A055783 -id:A055783 - cp's OEIS frontend

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

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 *)

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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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Mathematica