A254752 -id:A254752 - 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);

A254754 Prime numbers such that, in base 10, all their proper prefixes and suffixes represent composites.

Original entry on oeis.org

89, 409, 449, 499, 809, 4049, 4549, 4649, 4909, 4969, 6299, 6469, 6869, 6899, 6949, 8009, 8039, 8069, 8209, 8609, 8669, 8699, 8849, 9049, 9209, 9649, 9949, 40009, 40099, 40609, 40639, 40699, 40849, 42209, 42649, 44249, 44699, 45949, 46049, 46099

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: Slice the decimal expansion of the prime number a(n) in any way into two nonempty parts; then both parts represent a composite number.
This sequence is a subset of A254750. Each member a(n) must start with one of the digits {4,6,8,9} and end with 9.
Every proper prefix of each member a(n) is a member of A202260, and every proper suffix is a member of A254755.
These numbers are rare and tend to become rarer with increasing n, but the sequence does not seem to terminate (for example, 4*10^28 + 9 is a member).

			7 is not a member because its expansion cannot be sliced in two.
The prime 4969 is a member because it is a prime and the slices (4, 969, 49, 69, 496, and 9) are all composites.

Stanislav Sykora, Table of n, a(n) for n = 1..20000

Cf. A202260, A254750, A254751, A254752, A254753, A254755.

Select[Prime[Range[5,5000]],AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#],n],{n,IntegerLength[ #]-1}]],CompositeQ]&] (* Harvey P. Dale, Sep 22 2024 *)

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);}
isPrimeSlicingIntoComposites(n,b=10) = isprime(n) && slicesIntoComposites(n,b);

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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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A254754 Prime numbers such that, in base 10, all their proper prefixes and suffixes represent composites.

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