A125524 -id:A125524 - cp's OEIS frontend

A125525 Centrist primes: primes such that both the right half and the left half of the prime are prime.

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 263, 277, 283, 293, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 503, 523, 547, 557, 563, 577, 587, 593, 727, 733, 743, 757, 773, 787, 797, 1103, 1117, 1123, 1129, 1153, 1171, 1303, 1307, 1319, 1361, 1367

Offset: 1

Cino Hilliard, Jan 22 2007

If the length of n > 9 is odd then the central number is not used in the calculation. So neither the left half nor the right half will contain the central digit. If the length of n is even, then all numbers are used. My guess is there are infinitely many of these numbers.

Number of n-digit terms for n=1..9: {4, 4, 33, 92, 1100, 3223, 37611, 130607, 1590017}. - Zak Seidov, Feb 19 2015

			The left half of 23 is 2 which is prime. The right half is 3 which is also prime so 23 is a centrist prime. [Corrected by _N. J. A. Sloane_, Jan 12 2019]

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A125523, A125524.

left(str, n) = /* Get the left n characters from string str */ { my(v, tmp, x); v =""; tmp = Vec(str); ln=length(tmp); if(n > ln, n=ln); for(x=1, n, v=concat(v, tmp[x]); ); return(v) }
right(str, n) = /* Get the right n characters from string str.*/ { my(v, ln, s, x); v =""; tmp = Vec(str); ln=length(tmp); if(n > ln, n=ln); s = ln-n+1; for(x=s, ln, v=concat(v, tmp[x]); ); return(v) }
/* Political primes, Centrist case */ rep(n) = { my(x,ln,y,lp,rp); forprime(x=2,n, y=Str(x); if(x > 9, ln=floor(length(y)/2), ln=1); lp = eval(left(y,ln)); rp = eval(right(y,ln)); if(isprime(lp)&& isprime(rp),print1(x",") ) ) }

Offset changed to 1 by Zak Seidov, Feb 19 2015

A125523 Democratic primes: primes such that the left half of the prime is prime and the right half is not.

Original entry on oeis.org

29, 31, 59, 71, 79, 211, 229, 239, 241, 251, 269, 271, 281, 311, 331, 349, 359, 379, 389, 509, 521, 541, 569, 571, 599, 701, 709, 719, 739, 751, 761, 769, 1109, 1151, 1163, 1181, 1187, 1193, 1301, 1321, 1327, 1381, 1399, 1709, 1721, 1733, 1777, 1787, 1901

Offset: 2

Cino Hilliard, Jan 22 2007

If the length of n is odd then the central number is not used in the calculation. So neither the left half nor the right half will contain the central digit. If the length of n is even, then all numbers are used.

			The left half of 29 is 2, which is prime. The right half is 9, which is not prime.
The left half of 211 is 2, which is prime. The right half is 1, which is not prime.

Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)

Cf. A125524, A125525.

/* Political primes, democratic case. */ dem(n) = { local(x,ln,y,lp,rp); forprime(x=2,n, y=Str(x); if(x > 9, ln=floor(length(y)/2), ln=1); lp = eval(left(y,ln)); rp = eval(right(y,ln)); if(isprime(lp)&& !isprime(rp),print1(x",") ) ) }

A125664 Numbers such that the right half of the digits form a prime and the left half do not.

Original entry on oeis.org

12, 13, 15, 17, 42, 43, 45, 47, 62, 63, 65, 67, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 122, 123, 125, 127, 132, 133, 135, 137, 142, 143, 145, 147, 152, 153, 155, 157, 162, 163, 165, 167, 172, 173, 175, 177, 182, 183, 185, 187

Offset: 1

Cino Hilliard, Jan 29 2007

If the number of digits in the number is odd > 1, then the middle digit is ignored.

			12 is the first number with this property.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A125524.

rightprime(n) = { local(x,ln,y,lp,rp); for(x=1,n, y=Str(x); if(x > 9, ln=floor(length(y)/2), ln=1); lp = eval(left(y,ln)); rp = eval(right(y,ln)); if(!isprime(lp)&& isprime(rp),print1(x",") ) ) }

from sympy import isprime
def ok(n):
    if n < 10: return False
    s = str(n)
    m = len(s)//2
    return isprime(int(s[-m:])) and not isprime(int(s[:m]))
print([k for k in range(188) if ok(k)]) # Michael S. Branicky, Dec 13 2021

The left half of an n-digit number is the first floor(n/2) digits. The right half of an n-digit number is the last floor(n/2) digits.

A337508 Primes such that neither the left half nor the right half of the prime is prime.

Original entry on oeis.org

11, 19, 41, 61, 89, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 601, 619, 631, 641, 659, 661, 691, 809, 811, 821, 829, 839, 859, 881, 911, 919, 929, 941, 971, 991, 1009, 1021, 1033, 1039, 1049, 1051

Offset: 1

Iain Fox, Aug 30 2020

For n > 9, the center digit is not considered when making the calculation. For a prime number to be in this sequence, both the substring to the left of the center and the substring to the right of the center must be nonprime.
If a number appears in this sequence, it will not appear in A125523, A125524, or A125525.
A000040 is the union of this sequence, A125523, A125524, and A125525.

			479 is prime. The left part of (4)79 is not prime. The right part of 47(9) is not prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A125523, A125524, A125525.

q:= n-> isprime(n) and (s-> (h-> not ormap(x-> isprime(parse(x)),
        [s[1..h], s[-h..-1]]))(iquo(length(s), 2)))(""||n):
select(q, [$11..2000])[];  # Alois P. Heinz, Sep 14 2020

lhrhQ[p_]:=Module[{idp=IntegerDigits[p],c},c=Floor[Length[idp]/2];AllTrue[ {FromDigits[ Take[idp,c]],FromDigits[Take[idp,-c]]},!PrimeQ[#]&]]; Select[Prime[Range[5,200]],lhrhQ] (* Harvey P. Dale, Aug 09 2023 *)

lista(nn) = forprime(p=11, nn, my(l=#Str(p), e=floor(l/2), left=floor(p/10^(e+l%2)), right=p-floor(p/10^e)*10^e); if(!isprime(left) && !isprime(right), print1(p, ", ")))

from sympy import nextprime, isprime
A337508_list, p = [], 11
while p < 10**6:
    s = str(p)
    l = len(s)//2
    if not (isprime(int(s[:l])) or isprime(int(s[-l:]))):
        A337508_list.append(p)
    p = nextprime(p) # Chai Wah Wu, Sep 14 2020

A385404 Numbers that can be split into two at any place between their digits such that the resulting numbers are always a nonprime on the left and a prime on the right.

Original entry on oeis.org

12, 13, 15, 17, 42, 43, 45, 47, 62, 63, 65, 67, 82, 83, 85, 87, 92, 93, 95, 97, 123, 143, 147, 153, 167, 183, 423, 443, 447, 453, 467, 483, 497, 623, 637, 643, 647, 653, 667, 683, 697, 813, 817, 823, 843, 847, 853, 867, 873, 883, 913, 917, 923, 937, 943, 947, 953, 967, 983, 997

Offset: 1

Tamas Sandor Nagy, Jun 27 2025

As no leading zeros are allowed, all terms are zeroless.
From Michael S. Branicky, Jun 27 2025: (Start)
Finite since each term with first digit removed must be a member of A024785, which is finite.
Last term here is a(7407) = 6357686312646216567629137. (End)

			637 is a term because when it is split in two in all possible ways, it first results in 63, a nonprime, and 3, a prime. When split in the second and final possible way, it results in 6, a nonprime, and 37, a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..7407

Cf. A125664, A125524, A024785.

q[n_] := !MemberQ[IntegerDigits[n], 0] && AllTrue[Range[IntegerLength[n]-1], PrimeQ[QuotientRemainder[n, 10^#]] == {False, True} &]; Select[Range[10, 1000], q] (* Amiram Eldar, Jun 27 2025 *)

from sympy import isprime
def ok(n): return '0' not in (s:=str(n)) and len(s) > 1 and all(not isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, Jun 27 2025

# uses import and function ok above
from itertools import count, islice, product
def agen():  # generator of terms
    tp = list("23579")  # set of left-truncatable primes
    for d in count(2):
        tpnew = []
        for f in "123456789":
            for e in tp:
                if isprime(int(s:=f+e)):
                    tpnew.append(s)
                if ok(t:=int(f+e)):
                    yield t
        tp = tpnew
        if len(tp) == 0:
            return
afull = list(agen())
print(afull[:60]) # Michael S. Branicky, Jun 27 2025

cp's OEIS Frontend

A125525 Centrist primes: primes such that both the right half and the left half of the prime are prime.

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A125523 Democratic primes: primes such that the left half of the prime is prime and the right half is not.

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A125664 Numbers such that the right half of the digits form a prime and the left half do not.

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A337508 Primes such that neither the left half nor the right half of the prime is prime.

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Maple

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A385404 Numbers that can be split into two at any place between their digits such that the resulting numbers are always a nonprime on the left and a prime on the right.

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Mathematica

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Python