A024770 -id:A024770 - cp's OEIS frontend

A069866 Primes in which repeatedly deleting the most significant digit then the least significant digit gives a prime at every step until a single-digit prime remains.

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 131, 137, 173, 179, 223, 229, 271, 331, 337, 353, 359, 373, 379, 431, 479, 523, 571, 631, 653, 659, 673, 773, 823, 829, 853, 859, 929, 937, 953, 971, 1031, 1373, 1433, 1439, 1733, 2029, 2053, 2131, 2137

Offset: 1

Amarnath Murthy, Apr 21 2002

Michael De Vlieger, Table of n, a(n) for n = 1..645 (Primes p <= 10^7).

Cf. A024770, A033664, A069867.

Select[Prime@ Range@ 400, AllTrue[FromDigits /@ Rest@ Fold[Append[#1, Delete[Last[#1], 1 - 2 Boole[EvenQ@ #2]]] &, {#}, Range[Length@ # - 1]] &@ IntegerDigits[#], PrimeQ] &] (* Michael De Vlieger, Jan 20 2018 *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Sep 24 2002

A202263 Primes in which all substrings and reversal substrings are primes.

Original entry on oeis.org

2, 3, 5, 7, 37, 73, 373

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 7 terms.

Subsequence of A085823, A068669, A024770, A012883.

			All substrings and reversal substrings of 373 are primes:3, 7, 37, 73, 373.

Cf. A202264 (noncomposite numbers in which all substrings and reversal substrings are noncomposite), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

A232125 Smallest prime such that the n numbers obtained by removing 1 digit on the right are also prime, while no digit can be added on the right to get another prime.

Original entry on oeis.org

53, 53, 317, 2393, 23333, 373393, 2399333, 23399339, 1979339333, 103997939939, 4099339193933, 145701173999399393, 2744903797739993993333, 52327811119399399313393, 13302806296379339933399333

Offset: 0

Michel Marcus, Nov 19 2013

Inspired by article on 43 in Archimedes' Lab link.

			a(0)=53 because 53 is the smallest prime such that all numbers obtained by adding a digit to the right are composite.
a(1)=53 because 5 and 53 are primes.
a(2)=317 because 3, 31, 317 are all primes, and 317 has the same property as 53 when adding a digit to the right.

G. A. Sarcone and M. J. Waeber, What's Special About This Number?, Archimedes' Lab website.

Cf. A024770, A119289, A227919, A239747.

a(n) = {n++; v = vector(n); i = 1; ok = 0; until (ok, while ((i>1) && (v[i] == 9), v[i] = 0; i--); if (i == 1, v[i] = nextprime(v[i]+1), v[i] = v[i]+1); curp = sum (j=1, i, v[j]*(10^(i-j))); if (isprime(curp), if (i != n, i++, nbp = 0; for (z=1, 9, if (isprime(10*curp+z), nbp++);); if (nbp == 0, ok = 1);););); sum (j=1, n, v[j]*(10^(n-j)));}

from sympy import isprime, nextprime
def a(n):
    p, oo = 2, float('inf')
    while True:
        extends, reach, r1 = 0, [str(p)], []
        while len(reach) > 0 and extends <= n:
            minnotext = oo
            for s in reach:
                wasextended = False
                for d in "1379":
                    if isprime(int(s+d)): r1.append(s+d); wasextended = True
                if not wasextended: minnotext = min(minnotext, int(s))
            if extends == n and minnotext < oo: return minnotext
            if len(r1) > 0: extends += 1
            reach, r1 = r1, []
        p = nextprime(p)
for n in range(12): print(a(n), end=", ") # Michael S. Branicky, Aug 08 2021

a(12)-a(13) from Michael S. Branicky, Aug 08 2021

a(14) from Michael S. Branicky, Aug 23 2021

A127889 Smallest n-digit right-truncatable prime.

Original entry on oeis.org

2, 23, 233, 2333, 23333, 233993, 2339933, 23399339

Offset: 1

Ray Chandler, Feb 04 2007

Agrees with A088603 for 8 terms, but this sequence ends there while A088603 continues.

Right-truncatable means that the integer part of successive divisions by 10 always yields primes (or zero). - M. F. Hasler, Nov 07 2018

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Index entries for sequences related to truncatable primes

Cf. A024770, A050986, A088603, A127890.

A127889=vector(8, n, p=concat(apply(t->primes([t, t+1]*10), if(n>1, p))); p[1]) \\ M. F. Hasler, Nov 07 2018

A210498 Prime numbers that become emirps when their least-significant-digit is deleted.

Original entry on oeis.org

131, 137, 139, 173, 179, 311, 313, 317, 373, 379, 719, 733, 739, 797, 971, 977, 1493, 1499, 1571, 1579, 1993, 1997, 1999, 3119, 3371, 3373, 3593, 7013, 7019, 7331, 7333, 7393, 7433, 7517, 7691, 7699, 9371, 9377, 9413, 9419, 9533, 9539, 9677, 9679, 9719, 9833

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 03 2013

This sequence (like the emirps) experiences large gaps when the most-significant-digit is {2,4,5,6,8}.

			Example: a(1)=131, which becomes 13 upon deletion, which is an emirp.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A006567, A024770, A000040, A225235

library(gmp); isemirp<-function(x) isprime(x) & (j=paste(rev(unlist(strsplit(as.character(x),split=""))),collapse=""))!=x & isprime(j);i=as.bigz(0); y=as.bigz(rep(0,100)); len=0;
while(len<100)
    if(isprime((i=nextprime(i))))
        if(isemirp(as.bigz(substr(i,1,nchar(as.character(i))-1))))
            y[(len=len+1)]=i

A226354 Squares that become cubes when their rightmost digit is removed.

Original entry on oeis.org

1, 4, 9, 16, 81, 10000, 640000, 7290000, 40960000, 156250000, 188210961, 466560000, 1176490000, 2621440000, 5314410000, 10000000000, 17715610000, 29859840000, 48268090000, 75295360000, 113906250000, 167772160000, 241375690000, 340122240000, 470458810000

Offset: 1

Christian N. K. Anderson and Kevin L. Schwartz, Jun 04 2013

			188210961=13719^2, while 18821096=266^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..500
Christian N. K. Anderson, Decomposition of a(n) into its square root, and truncated cube root for n=0..500.

Cf. A024770, A225873, A225885.

cQ[n_]:=IntegerQ[Surd[FromDigits[Most[IntegerDigits[n]]],3]]; Select[Range[ 700000]^2,cQ] (* Harvey P. Dale, Feb 21 2014 *)

trimR=function(x) { x=as.character(x); ifelse(nchar(x)<2,0,substr(x,1,nchar(x)-1)) }
iscube<-function(x) ifelse(as.bigz(x)<2,T,all(table(as.numeric(factorize(x)))%%3==0))
which(sapply(1:6400,function(x) iscube(trimR(x^2))))^2

For n > 11: a(n)=(100*(n-6)^3)^2 (188210961 is the last "exception" as is easy to prove with the help of the Nagell-Lutz theorem). - Reiner Moewald, Dec 30 2013

A024768 Prefix primes in base 8 (written in base 8).

Original entry on oeis.org

2, 3, 5, 7, 21, 23, 27, 35, 37, 51, 53, 57, 73, 75, 211, 213, 235, 277, 351, 357, 373, 513, 533, 535, 573, 577, 737, 753, 2111, 2117, 2135, 2353, 2773, 3513, 3517, 3571, 3733, 5331, 5355, 5735, 5773, 7371, 7531, 7533, 21113, 21117, 21177, 21355, 23537, 27733

Offset: 1

David W. Wilson

The final term is a(68) = 21117717.

Nathaniel Johnston, Table of n, a(n) for n = 1..68 (full sequence)

Cf. A024763, A024764, A024765, A024766, A024767, A024769, A024770.

s:=[1,3,5,7]: a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: do for j from l1 to l2 do for k from 1 to 4 do d:=[s[k], op(a[j])]: if(isprime(op(convert(d, base, 8, 8^nops(d)))))then a:=[op(a), d]: fi: od: od: l1:=l2+1: l2:=nops(a): if(l1>l2)then break: fi: od: for j from 1 to nops(a) do printf("%d, ", op(convert(a[j], base, 10, 10^nops(a[j])))); od: # Nathaniel Johnston, Jun 22 2011

A069867 Primes in which repeatedly deleting the least significant digit then the most significant digit gives a prime at every step until a single-digit prime remains.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 131, 137, 139, 173, 179, 233, 239, 373, 379, 431, 433, 439, 479, 673, 677, 733, 739, 839, 971, 977, 1319, 1373, 1733, 2237, 2239, 2293, 2297, 2711, 2713, 2719, 3313, 3319, 3371, 3373, 3533, 3539, 3593, 3733

Offset: 1

Amarnath Murthy, Apr 21 2002

Michael De Vlieger, Table of n, a(n) for n = 1..440 (Primes p <= 5 * 10^7).

Cf. A033664, A024770, A069866.

Select[Prime@ Range@ 512, AllTrue[FromDigits /@ Rest@ Fold[Append[#1, Delete[Last[#1], 1 - 2 Boole[OddQ@ #2]]] &, {#}, Range[Length@ # - 1]] &@ IntegerDigits[#], PrimeQ] &] (* Michael De Vlieger, Jan 20 2018 *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Sep 24 2002

A127890 Largest n-digit right-truncatable prime.

Original entry on oeis.org

7, 79, 797, 7393, 73939, 739399, 7393933, 73939133

Offset: 1

Ray Chandler, Feb 04 2007

For a variant see the Howard reference. - Alexander R. Povolotsky, Dec 23 2007

Right-truncatable means that the integer part of successive divisions by 10 always yields primes (or zero). - M. F. Hasler, Nov 07 2018

Toby Howard, "Magic Pi - The Magic of Numbers", PC Advisor magazine, May 1998.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Jens Fehlau, 73939133 - Probably the Most Interesting Prime Number [Part 1] and [Part 2], Flammable Maths videos (2020).
Index entries for sequences related to truncatable primes

Cf. A024770, A050986, A127889.

A127890=vector(8, n, p=concat(apply(t->primes([t, t+1]*10), if(n>1, p)));p[#p]) \\ M. F. Hasler, Nov 07 2018

A239747 Super-prime leaders: right-truncatable primes p with property that appending any single decimal digit to p does not produce a prime.

Original entry on oeis.org

53, 317, 599, 797, 2393, 3793, 3797, 7331, 23333, 23339, 31193, 31379, 37397, 73331, 373393, 593993, 719333, 739397, 739399, 2399333, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133

Offset: 1

Arkadiusz Wesolowski, Mar 26 2014

The name "super-prime leaders" is not due to the author.

			2393 belongs to this sequence because 2393, 239, 23 and 2 are all prime; 10*2393 + k, for k = 0 to 9, are all composite.

Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, p. 292.

Chris Caldwell, The Prime Glossary, Right-truncatable prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 53
Kvant magazine, The simplest prime numbers, (in Russian) No 11, 1979. (beware of typo)
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Subsequence of A024770 and of A119289.

f=1; for(n=2, 73939133, v=n; t=1; while(isprime(n), if(!Mod(f, n^2)==0, t=t*n); c=n; n=(c-lift(Mod(c, 10)))/10); if(n==0, f=f*t); n=v); s=Set(factor(f)[, 1]); for(k=1, #s, p=s[k]; if(!Mod(f, p^2)==0, print1(p, ", ")));

A024770 INTERSECT A119289.

cp's OEIS Frontend

A069866 Primes in which repeatedly deleting the most significant digit then the least significant digit gives a prime at every step until a single-digit prime remains.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A202263 Primes in which all substrings and reversal substrings are primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

A232125 Smallest prime such that the n numbers obtained by removing 1 digit on the right are also prime, while no digit can be added on the right to get another prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Extensions

A127889 Smallest n-digit right-truncatable prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A210498 Prime numbers that become emirps when their least-significant-digit is deleted.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

A226354 Squares that become cubes when their rightmost digit is removed.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

R

Formula

A024768 Prefix primes in base 8 (written in base 8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A069867 Primes in which repeatedly deleting the least significant digit then the most significant digit gives a prime at every step until a single-digit prime remains.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A127890 Largest n-digit right-truncatable prime.

Views

Author