A069090 -id:A069090 - cp's OEIS frontend

A276707 Number of terms of A069090 with exactly n digits.

4, 12, 60, 381, 2522, 19094, 151286, 1237792, 10354144, 88407746, 766869330

Offset: 1

Barry Carter, Sep 15 2016

Barry Carter, Mathematica work on first n for which randomly rolled digits will yield a prime

A276707 := proc(n)
    local a,k;
    a := 0 ;
    k := nextprime(10^(n-1)) ;
    while k < 10^n do
        if isA069090(k) then
            a := a+1 ;
        end if;
        k := nextprime(k) ;
    end do:
    a ;
end proc: # R. J. Mathar, Dec 15 2016

from sympy import primerange, isprime
def ok(p):
    s = str(p)
    if len(s) == 1: return True
    return all(not isprime(int(s[:i])) for i in range(1, len(s)))
def a(n): return sum(ok(p) for p in primerange(10**(n-1), 10**n))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Jul 03 2021

# faster version skipping bad prefixes
from sympy import isprime, nextprime
def a(n):
    if n == 1: return 4
    p, c = nextprime(10**(n-1)), 0
    while p < 10**n:
        s, fail = str(p), False
        for i in range(1, n):
            ti = int(s[:i])
            if isprime(ti): fail = i; break
        if fail: p = nextprime((ti+1)*10**(n-i))
        else: p, c = nextprime(p), c+1
    return c
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Jul 03 2021

a(9)-a(11) from Michael S. Branicky, Jul 03 2021

A074721 Concatenate the primes as 2357111317192329313..., then insert commas from left to right so that between each pair of successive commas is a prime, always making the new prime as small as possible.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 2, 93137414347535961677173798389971011031071091131, 2, 7, 13, 11, 3, 7, 13, 91491511, 5, 7, 163, 167, 17, 3, 17, 9181, 19, 11, 9319, 7, 19, 9211223227229233239241251257, 2, 6326927, 127, 7, 2, 81283, 2, 93307, 3, 11, 3, 13, 3, 17, 3, 3, 13, 3, 7, 3, 47, 3, 493533593673733

Offset: 1

Reinhard Zumkeller, Sep 04 2002

Note that leading zeros are dropped. Example: When the primes 691, 701, 709, and 719 get concatenated and digitized, they become {..., 6, 9, 1, 7, 0, 1, 7, 0, 9, 7, 1, 9, ...}. These will end up in A074721 as: a(98)=691, a(99)=7, a(100)=17, a(101)=97, a(102)=19, ..., . Terms a(100) & a(101) have associated with them unstated leading zeros. - Hans Havermann, Jun 26 2009
Large terms in the links are probable primes only. For example, a(1290) has 24744 digits and a(4050), 32676 digits. If of course any probable primes were not actual primes, the indexing of subsequent terms would be altered. - Hans Havermann, Dec 28 2010
What is the next term after {2, 3, 5, 7, 11, 13, 17, 19}, if any, giving a(k)=A000040(k)?

Robert G. Wilson v, Table of n, a(n) for n = 1..329 [a(330) is too large to be included in a b-file: see the a-file]
Hans Havermann, Two-color listing of 5359 terms
Robert G. Wilson v, Table of n, a(n) for n = 1..1289

Cf. A073034, A047777, A053648, A069090.

Cf. A033308.

a074721 n = a074721_list !! (n-1)
a074721_list = f 0 $ map toInteger a033308_list where
   f c ds'@(d:ds) | a010051'' c == 1 = c : f 0 ds'
                  | otherwise = f (10 * c + d) ds
-- Reinhard Zumkeller, Mar 11 2014

id = IntegerDigits@ Array[ Prime, 3000] // Flatten; lst = {}; Do[ k = 1; While[ p = FromDigits@ Take[ id, k]; !PrimeQ@p || p == 1, k++ ]; AppendTo[lst, p]; id = Drop[id, k], {n, 1289}]

a=0;
tryd(d) = { a=a*10+d; if(isprime(a),print(a);a=0); }
try(p) = { if(p>=10,try(p\10)); tryd(p%10); }
forprime(p=2,1000,try(p)); \\ Jack Brennen, Jun 25 2009

Edited by Robert G. Wilson v, Jun 26 2009

Further edited by N. J. A. Sloane, Jun 27 2009, incorporating comments from Leroy Quet, Hans Havermann, Jack Brennen and Franklin T. Adams-Watters

A331045 a(n) is the least prime number of the form floor(n/10^k) for some k >= 0, or 0 if no such prime number exists.

Original entry on oeis.org

0, 0, 2, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0, 41, 0, 43, 0, 0, 0, 47, 0, 0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 61, 0, 0, 0, 0, 0, 67, 0, 0, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 0, 0, 0, 83

Offset: 0

Rémy Sigrist, Jan 08 2020

In other words, a(n) is the least prime prefix of n, or 0 if every prefix of n is nonprime.

This sequence is a variant of A331044.

			For n = 23:
- 2 is a prime number,
- hence a(23) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A069090, A202259, A331044.

a(n, base=10) = my (d=digits(n, base), p=0); for (k=1, #d, if (isprime(p=base*p+d[k]), return (p))); return (0)

a(n) <= n with equality iff n = 0 or n belongs to A069090.
a(n) >= 0 with equality iff n belongs to A202259.
a(n) <= A331044(n).

A223458 Primes whose first digit is a composite number.

Original entry on oeis.org

41, 43, 47, 61, 67, 83, 89, 97, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881

Offset: 1

K. D. Bajpai, Aug 24 2013

			409 is a prime number whose first digit is 4, a composite number, so 409 is a term.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A069090 (primes none of whose proper initial segments are primes).

KD := proc() local a,b,d,e;  a:= ithprime(n);  b:=length(a);  d:=a/(10^(b-1));  e:=floor(d); if isprime(e)=false and e>1 then RETURN (a): fi; end: seq(KD(),n=1..200);

from sympy import primerange
from itertools import count, islice
def agen(): yield from (p for e in count(1) for k in [4, 6, 8, 9] for p in primerange(k*10**e, (k+1)*10**e))
print(list(islice(agen(), 54))) # Michael S. Branicky, Jun 25 2022

A383779 Primes where successively deleting the least significant digit yields a sequence that alternates between a prime and a nonprime at every step until a single-digit number remains.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 41, 43, 47, 61, 67, 83, 89, 97, 211, 223, 227, 229, 241, 251, 257, 263, 269, 271, 277, 281, 283, 307, 331, 337, 347, 349, 353, 359, 367, 383, 389, 397, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 701, 709, 727, 743, 751, 757, 761, 769, 773, 787

Offset: 1

Stefano Spezia, May 09 2025

Is this sequence infinite?
Likely no, see A383780 and comment there. - Michael S. Branicky, May 11 2025
From Michael S. Branicky, May 16 2025: (Start)
Sequence is finite with 3356513448 terms (cf. A383780).
Last term: 70123916363515199416199518301698321195339012727994799190371992151279729974757397909992327936943877127375781091143 (End)

			211 is a term since 211 is a prime, 21 is a nonprime, and 2 is a prime;
23 is not a term since 23 and 2 are both prime.

Michael S. Branicky, Table of n, a(n) for n = 1..20000
Michael S. Branicky, Largest even-length (= 108 digits) then odd-length (= 113 digits) terms

Cf. A024770, A069090, A383780, A383781.

Unprotect[CompositeQ]; CompositeQ[1]:=True; Protect[CompositeQ]; Q[n_]:=And[AllTrue[FromDigits/@Table[Take[IntegerDigits[n], i], {i,IntegerLength[n],1,-2}], PrimeQ], AllTrue[FromDigits/@Table[Take[IntegerDigits[n], i], {i,IntegerLength[n]-1,1,-2}],CompositeQ]]; Select[Prime[Range[140]], Q]

from gmpy2 import is_prime, mpz
from itertools import count, islice
def agen():
    olst, elst = [2, 3, 5, 7], [11, 13, 17, 19, 41, 43, 47, 61, 67, 83, 89, 97]
    yield from olst + elst
    for n in count(1):
        olst2, elst2 = [], []
        for o in olst:
            for i in range(1, 100, 2):
                t = 100*o + i
                if is_prime(t) and not is_prime(t//10):
                    olst2.append(t)
        yield from olst2
        for e in elst:
            for i in range(100):
                t = 100*e + i
                if is_prime(t) and not is_prime(t//10):
                    elst2.append(t)
        yield from elst2
        olst, elst = olst2, elst2
print(list(islice(agen(), 70))) # Michael S. Branicky, May 11 2025

# predicate test useful for large n (cf. a-file of largest terms)
from gmpy2 import digits, is_prime, mpz
def ok(n):
    s = digits(n)
    return is_prime(n) and all(int(is_prime(mpz(s[:-i]))) == 1-i&1 for i in range(1, len(s)))
    # Michael S. Branicky, May 16 2025

A331046 Numbers k such that floor(k/10^m) is a prime number for some m >= 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 83, 89, 97, 101, 103, 107, 109, 110, 111, 112, 113

Offset: 1

Rémy Sigrist, Jan 08 2020

In other words, these are the numbers with a prime prefix.
For any m > 0:
- let f(m) be the proportion of positive numbers <= 10^m belonging to this sequence,
- we have f(m) = Sum_{p < 10^m in A069090} 1/10^A055642(p),
- also f(m) <= f(m+1) <= 1,
- so {f(m)} has a limit, say F, as m tends to infinity,
- what is the value of F?

			The number 2 is prime, so every number in A217394 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A055642, A069090, A202259 (complement), A217394, A331044, A331045.

is(n,base=10) = while (n, if (isprime(n), return (1), n\=base)); return (0)

cp's OEIS Frontend

A276707 Number of terms of A069090 with exactly n digits.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Python

Python

Extensions

A074721 Concatenate the primes as 2357111317192329313..., then insert commas from left to right so that between each pair of successive commas is a prime, always making the new prime as small as possible.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A331045 a(n) is the least prime number of the form floor(n/10^k) for some k >= 0, or 0 if no such prime number exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A223458 Primes whose first digit is a composite number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Python

A383779 Primes where successively deleting the least significant digit yields a sequence that alternates between a prime and a nonprime at every step until a single-digit number remains.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

A331046 Numbers k such that floor(k/10^m) is a prime number for some m >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI