A105184 -id:A105184 - cp's OEIS frontend

A248208 Primes p such that p^3 is the concatenation of two k-digit primes where k is half the number of decimal digits in p^3.

3, 11, 47, 83, 1063, 1637, 1699, 7529, 7673, 23059, 28097, 29573, 34157, 34961, 36587, 40897, 43609, 44711, 101839, 102763, 103423, 104087, 104393, 106363, 117437, 117499, 124471, 125407, 126011, 129419, 134753, 135007, 137393, 139487, 143879, 143971, 145037

Offset: 1

Derek Orr, Oct 03 2014

			47 is prime and 47^3 = 103823 is the concatenation of two primes (103 and 823) that are of the same length (here, their length is 3). So, 47 is a member of this sequence.
73 is not in the sequence since 73^3 = 389017, where 389 is a 3-digit prime but 017 is a 2-digit prime. - _Jens Kruse Andersen_, Oct 06 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A030078, A105184, A030461, A153048, A080906, A248046.

forprime(p=1,10^6,d=digits(p^3);if((#d)%2==0,if(isprime((p^3)\(10^(#d/2)))&&isprime((p^3)%(10^(#d/2)))&&#Str((p^3)%(10^(#d/2)))==#d/2,print1(p,", "))))

Terms and PARI program corrected by Jens Kruse Andersen, Oct 06 2014

A173935 a(n) is the least prime that is the concatenation of two primes in exactly n different ways.

Original entry on oeis.org

2, 23, 313, 3137, 233347, 739397, 379837313, 73932013313, 7399973479337

Offset: 0

Robert G. Wilson v, Mar 02 2010

			23 = 2 & 3, 313 = 3 & 13 and 31 & 3, etc.

Cf. A105184, A106582.

f[n_] := Block[{c = 0, id = IntegerDigits@n, k = 0}, len = Length@ id; While[ k < len, If[ Union@ PrimeQ[ FromDigits@# & /@ {id[[;; k + 1]], id[[k + 2 ;;]]}] == {True}, c++ ]; k++ ]; c]; t = Table[0, {10}]; p = 2; While[p < 10^8, a = f@p; If[ t[[a]] == 0, t[[a]] = p; Print[{a, p}]]; p = NextPrime@ p]

a(6) from Robert G. Wilson v, Mar 04 2010
a(7) from Donovan Johnson, Nov 09 2010
a(8) from Giovanni Resta, Mar 04 2014

A238647 Primes which are not the concatenation of two primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 47, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 127, 131, 139, 149, 151, 157, 163, 167, 179, 181, 191, 199, 227, 239, 251, 257, 263, 269, 277, 281, 307, 349, 401, 409, 419, 421, 431, 439, 443, 449, 457, 461

Offset: 1

Colin Barker, Mar 02 2014

223 is the first term in A141409 which is not in this sequence.

In this sequence, a prime preceded by one or more zeros is not considered to be a prime.

			59 is in the sequence because 5 is prime but 9 is not prime.
223 is not in the sequence because both 2 and 23 are primes.

Cf. A141409, A105184 (complement), A238056, A238057, A238499.

A255976 Primes that are the concatenation of two 3-digit primes.

Original entry on oeis.org

101107, 101113, 101149, 101173, 101197, 101281, 101293, 101347, 101359, 101383, 101419, 101449, 101467, 101503, 101599, 101641, 101653, 101701, 101719, 101797, 101839, 101863, 101929, 101977, 103307, 103349, 103409, 103421, 103457, 103577, 103613

Offset: 1

Zak Seidov, Mar 12 2015

The last term is a(2753)=997991.

Zak Seidov, Table of n, a(n) for n = 1..2753

A168529 is a subsequence.

Subsequence of A105184.

Select[1000 First@ # + Last@ # & /@ Permutations[Select[Range[100, 999], PrimeQ], {2}], PrimeQ] (* Michael De Vlieger, Mar 13 2015 *)

A276803 Semiprimes k such that the concatenation of its prime factors is prime.

Original entry on oeis.org

6, 21, 22, 33, 39, 46, 51, 58, 82, 93, 111, 115, 133, 141, 142, 159, 166, 177, 187, 201, 205, 219, 226, 235, 237, 247, 249, 253, 262, 267, 274, 291, 301, 319, 327, 355, 358, 391, 411, 427, 478, 489, 501, 502, 505, 511, 535, 538, 543, 562, 565, 573, 583, 586, 589

Offset: 1

K. D. Bajpai, Sep 17 2016

Alternatively: Semiprimes p*q, with p

Corresponding primes are at A105184.

			21 is a term because 21 = 3 * 7 that is a semiprime : concatenation of 3 and 7 = 37  which is prime.
142 is a term because 142 = 2 * 71 that is a semiprime : concatenation of 2 and 71 = 271 which is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A100607, A105184.

Select[Select[Range[1000], PrimeOmega[#] == 2 &], PrimeQ[FromDigits[Join[IntegerDigits [First@First[FactorInteger[#]]], IntegerDigits[First@Last[FactorInteger[#]]]]]] &]
Select[Range[1000],PrimeOmega[#]==PrimeNu[#]==2&&PrimeQ[FromDigits[ Flatten[ IntegerDigits/@FactorInteger[#][[All,1]]]]]&] (* Harvey P. Dale, Aug 03 2022 *)

list(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2, min(p,lim\p), if(isprime(eval(Str(q,p))), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Sep 17 2016

A361530 Primes that can be written as the result of shuffling the decimal digits of two primes.

Original entry on oeis.org

23, 37, 53, 73, 113, 127, 131, 137, 139, 151, 157, 173, 179, 193, 197, 211, 223, 229, 233, 239, 241, 271, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 421, 431, 433, 457, 523, 541, 547, 571, 593, 613, 617, 631, 673, 677, 719

Offset: 1

Robert C. Lyons, Mar 14 2023

Each term is essentially an element of the shuffle product of the decimal digits of two primes (possibly equal).

			37 and 73 are in the sequence because they are both the result of shuffling 3 and 7.
127 is in the sequence because it is the result of shuffling 2 and the digits of 17.
1193 is in the sequence because it is the result of shuffling the digits of 13 and the digits of 19.
163 is not in the sequence because it is not the result of shuffling the digits of two primes. 163 is the result of permuting the digits of 3 and 61; however, 163 contains the digits of 61 in the wrong order.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
John D. Cook, Shuffle product.
Wikipedia, Shuffle product.

Cf. A019549, A083427, A105184.

import sympy
def get_shuffle_product(list_1, list_2):
    shuffle_product = set()
    shuffle = []
    _get_shuffle_product(list_1, list_2, shuffle, shuffle_product)
    return shuffle_product
def _get_shuffle_product(list_1, list_2, shuffle, shuffle_product):
    if len(list_1) == 0 and len(list_2) == 0:
        shuffle_product.add(tuple(shuffle))
        return
    else:
        if len(list_1) == 0:
            shuffle.append(list_2[0])
            _get_shuffle_product(list_1, list_2[1:], shuffle, shuffle_product)
            shuffle.pop()
        elif len(list_2) == 0:
            shuffle.append(list_1[0])
            _get_shuffle_product(list_1[1:], list_2, shuffle, shuffle_product)
            shuffle.pop()
        else:
            shuffle.append(list_1[0])
            _get_shuffle_product(list_1[1:], list_2, shuffle, shuffle_product)
            shuffle.pop()
            shuffle.append(list_2[0])
            _get_shuffle_product(list_1, list_2[1:], shuffle, shuffle_product)
            shuffle.pop()
max_prime_index = 25 # one and two digit primes.
max_element = 999
prime_set = set()
for p_index in range(1, max_prime_index+1):
    p = sympy.prime(p_index)
    for q_index in range(p_index, max_prime_index+1):
        q = sympy.prime(q_index)
        list_p = list(str(p))
        list_q = list(str(q))
        shuffle_product = get_shuffle_product(list_p, list_q)
        for s in shuffle_product:
            candidate = int(''.join(s))
            if sympy.isprime(candidate) and candidate <= max_element:
                prime_set.add(candidate)
print(sorted(prime_set))

from sympy import isprime
from itertools import chain, combinations
def powerset(s): # skipping empty set and entire set
    return chain.from_iterable(combinations(s, r) for r in range(1, len(s)))
def ok(n):
    if not isprime(n): return False
    s = str(n)
    for indices in powerset(range(len(s))):
        t1 = "".join(s[i] for i in indices)
        t2 = "".join(s[i] for i in range(len(s)) if i not in indices)
        if t1[0] != "0" and t2[0] != "0" and isprime(int(t1)) and isprime(int(t2)):
            return True
print([k for k in range(720) if ok(k)]) # Michael S. Branicky, Apr 16 2023

A165631 Numbers whose cube is a concatenation of primes, i.e., in A152242.

Original entry on oeis.org

3, 7, 9, 11, 13, 15, 17, 18, 27, 28, 29, 31, 33, 38, 39, 45, 47, 48, 49, 53, 55, 58, 59, 61, 63, 68, 71, 73, 75, 83, 85, 88, 91, 95, 98, 103, 108, 111, 113, 117, 121, 125, 127, 131, 133, 135, 137, 138, 148, 153, 157, 159, 161, 163, 167, 168, 173, 175, 177, 178, 179

Offset: 1

Zak Seidov and M. F. Hasler, Oct 16 2009

Cf. A038692, A019549, A105184, A166503.

for(n=1,999, is_A152242(n^3) & print1(n", "))

Edited by Charles R Greathouse IV, Apr 24 2010

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A248208 Primes p such that p^3 is the concatenation of two k-digit primes where k is half the number of decimal digits in p^3.

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A173935 a(n) is the least prime that is the concatenation of two primes in exactly n different ways.

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Mathematica

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A238647 Primes which are not the concatenation of two primes.

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A255976 Primes that are the concatenation of two 3-digit primes.

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Mathematica

A276803 Semiprimes k such that the concatenation of its prime factors is prime.

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Mathematica

PARI

A361530 Primes that can be written as the result of shuffling the decimal digits of two primes.

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Python

Python

A165631 Numbers whose cube is a concatenation of primes, i.e., in A152242.

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Author

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Programs

PARI

Extensions