A020994 -id:A020994 - cp's OEIS frontend

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

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);

A323137 Largest prime that is both left-truncatable and right-truncatable in base n.

Original entry on oeis.org

23, 11, 67, 839, 37, 1867, 173, 739397, 79, 105691, 379, 37573, 647, 3389, 631, 202715129, 211, 155863, 1283, 787817, 439, 109893629, 577, 4195880189, 1811, 14474071, 379, 21335388527, 2203, 1043557, 2939, 42741029, 2767, 50764713107, 853, 65467229, 4409, 8524002457

Offset: 3

Felix Fröhlich, Jan 05 2019

			For n = 12: 105691 is 511B7 in base 12. Successively removing the leftmost digit yields the base-12 numbers 11B7, 1B7, B7 and 7. When converted to base 10, these are 2011, 283, 139 and 7, respectively, all primes. Successively removing the rightmost digit yields the base-12 numbers 511B, 511, 51 and 5. When converted to base 10, these are 8807, 733, 61 and 5, respectively, all primes. Since no larger prime with this property in base 12 exists (as proven by Daniel Suteu), a(12) = 105691.

Daniel Suteu, Table of n, a(n) for n = 3..64
Wikipedia, Truncatable prime

Cf. A020994, A023107, A076586, A076623, A103443, A323390, A323396.

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
a(n) = vecmax(bothTruncatablePrimesInBase(n)); \\ for n>=3; Daniel Suteu, Jan 22 2019

a(n) <= min(A023107(n), A103443(n)). - Daniel Suteu, Feb 24 2019

a(17)-a(40) from Daniel Suteu, Jan 11 2019

A323396 Irregular array read by rows, where T(n, k) is the k-th prime that is both left-truncatable and right-truncatable in base n.

Original entry on oeis.org

2, 23, 2, 3, 11, 2, 3, 13, 17, 67, 2, 3, 5, 17, 23, 83, 191, 479, 839, 2, 3, 5, 17, 19, 23, 37, 2, 3, 5, 7, 19, 23, 29, 31, 43, 47, 59, 61, 139, 157, 239, 251, 331, 349, 379, 479, 491, 1867, 2, 3, 5, 7, 23, 29, 47, 173, 2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397

Offset: 3

Daniel Suteu and Felix Fröhlich, Jan 13 2019

The n-th row contains A323390(n) terms.

The largest term in the n-th row is given by A323137(n).

			Rows for n = 3..7:
  [2, 23]
  [2,  3, 11]
  [2,  3, 13, 17, 67]
  [2,  3,  5, 17, 23, 83, 191, 479, 839]
  [2,  3,  5, 17, 19, 23,  37]

Daniel Suteu, Table of n, a(n) for n = 3..6587
Wikipedia, Truncatable prime

Cf. A020994, A076586, A076623, A323137, A323390.

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
row(n) = vecsort(bothTruncatablePrimesInBase(n));
T(n,k) = row(n)[k];

A323390 Total number of primes that are both left-truncatable and right-truncatable in base n.

Original entry on oeis.org

0, 2, 3, 5, 9, 7, 22, 8, 15, 6, 35, 11, 37, 17, 22, 12, 69, 12, 68, 18, 44, 13, 145, 16, 47, 20, 77, 13, 291, 15, 89, 27, 74, 20, 241, 18, 106, 25, 134, 15, 450, 23, 144, 33, 131, 24, 491, 27, 235, 29, 187, 23, 575, 30, 218, 31, 183, 25, 1377, 26, 247, 37, 231

Offset: 2

Daniel Suteu, Jan 13 2019

			For n = 2, there are no both-truncatable primes, therefore a(2) = 0.
For n = 3, there are 2 both-truncatable primes: 2, 23.
For n = 4, there are 3 both-truncatable primes: 2, 3, 11.
For n = 5, there are 5 both-truncatable primes: 2, 3, 13, 17, 67.
For n = 6, there are 9 both-truncatable primes: 2, 3, 5, 17, 23, 83, 191, 479, 839.

Chris Caldwell, right-truncatable prime, The Prime Glossary.
Eric Weisstein's World of Mathematics, Truncatable Prime

Cf. A020994, A076623, A076586, A323137, A323396.

digitsToNum(d, base) = sum(k=1, #d, base^(k-1) * d[k]);
isLeftTruncatable(d, base) = my(ok=1); for(k=1, #d, if(!isprime(digitsToNum(d[1..k], base)), ok=0; break)); ok;
generateFromPrefix(p, base) = my(seq = [p]); for(n=1, base-1, my(t=concat(n, p)); if(isprime(digitsToNum(t, base)), seq=concat(seq, select(v -> isLeftTruncatable(v, base), generateFromPrefix(t, base))))); seq;
bothTruncatablePrimesInBase(base) = my(t=[]); my(P=primes(primepi(base-1))); for(k=1, #P, t=concat(t, generateFromPrefix([P[k]], base))); vector(#t, k, digitsToNum(t[k], base));
a(n) = #(bothTruncatablePrimesInBase(n));

A144714 Left-truncatable primes that contain one or more zero digits.

Original entry on oeis.org

103, 107, 307, 503, 607, 907, 1013, 1097, 1103, 1307, 1607, 1907, 2003, 2017, 2053, 2083, 2503, 3023, 3037, 3067, 3083, 3307, 3607, 3907, 4003, 4007, 4013, 4073, 5003, 5023, 5107, 5503, 6007, 6037, 6043, 6047, 6053, 6067, 6073, 6607, 6907, 7013, 7043

Offset: 1

Harry J. Smith, Oct 08 2008

These are the terms in sequence A033664 that are not in A024785. This sequence is infinitely long.

H. J. Smith, Table of n, a(n) for n = 1..10000
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
H. J. Smith, Link to programs to generate truncatable primes.
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Cf. A020994, A024770, A024785, A033664, A132394.

zeroin(z)={until(z==0,q=z\10;r=z-10*q;if(r==0,return(1));z=q;);return(0);}
{fileO="b144714.txt";v=vector(15000);v[1]=2;v[2]=3;v[3]=5;v[4]=7;j=4;m=0;
p10=1;until(0,p10*=10;j0=j;for(k=1,9,k10=k*p10;for(i=1,j0,z=k10+v[i];
if(isprime(z),j++;v[j]=z;if(zeroin(z),m++;
write(fileO,m," ",z);if(m==10000,break(3));)))));}

A173057 Partial sums of A024770.

Original entry on oeis.org

2, 5, 10, 17, 40, 69, 100, 137, 190, 249, 320, 393, 472, 705, 944, 1237, 1548, 1861, 2178, 2551, 2930, 3523, 4122, 4841, 5574, 6313, 7110, 9443, 11782, 14175, 16574, 19513, 22632, 25769, 29502, 33241, 37034, 40831, 46770, 53963, 61294, 68627

Offset: 1

Jonathan Vos Post, Feb 08 2010

Partial sums of right-truncatable primes, primes whose every prefix is prime (in decimal representation). The sequence has 83 terms. The subsequence of prime partial sums of right-truncatable primes begins: 2, 5, 17, 137, 1237, 1861, 2551, 199483. What is the largest value in the subsubsequence of right-truncatable prime partial sums of right-truncatable primes?

			a(50) = 2 + 3 + 5 + 7 + 23 + 29 + 31 + 37 + 53 + 59 + 71 + 73 + 79 + 233 + 239 + 293 + 311 + 313 + 317 + 373 + 379 + 593 + 599 + 719 + 733 + 739 + 797 + 2333 + 2339 + 2393 + 2399 + 2939 + 3119 + 3137 + 3733 + 3739 + 3793 + 3797 + 5939 + 7193 + 7331 + 7333 + 7393 + 23333 + 23339 + 23399 + 23993 + 29399 + 31193 + 31379.

Cf. A000040, A024770, A033664, A024785 (left-truncatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866.

A173060 Partial sums of A024785.

Original entry on oeis.org

2, 5, 10, 17, 30, 47, 70, 107, 150, 197, 250, 317, 390, 473, 570, 683, 820, 987, 1160, 1357, 1580, 1863, 2176, 2493, 2830, 3177, 3530, 3897, 4270, 4653, 5050, 5493, 5960, 6483, 7030, 7643, 8260, 8903, 9550, 10203, 10876, 11559, 12302, 13075, 13872

Offset: 1

Jonathan Vos Post, Feb 08 2010

Partial sums of left-truncatable primes. This sequence has 4260 terms. The subsequence of prime partial sums of left-truncatable primes begins 2, 5, 17, 47, 107, 197, 317, 683, 7643. The subsubsequence of left-truncatable prime partial sums of left-truncatable primes begins 2, 5, 197, 317.

			a(57) = 2 + 3 + 5 + 7 + 13 + 17 + 23 + 37 + 43 + 47 + 53 + 67 + 73 + 83 + 97 + 113 + 137 + 167 + 173 + 197 + 223 + 283 + 313 + 317 + 337 + 347 + 353 + 367 + 373 + 383 + 397 + 443 + 467 + 523 + 547 + 613 + 617 + 643 + 647 + 653 + 673 + 683 + 743 + 773 + 797 + 823 + 853 + 883 + 937 + 947 + 953 + 967 + 983 + 997 + 1223 + 1283 + 1367.

Cf. A000040, A024770 (right-truncatable primes), A024785, A033664, A032437, A020994, A052023, A052024, A052025, A050986, A050987, A173057.

a(n) = SUM[i=1..n] A024785(i) = SUM[i=1..n] {p prime, and every suffix of p in decimal expansion is prime, and no digits are zero}.

A284060 Primes that are left-, left/right-, and right-truncatable.

Original entry on oeis.org

2, 3, 5, 7, 23, 37, 53, 73, 373, 3137, 3797

Offset: 1

Rick L. Shepherd, Mar 19 2017

Intersection of A020994 and A077390. Only the last three terms exhibit all three properties nontrivially.

			The prime 3797 is a term because it is a term of A024785 (truncating from the left: 797, 97, 7 are primes), of A077390 (truncating the same number of digits from left and from right: 79 is a prime), and of A024770 (truncating from the right: 379, 37, 3 are primes). The digit 9 is not a prime, so 3797 is not also a term of A085823.

Cf. A020994, A024770, A024785, A077390, A085823 (a subsequence).

cp's OEIS Frontend

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

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A323137 Largest prime that is both left-truncatable and right-truncatable in base n.

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A323396 Irregular array read by rows, where T(n, k) is the k-th prime that is both left-truncatable and right-truncatable in base n.

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A323390 Total number of primes that are both left-truncatable and right-truncatable in base n.

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A144714 Left-truncatable primes that contain one or more zero digits.

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A173057 Partial sums of A024770.

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A173060 Partial sums of A024785.

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A284060 Primes that are left-, left/right-, and right-truncatable.

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