A254754 -id:A254754 - cp's OEIS frontend

A202260 Right-truncatable composites: every decimal prefix is a composite number.

4, 6, 8, 9, 40, 42, 44, 45, 46, 48, 49, 60, 62, 63, 64, 65, 66, 68, 69, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 400, 402, 403, 404, 405, 406, 407, 408, 420, 422, 423, 424, 425, 426, 427, 428, 429, 440, 441, 442, 444, 445, 446, 447, 448

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A202259.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A012883 (right-truncatable noncomposites), A202259 (right-truncatable nonprimes), A024770 (right-truncatable primes).

Cf. A254750, A254752, A254754, A254755 (left-truncatable composites).

isComposite(n) = (n>2)&&(!isprime(n));
isRightTruncatableComposite(n,b=10) = {my(k=b);if(!isComposite(n),return(0););while(n\k>0,if(!isComposite(n\k),return(0););k*=b);return(1);} \\ Stanislav Sykora, Feb 15 2015

A254750 Numbers such that, in base 10, all their proper prefixes and suffixes represent composites.

Original entry on oeis.org

44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 89, 94, 96, 98, 99, 404, 406, 408, 409, 424, 426, 428, 444, 446, 448, 449, 454, 456, 458, 464, 466, 468, 469, 484, 486, 488, 494, 496, 498, 499, 604, 606, 608, 609, 624, 626, 628, 634, 636, 638

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of a(n) in any way into two nonempty parts, each part represents a composite number.
The list is infinite because any string of two or more digits selected from {4,6,8} represents a member.
Each member a(n) starts, as well as ends, with one of the digits {4,6,8,9}.
Every proper prefix of each member a(n) is a member of A202260, and every proper suffix is a member of A254755.
The sequence is a union of A254752 and A254754.

			6 is not a member because its expansion cannot be sliced in two.
638 is a member because (6, 38, 63, and 8) are all composites.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A202260, A254751, A254752, A254753, A254754, A254755.

isComposite(n) = (n>2)&&(!isprime(n));
slicesIntoComposites(n,b=10) = {my(k=b);if(n0,if(!isComposite(n\k)||!isComposite(n%k),return(0););k*=b);return(1);}

A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

Original entry on oeis.org

22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 237, 297, 313, 317, 373, 537, 597, 713, 717, 737, 797, 2337, 2397, 2937, 3113, 3137, 3173, 3797, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313, 739397

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of a(n) in any way into two nonempty parts, each part represents a prime number.
Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since these are finite sequences, a(n) is also finite. It has 45 members, the largest of which is 739397 and happens to be a prime.
The sequence is a union of A254753 and A020994.
A subsequence of A260181. - M. F. Hasler, Sep 16 2016

			6 is not a member because its expansion cannot be sliced in two.
597 is a member because (5,97,59, and 7) are all primes.
2331 is excluded because 233 is prime, but 1 is not. - _Gordon Hamilton_, Feb 20 2015

Cf. A020994, A024770, A024785, A254750, A254752, A254753, A254754, A254756.

Cf. A260181.

fQ[n_] := (p = {2, 3, 5, 7}; If[ Union@ Join[p, {Mod[n, 10]}] != p, {False}, Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, Union@ PrimeQ@ Flatten@ Table[{FromDigits[ Take[idn, i]], FromDigits[ Take[idn, -lng + i - 1]]}, {i, lng}] == {True}]]); Select[ Range@1000000, fQ] (* Robert G. Wilson v, Feb 21 2015 *)
Select[Range[10,750000],AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#],n],{n,IntegerLength[#]-1}]],PrimeQ]&] (* Harvey P. Dale, Feb 13 2024 *)

slicesIntoPrimes(n,b=10) = {my(k=b);if(n0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);return(1);}

def breakIntoPrimes(n):
    D=n.digits()
    for i in [1..len(D)-1]:
        if not(is_prime(sum(D[i:][j]*10^j for j in range(len(D[i:])))) and is_prime(sum(D[:i][j]*10^j for j in range(len(D[:i]))))):
            return False
        else:
            continue
    return True
[n for n in [10..1000] if breakIntoPrimes(n)] # Tom Edgar, Feb 20 2015

A254753 Composite numbers with only prime proper prefixes and suffixes in base 10.

Original entry on oeis.org

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 297, 537, 597, 713, 717, 737, 2337, 2397, 2937, 3113, 3173, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of a composite a(n) in any way into two nonempty parts, each part represents a prime number.
This sequence is a subset of A254751. Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since the latter are finite sequences, a(n) is also a finite sequence. It has 34 members, the largest of which is the composite number 73313.
Should one change the definition to 'prime numbers such that, in base 10, all their proper prefixes and suffixes represent primes', the result would be the sequence A020994.

			6 is not a member because its expansion cannot be sliced in two.
The composite 73313 is a member because (7, 3313, 73, 313, 733, 13, 7331, 3) are all primes.

Cf. A020994, A024770, A024785, A254750, A254751, A254752, A254754.

apQ[n_]:=Module[{idn=IntegerDigits[n],c1,c2},c1=FromDigits/@ Table[ Take[ idn,k],{k,Length[idn]-1}];c2=FromDigits/@Table[Take[idn,k],{k,-(Length[ idn]-1), -1}]; AllTrue[ Join[c1,c2],PrimeQ]]; Select[Range[ 10,80000], CompositeQ[#] && apQ[#]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 29 2018 *)

isComposite(n) = (n>2)&&(!isprime(n));
slicesIntoPrimes(n,b=10) = {my(k=b);if(n0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);1;}
isCompositeSlicingIntoPrimes(n,b=10) = isComposite(n) && slicesIntoPrimes(n,b);

A254752 Composite numbers such that, in base 10, all their proper prefixes and suffixes represent composites.

Original entry on oeis.org

44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 404, 406, 408, 424, 426, 428, 444, 446, 448, 454, 456, 458, 464, 466, 468, 469, 484, 486, 488, 494, 496, 498, 604, 606, 608, 609, 624, 626, 628, 634, 636, 638, 639, 644, 646, 648, 649, 654, 656, 658, 664, 666, 668, 669, 684, 686, 688, 694, 696, 698, 699, 804, 806, 808, 814, 816, 818, 824, 826, 828

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of the composite number a(n) in any way into two nonempty parts, each part represents a composite number.
This list is infinite because any string of two or more digits selected from {4,6,8} is a member.
It is a subsequence of A254750 and shares with it these properties: Each member of a(n) must start, as well as end, with one of the digits {4,6,8,9}. Every proper prefix of each member a(n) is a member of A202260, and every proper suffix is a member of A254755.

			6 is not a member because its expansion cannot be sliced in two.
The composite 469 is a member because it is a composite and the slices (4, 69, 46, and 9) are all composites.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A202260, A254750, A254751, A254753, A254754, A254755.

isComposite(n) = (n>2)&&(!isprime(n));
slicesIntoComposites(n,b=10) = {my(k=b);if(n0,if(!isComposite(n\k)||!isComposite(n%k),return(0););k*=b);return(1);}
isCompositeSlicingIntoComposites(n,b=10) = isComposite(n) && slicesIntoComposites(n,b);

A279366 Primes whose proper substrings of consecutive digits are all composite.

Original entry on oeis.org

89, 449, 499, 4649, 4969, 6469, 6869, 6949, 8669, 8699, 8849, 9649, 9949, 44699, 46649, 48649, 48869, 49669, 64849, 84869, 86969, 88469, 94849, 94949, 98869, 99469, 444469, 444869, 446969, 466649, 468869, 469849, 469969, 494699, 496669, 496849, 498469, 644669

Offset: 1

Rodrigo de O. Leite, Dec 10 2016

All digits are composite. Each term ends with the digit '9'. Since each term is prime, it never serves as the suffix of any subsequent term; e.g., no term beyond 89 ends with the digits '89', so the only remaining allowed two-digit endings are '49', '69', and '99'; no terms beyond 449 and 499 end with '449' or '499' (and '899' is ruled out because of 89), so the only remaining allowed three-digit endings are '469', '649', '669', '699', '849', '869', '949', '969', and '999' (and each of these appears as the ending of at least one four-digit term, except '999', which doesn't appear as the ending of any term until a(75) = 4696999). - Jon E. Schoenfield, Dec 10 2016

Number of terms < 10^k, k=1,2,3,...: 0, 1, 2, 10, 13, 38, 66, 197, 410, 1053, 2542, 7159, 18182, 49388, ..., . Robert G. Wilson v, Jan 15 2017

			44699 is in the sequence because 4, 6, 9, 44, 46, 69, 99, 446, 469, 669, 4469 and 4699 are composite numbers. However, 846499 is not included because 4649 is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 63 terms from Rodrigo de O. Leite)

Subsequence of A051416.

Cf. A033274, A061372, A254754.

Select[Prime@ Range[5, 10^5], Function[n, Times @@ Boole@ Map[CompositeQ, Flatten@ Map[FromDigits /@ Partition[n, #, 1] &, Range[Length@ n - 1]]] == 1]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 10 2016 *)
Select[Flatten[Table[FromDigits/@Tuples[{4,6,8,9},d],{d,6}]],PrimeQ[#]&&AllTrue[ FromDigits /@ Union[Flatten[Table[Partition[IntegerDigits[#],n,1],{n,IntegerLength[#]-1}],1]], CompositeQ]&] (* Harvey P. Dale, Jul 15 2023 *)

from sympy import isprime
from itertools import count, islice, product
def ok(n):
    s = str(n)
    if set(s) & {"1", "2", "3", "5", "7"} or not isprime(n): return False
    ss2 = set(s[i:i+l] for i in range(len(s)-1) for l in range(2, len(s)))
    return not any(isprime(int(ss)) for ss in ss2)
def agen():
    for d in count(2):
        for p in product("4689", repeat=d-1):
            t = int("".join(p)+"9")
            if ok(t): yield t
print(list(islice(agen(), 38))) # Michael S. Branicky, Oct 07 2022

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A202260 Right-truncatable composites: every decimal prefix is a composite number.

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A254750 Numbers such that, in base 10, all their proper prefixes and suffixes represent composites.

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A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

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A254753 Composite numbers with only prime proper prefixes and suffixes in base 10.

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A254752 Composite numbers such that, in base 10, all their proper prefixes and suffixes represent composites.

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A279366 Primes whose proper substrings of consecutive digits are all composite.

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