A254755 -id:A254755 - cp's OEIS frontend

A103443 Largest left-truncatable prime in base n (decimal expansion).

23, 4091, 7817, 4836525320399, 817337, 14005650767869, 1676456897, 357686312646216567629137, 2276005673, 13092430647736190817303130065827539, 812751503, 615419590422100474355767356763

Offset: 3

Martin Renner, Mar 21 2005, Sep 24 2007, Apr 20 2008

Martin Fuller, Table of n, a(n) for n = 3..29
I. O. Angell, and H. J. Godwin, On Truncatable Primes Math. Comput. 31, 265-267, 1977. (See also Sloane link below)
James Grime and Brady Haran, 357686312646216567629137, Numberphile video (2018).
Martin Renner, Table of n, a(n) for n = 3..53 (with some question marks). Corrected and expanded by Hans Havermann, Jan 25 2014.
N. J. A. Sloane, The Angell-Godwin table of initial terms of A023107 and A103443
Eric Weisstein's World of Mathematics, Truncatable Prime.
Index entries for sequences related to truncatable primes

Cf. A076623, A023107, A103463, A254755.

a(n)=my(v=primes(primepi(n-1)),u,t,b=1,best); while(#v, best=vecmax(v); b*=n; u=List(); for(i=1,#v,for(k=1,n-1,if(isprime(t=v[i]+k*b), listput(u,t)))); v=Vec(u)); best \\ Charles R Greathouse IV, Feb 05 2013

Base-14 entry corrected by Hans Havermann, May 30 2011
Corresponding entry in a-file corrected by N. J. A. Sloane, Jun 02 2011
a-file corrected and expanded by Hans Havermann, Jan 25 2014

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

Original entry on oeis.org

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);}

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);

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);

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A103443 Largest left-truncatable prime in base n (decimal expansion).

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

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

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PARI