilson v, Apr 24 2015 *)
m3[n_]:=Module[{k=10n+3},While[!PrimeQ[k],k=10k+3];IntegerLength[k]-IntegerLength[ n]]; Join[{1,0},m3/@Prime[Range[3,90]]] (* Harvey P. Dale, Feb 11 2018 *)

a(n) = {if (n==2, return (0)); p = prime(n); k = 1; while (! isprime(p = p*10+3), k++); k;} \\ Michel Marcus, Sep 13 2014

More terms from Peter J. C. Moses, Sep 13 2014

A038800 Number of primes between 10n and 10n+9.

Original entry on oeis.org

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

Offset: 0

Jeff Burch

If n runs through the primes, the subsequence 2, 2, 2, 3, 1, 3, 2, 4, 2, 1, 3, 2, 1, 3, 1, 0, 2, 3, 2,... is created. - R. J. Mathar, Jul 19 2012

Since 431, 433, and 439 are all prime, a(43)=3. - Bobby Jacobs, Sep 25 2016

Michael De Vlieger, Table of n, a(n) for n = 0..10000
A. Frank and P. Jacqueroux, International Contest, 2001. Numerators of Item 23

Cf. A055781, A055782, A055783, A055784.
Positions of 4's: {0} U A007811.
Cf. A098592.

Table[Count[Range[10 n, 10 n + 9], p_ /; PrimeQ@ p], {n, 0, 105}] (* Michael De Vlieger, Sep 25 2016 *)
Table[PrimePi[10n+9]-PrimePi[10n],{n,0,120}] (* Harvey P. Dale, May 04 2025 *)

a(n) = primepi(10*n+9) - primepi(10*n); \\ Michel Marcus, Sep 26 2016

a(43) corrected by Bobby Jacobs, Sep 25 2016

a(101) and a(104) corrected by Michael De Vlieger, Sep 25 2016

A214342 Count of the decimal descendants of the n-th prime.

Original entry on oeis.org

23, 22, 11, 23, 1, 14, 4, 40, 15, 6, 7, 13, 1, 14, 5, 0, 9, 16, 11, 4, 15, 1, 1, 0, 3, 10, 28, 0, 12, 0, 8, 1, 1, 9, 5, 1, 4, 1, 0, 2, 0, 6, 2, 5, 10, 19, 3, 5, 5, 6, 8, 5, 7, 0, 5, 3, 5, 8, 4, 1, 2, 5, 1, 2, 2, 0, 9, 5, 0, 7, 7, 2, 11, 9, 2, 2, 0, 0, 4, 28, 0, 7

Offset: 1

Alex Ratushnyak, Jul 12 2012

Prime q is a decimal descendant of prime p if q = p*10+k and 0<=k<=9.
The number of direct decimal descendants is A038800(p).
a(n) is the total count of direct decimal descendants of the n-th prime that are also prime, plus their decimal descendants that are prime, and so on.
Conjecture: no terms bigger than 35 after a(8)=40.

			prime(3)=5 has eleven descendants: 53, 59, 593, 599, 5939, 59393, 59399, 593933, 593993, 5939333, 59393339. So a(3)=11. All candidates of the form 5nnn1 and 5nnn7 are divisible by 3.
prime(5)=11, the only decimal descendant of 11 that is prime is 113, and because there are no primes between 1130 and 1140, a(5)=1.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A214280, A055781, A055782, A055783, A055784.

A214342 := proc(n)
    option remember;
    local a,p,k,d ;
    a := 0 ;
    p := ithprime(n) ;
    for k from 0 to 9 do
        d := 10*p+k ;
        if isprime(d) then
            a := a+1+procname(numtheory[pi](d)) ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Jul 19 2012

Table[t = {Prime[n]}; cnt = 0; While[t = Select[Flatten[Table[10*i + {1, 3, 7, 9}, {i, t}]], PrimeQ]; t != {}, cnt = cnt + Length[t]]; cnt, {n, 100}] (* T. D. Noe, Jul 24 2012 *)

A327920 The 16 pure prime dates of each year of the form concatenate(day,month) with month and day also prime numbers.

Original entry on oeis.org

23, 53, 73, 113, 173, 193, 233, 293, 313, 37, 137, 197, 317, 211, 311, 2311

Offset: 1

Wolfdieter Lang, Oct 08 2019

For the case of the 16, respectively 17, pure prime dates of the form concatenate(month,day) see A327918 (non-leap years) and A327919 (leap years).
Only the three months m = 3, 7, and 11 qualify. The qualifying days are for November (m = 11): d = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and for months March and July (m = 3 and 7): d = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
The months m = 3, 7, 11 contribute 9, 4, 3 dates, respectively, adding to 16.

Cf. A055782 (first nine terms), A055783 (first four terms), A114007 (three terms starting with n = 8). A327346 (with d and m also nonprime allowed), A327918, A327919.

Select[Flatten[Table[d*10^IntegerLength[m]+m,{m,{3,7,11}},{d,Prime[ Range[ 11]]}]],PrimeQ] (* Harvey P. Dale, May 24 2022 *)

A057667 Primes q of form q = 10p + 3, where p is an odd prime.

Original entry on oeis.org

53, 73, 113, 173, 193, 233, 293, 313, 373, 433, 593, 613, 673, 733, 1013, 1033, 1093, 1373, 1493, 1733, 1913, 1933, 1973, 1993, 2113, 2273, 2293, 2333, 2393, 2633, 2693, 2713, 2833, 3313, 3373, 3533, 3593, 3673, 3733, 3793, 3833, 4013, 4093, 4493

Offset: 1

Richard Murton (LeonardoFink(AT)aol.com), Oct 17 2000, assisted by G. L. Honaker, Jr. and Jud McCranie

nonn

All terms are sums of two distinct squares.

			5413 = 541*10 + 3, 3 appended to 541.

Cf. A005384, A005385. Apart from initial term, same as A055782.

Select[Table[10*p+3, {p, Prime[Range[2,81]]}], PrimeQ] (* Metin Sariyar, Aug 09 2019 *)

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A232210 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation prime(n)b_k is prime, or a(n)=0 if there is no such prime.

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A038800 Number of primes between 10n and 10n+9.

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A214342 Count of the decimal descendants of the n-th prime.

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A327920 The 16 pure prime dates of each year of the form concatenate(day,month) with month and day also prime numbers.

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A057667 Primes q of form q = 10p + 3, where p is an odd prime.

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