A248349 -id:A248349 - cp's OEIS frontend

A259350 Numbers n such that n-1, n, and n+1 are all products of 7 distinct primes.

41704979954, 124731595066, 365993436094, 366230785766, 367810728790, 368695198806, 589316590786, 598986161410, 607638803134, 673917791834, 710756189898, 753389272714, 762118572046, 772416848554, 806996241806, 832216749090, 874567856590, 905173650094, 933893335166, 958872775134, 970959170390, 985722818366, 997785568130

Offset: 1

James G. Merickel, Jun 24 2015

nonn

A subsequence of A169834. A093550(7)=a(1), that sequence with offset 2 (so actually its 6th term) holding first terms of sequences of this kind.
Other than a(4)=366230785766 and a(18)=905173650094 (with minimax prime factor 1867 for it and its neighbors), the terms were initially discovered by increasing value of the trios' smallest large prime factors. An exhaustive search running multiple (suitably modified) copies of a pre-acceptance PARI program that disposed of fails in a somewhat efficient way and ran about an order of magnitude faster than the analog of the simple program at A259349 required about 1000 window-hours to produce the list given (adding two terms, including one that was unachievable by the increasing-minimax-prime method). Then the much faster program--15 minutes in just one PARI window--shown was developed and edited in here in its place. By specifying the 4 largest prime factors secondary to setting the product of the smallest 3 such that this is at least 627--as must be true for one of 3 relatively prime sphenic numbers--a speedup of over 3.5 orders of magnitude more (over the single order of magnitude that the replaced program managed, for a total of about 10^4.5 in time ratio over the program used for 6 primes) was achieved.
Note: The PARI program avoids duplicates but does not order terms.

			41704979953 = 7*13*29*41*47*59*139,
41704979954 = 2*11*23*31*83*103*311, and
41704979955 = 3*5*17*19*109*157*503; and no smaller such trio exists, so that a(1)=41704979954.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2510

Cf. A093550, A169834, A248201, A248202, A248203, A248204, A248349, A259801.

{
\\Program runs for arbitrary B.\\
B=10^12;N=primepi(B/(627*17*19*23));
p=vector(N,n,prime(n));
in=primepi((B/210)^(1/3));
P=prod(i=1,27,p[i]);Q=prod(i=28,in,p[i]);
v=28;d=[[1,2],[-1,1],[-2,-1]];i3=6;
while(6*p[i3]^5626,
    if(k1*p[i3+1]*p[i3+2]*p[i3+3]*p[i3+4]=k1,v--;Q*=p[v];P/=p[v]));
     r=(B\k1)^(1/4);j1=i3+1;
     while(p[j1]2,
          f=1;if(y1==3,if(a1>j1,f=0));
          if(f,
           b1=gcd(P,b);z1=omega(b1);
           if(z1>2,
            if(z1==3,if(b1>j1,f=0));
            if(f,
             a2=a/a1;
             if(gcd(a1,a2)==1,
              b2=b/b1;
              if(gcd(b1,b2)==1,
               a21=gcd(a2,Q);a22=a2/a21;
               if(gcd(a21,a22)==1,
                y=y1+omega(a21);
                if(y>4,
                 if(y<8,
                  b21=gcd(Q,b2);b22=b2/b21;
                  if(gcd(b21,b22)==1,
                   z=z1+omega(b21);
                   if(z>4,
                    if(z<8,
                     if(y+omega(a22)==7,
                      if(z+omega(b22)==7,
                       f1=factor(a1);
                       if(f1[1,1]*f1[2,1]*f1[3,1]

A248350 Numbers n such that 10^n - 123456789 is prime.

Original entry on oeis.org

9, 10, 13, 19, 26, 68, 73, 115, 190, 195, 232, 549, 742, 1502, 2239, 2618, 5143, 8081, 9442, 31402, 77919, 93790, 99434, 120841

Offset: 1

Derek Orr, Oct 05 2014

a(26) > 200000. - Robert Price, Jun 06 2020

Cf. A248349, A248351, A248352, A050289.

for(n=1,10^4,if(ispseudoprime(10^n - 123456789),print1(n,", ")))

a(21)-a(25) from Robert Price, Feb 26 2020

A248351 Numbers k such that 10^k + 987654321 is prime.

Original entry on oeis.org

6, 11, 15, 27, 42, 113, 135, 186, 207, 503, 2999, 3005, 3487, 5718, 7265, 7629, 11987, 16063, 27379, 64770, 73579, 96504, 116557

Offset: 1

Derek Orr, Oct 05 2014

Note that 987654321 is the largest pandigital number in base-10, omitting 0.

Cf. A248349, A248350, A248352, A050289.

[n: n in [1..500] | IsPrime(10^n+987654321)]; // Vincenzo Librandi, Oct 12 2014

Select[Range[1000], PrimeQ[10^# + 987654321] &] (* Vincenzo Librandi, Oct 12 2014 *)

for(n=1,10^4,if(ispseudoprime(10^n+987654321),print1(n,", ")))

a(9) corrected and a(19)-a(23) added by Robert Price, Dec 05 2019

A248352 Numbers k such that 10^k - 987654321 is prime.

Original entry on oeis.org

986, 1240, 1928, 4054, 14252, 47528, 101728

Offset: 1

Derek Orr, Oct 05 2014

Note that 987654321 is the largest pandigital number in base-10, omitting 0.

Cf. A248349, A248350, A248351, A050289.

Select[Range[10000], PrimeQ[10^# - 987654321] &] (* Robert Price, Dec 05 2019 *)

for(n=1,10^4,if(ispseudoprime(10^n-987654321),print1(n,", ")))

a(6)-a(7) from Robert Price, Dec 05 2019

A258932 Numbers k such that 10^k + 103 is prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 27, 35, 85, 169, 209, 221, 321, 347, 603, 610, 1229, 1391, 2171, 2303, 2679, 3977, 4545, 5721, 7090, 35877

Offset: 1

Vincenzo Librandi, Jun 15 2015

a(29) > 60000. - Michael S. Branicky, Apr 27 2025

			For n = 3, a(3) = 10^3 + 103 = 1103, which is prime.

Sequences of the type 10^n+k: A049054 (k=3), A088274 (k=7), A088275 (k=9), A095688 (k=13), A108052 (k=19), A108050 (k=21), A108312 (k=27), A107083 (k=31), A107084 (k=33), A135109 (k=37), A135108 (k=39), A108049 (k=43), A108054 (k=49), A135118 (k=51), A135119 (k=57), A135116 (k=61), A135115 (k=63), A135113 (k=67), A135114 (k=69), A135132 (k=73), A135131 (k=79), A137848 (k=81), A135117 (k=87), A110918 (k=91), A135112 (k=93), A135107 (k=97), A110980 (k=99), this sequence (k=103), A258933 (k=109), A165508 (k=111), A248349 (k=123456789), A248351 (k=987654321).

[n: n in [1..600] | IsPrime(10^n+103)];

Select[Range[5000], PrimeQ[10^# + 103] &]

is(n)=ispseudoprime(10^n+103) \\ Charles R Greathouse IV, Jun 13 2017

a(26)-a(28) from Jens Kruse Andersen, Jun 23 2015

A322985 Numbers k such that 123456789*10^k+1 is prime.

Original entry on oeis.org

1, 5, 17, 23, 25, 28, 91, 187, 287, 398, 899, 1364, 2921, 5125, 5890, 8780, 14881, 35689, 46669, 71861, 111710

Offset: 1

Matthias Baur, Jan 01 2019

a(22) > 1.3*10^5. All numbers up to this bound were sieved using newpgen and sr1sieve. Remaining numbers were checked for primality using Jean Penné's LLR application (BLS (N-1/N+1) test).

			1 is a term because 1234567891 is prime.
2 is not a term because 12345678901 is composite (it is divisible by 857).

Cf. A248349, A248350, A321806.

Select[Range@ 1400, PrimeQ[123456789*10^# + 1] &] (* Michael De Vlieger, Jan 04 2019 *)

from sympy.ntheory.primetest import isprime
for n in range(1,1000):
    if isprime(123456789*10**n+1):
        print(n, end=', ') # Stefano Spezia, Jan 05 2019

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A259350 Numbers n such that n-1, n, and n+1 are all products of 7 distinct primes.

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A248350 Numbers n such that 10^n - 123456789 is prime.

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A248351 Numbers k such that 10^k + 987654321 is prime.

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A248352 Numbers k such that 10^k - 987654321 is prime.

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A258932 Numbers k such that 10^k + 103 is prime.

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Magma

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A322985 Numbers k such that 123456789*10^k+1 is prime.

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