This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
13 is a term because concatenation of 13 and 1 is prime.
All substrings and reversal substrings of 311 are noncomposites: 1, 3, 11, 13, 31, 113, 311.
113 and 1013 are both in the sequence, because upon deletion they become 13, which is an emirp.
library(gmp); isemirp<-function(x) isprime(x) & (j=paste(rev(unlist(strsplit(as.character(x),split=""))),collapse=""))!=x & isprime(j);
no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s}
i=as.bigz(0); y=as.bigz(rep(0,100)); len=0;
while(len<100)
if(isemirp(as.bigz(no0(substr((i=nextprime(i)),2,200)))))
y[(len=len+1)]=i
Example: 1171 has the emirp 17 as its internal digits.
library(gmp); isemirp<-function(x) isprime(x) & (j=paste(rev(unlist(strsplit(as.character(x),split=""))),collapse=""))!=x & isprime(j);
no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s}
i=as.bigz(0); y=as.bigz(rep(0,100)); len=0;
while(len<100) if(isemirp(as.bigz(no0(substr((i=nextprime(i)),2,nchar(as.character(i))-1))))) y[(len=len+1)]=i
4096 = 16^3, and becomes the prime number 409 when truncated.
Select[Range[1000]^3,PrimeQ[Floor[#/10]]&] (* Harvey P. Dale, May 28 2021 *)
library(gmp)trimR=function(x) { x=as.character(x); ifelse(nchar(x)<2,0,substr(x,1,nchar(x)-1)) }
y=as.bigz(rep(0,10000)); len=0; n=as.bigz(-1)
while(len<10000) if(isprime(trimR((n=n+1)^3))) y[(len=len+1)]=n^3
a(2) = 17 is prime: 17^3 = 4913. Removing rightmost digit gives 491 which is prime. a(3) = 53 is prime: 53^3 = 148877. Removing rightmost digit gives 14887 which is prime.
a258032 n = a258032_list !! (n-1) a258032_list = filter ((== 1) . a010051' . flip div 10. (^ 3)) a000040_list -- Reinhard Zumkeller, May 18 2015
[p: p in PrimesUpTo(6500) |IsPrime(Floor(p^3/10))]; // Vincenzo Librandi, May 17 2015
Select[Prime[Range[1000]], PrimeQ[Floor[(#^3)/10]] &]
forprime(p=1,10000, if(isprime(floor((p^3)/10)), print1(p,", ")))
211 is a term since 211 is a prime, 21 is a nonprime, and 2 is a prime; 23 is not a term since 23 and 2 are both prime.
Unprotect[CompositeQ]; CompositeQ[1]:=True; Protect[CompositeQ]; Q[n_]:=And[AllTrue[FromDigits/@Table[Take[IntegerDigits[n], i], {i,IntegerLength[n],1,-2}], PrimeQ], AllTrue[FromDigits/@Table[Take[IntegerDigits[n], i], {i,IntegerLength[n]-1,1,-2}],CompositeQ]]; Select[Prime[Range[140]], Q]
from gmpy2 import is_prime, mpz
from itertools import count, islice
def agen():
olst, elst = [2, 3, 5, 7], [11, 13, 17, 19, 41, 43, 47, 61, 67, 83, 89, 97]
yield from olst + elst
for n in count(1):
olst2, elst2 = [], []
for o in olst:
for i in range(1, 100, 2):
t = 100*o + i
if is_prime(t) and not is_prime(t//10):
olst2.append(t)
yield from olst2
for e in elst:
for i in range(100):
t = 100*e + i
if is_prime(t) and not is_prime(t//10):
elst2.append(t)
yield from elst2
olst, elst = olst2, elst2
print(list(islice(agen(), 70))) # Michael S. Branicky, May 11 2025
# predicate test useful for large n (cf. a-file of largest terms)
from gmpy2 import digits, is_prime, mpz
def ok(n):
s = digits(n)
return is_prime(n) and all(int(is_prime(mpz(s[:-i]))) == 1-i&1 for i in range(1, len(s)))
# Michael S. Branicky, May 16 2025
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));)))));}
2 is the leading single digit of e itself and is by the convention of A024770 considered truncatable; the leading digits of e^2, without decimal, are the right-truncatable 73; and e^75 is then the first to produce a 3-digit right-truncatable prime, also producing the 4-digit one (a(3)=373 and a(4)=3733, with e^75 beginning with these digits).
{
\\ R is the array of 8 by-length ordered lists of right-truncatable primes.\\
\\ a is the vector of list-sizes for R.\\
R=[[2,3,5,7],[23,29,31,37,53,59,71,73,79],[223,239,293,311,313,317,373,379,593,599,719,733,739,797],[2333,2339,2393,2399,2939,3119,3137,3733,3739,3793,3797,5939,7193,7331,7333,7393],[23333,23339,23399,23993,29399,31193,31379,37337,37339,37397,59393,59399,71933,73331,73939],[233993,239933,293999,373379,373393,593933,593993,719333,739393,739397,739399],[2339933,2399333,2939999,3733799,5939333,7393913,7393933],[23399339,29399999,37337999,59393339,73939133]];
a=[4,9,14,16,15,12,8,5];i=1;e=exp(1);e1=e/10;n=e;
for(j=1,8,
E=10^j;while(1,
m=floor(n);for(k=1,a[j],
if(m==R[j][k],print(m);n*=10;break(2)));
if(n>E/e,n*=e1,n*=e);i++))
}
14 is in the sequence because 14^2 = 196; deleting the last digit gives 19 which is prime. 26 is in the sequence because 26^2 = 676; deleting the last digit gives 67 which is prime.
[n : n in [1 .. 2000] | IsPrime (Floor (n^2/10))];
select(n -> isprime(floor(n^2/10)),[$1..2000]);
fQ[n_] := PrimeQ@Quotient[n^2, 10]; Select[Range[1, 2000], fQ]
isok(n) = isprime(n^2\10); \\ Michel Marcus, Jul 02 2017
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