A069837 -id:A069837 - cp's OEIS frontend

A098831 a(n) = A069837(n) - (10^n-1)*2/9.

0, 1, 1, 15, 51, 101, 51, 1, 5, 155, 1, 1051, 51, 105, 5, 311, 1335, 355, 105, 31, 55, 51, 105, 1031, 15, 105, 55, 5, 355, 115, 1001, 3115, 35, 551, 331, 1, 305, 51, 3305, 555, 1111, 305, 135, 1131, 1135, 1505, 51, 31, 10355, 5001, 1001, 1335, 1101, 331, 305

Offset: 1

David Wasserman, Nov 02 2004

This is a way to compress A069837.

Cf. A069837.

foo(n) = local(m); if (n < 4, m = q[n + 1], m = q[n%4 + 1] + 10*foo(n\4)); m;
print1(0, ", "); q = [0, 1, 3, 5]; for (n = 2, 70, x = (10^n - 1)*2/9; i = 1; found = 0; while (!found, m = foo(i); if (isprime(x + m), found = 1; print1(m, ", "), i += 2)));

A374376 Array read by downward antidiagonals: T(k,n) is the least number that has k prime factors (counted with multiplicity) and is the concatenation of n primes, or -1 if there is no such number.

Original entry on oeis.org

2, 23, -1, 223, 22, -1, 2237, 235, 27, -1, 22273, 2227, 222, 132, -1, 222323, 22223, 2222, 225, 32, -1, 2222273, 222223, 22222, 2223, 252, 729, -1, 22222223, 2222557, 222227, 22225, 2322, 352, 192, -1, 222222227, 22222237, 2222222, 222225, 22232, 2232, 2352, 2112, -1, 2222222377, 222222223

Offset: 1

Robert Israel, Jul 14 2024

			Array starts
  2  23   223   2237   22273  ...
 -1  22   235   2227   22223  ...
 -1  27   222   2222   22222  ...
 -1 132   225   2223   22225  ...
 -1  32   252   2322   22232  ...
A(4,3) = 225 because 225 = 3^2 * 5^2 is the product of 4 primes (with multiplicity) and is the concatenation of the 3 primes 2, 2 and 5, and is the least number that works.

T(k,1) = -1 for k > 1.

Cf. A001222, A069837 (first row), A374665 (main diagonal), A374669 (second column).

PD[1]:= [2,3,5,7]:
for i from 2 to 7 do PD[i]:= select(isprime,[seq(i,i=10^(i-1)+1..10^i-1,2)]) od:
dcat:= proc(a,b) 10^(ilog10(b)+1)*a+b end proc:
cp:= proc(m,n) option remember; local d,p,x,R;
  if n = 1 then return PD[m] fi;
  R:= {};
  for d from 1 to m-n+1 do
    R:= R union {seq(seq(dcat(p,x),p=PD[d]),x=procname(m-d,n-1))}
  od;
  R
end proc:
F:= proc(n,N)
local V,count,d,x,v;
if n = 1 then return <2,(-1)$(N-1)> fi;
V:= Vector(N); count:= 0;
for d from n while count < N do
  for x in sort(convert(cp(d,n),list)) while count < N do
    v:= numtheory:-bigomega(x);
    if v <= N and V[v] = 0 then
      V[v]:= x; count:= count+1;
    fi
od od:
V;
end proc:
N:= 10: M:= Matrix(N,N):
for i from 1 to N do
  V:= F(i,N+1-i);
  M[i,1..N+1-i]:= V;
od:
[seq(seq(M[t-i,i],i=1..t-1),t=2..N+1)];

A071060 Largest n-digit prime with only prime digits.

Original entry on oeis.org

7, 73, 773, 7757, 77773, 777737, 7777753, 77777377, 777777773, 7777777577, 77777777573, 777777777773, 7777777777573, 77777777777753, 777777777777773, 7777777777777753, 77777777777775557, 777777777777777737, 7777777777777777577, 77777777777777777257, 777777777777777777773, 7777777777777777773533, 77777777777777777775353

Offset: 1

Rick L. Shepherd, May 26 2002

Terms a(5) through a(23) have been certified prime with Primo.

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A069837.

genit(nstrt=1,cownt=23)={my(arr=List());for(n=nstrt, nstrt+cownt, my(cand=0); for(i=1,n,cand=10*cand+7); if(ispseudoprime(cand)==1, listput(arr,cand);next); for(j=1,+oo,cand=precprime(cand-1); my(v=digits(cand), pass=1); for(ptr=1, #v, my(q=v[ptr]); if(q==2||q==3||q==5||q==7,next);pass=0;break);if(pass>0,break)); listput(arr,cand)); Vec(arr)} \\ Bill McEachen, Apr 29 2023

from sympy import isprime
from itertools import product
def a(n): return next(t for t in (int("".join(p)+e) for p in product("7532", repeat=n-1) for e in "73") if isprime(t))
print([a(n) for n in range(1, 24)]) # Michael S. Branicky, Apr 29 2023

Conjecture: a(n) ~ floor((7/9) * 10^n). - Bill McEachen, Apr 07 2023

A083470 Smallest prime which is the concatenation of n distinct primes in increasing order.

Original entry on oeis.org

2, 23, 257, 2357, 235723, 23571143, 2357111371, 235711131767, 23571113172367, 2357111317192343, 235711131719233171, 23571113171923296797, 2357111317192329314189, 235711131719232931375979, 23571113171923293137414371, 2357111317192329313741436167

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 02 2003

Perhaps the n-th term begins with the concatenation of the first (n-1) primes, except for n = 3.

The 9th, 11th, 12th, 13th, .... terms are other counterexamples to the above "conjecture". On the other hand, it seems almost sure that the n-th term starts with the concatenation of the first (n-2) primes. - M. F. Hasler, Mar 20 2011

Cf. A069837, A083427.

More terms from David Wasserman, Nov 02 2004

cp's OEIS Frontend

A098831 a(n) = A069837(n) - (10^n-1)*2/9.

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A374376 Array read by downward antidiagonals: T(k,n) is the least number that has k prime factors (counted with multiplicity) and is the concatenation of n primes, or -1 if there is no such number.

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Maple

A071060 Largest n-digit prime with only prime digits.

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A083470 Smallest prime which is the concatenation of n distinct primes in increasing order.

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