A003617 -id:A003617 - cp's OEIS frontend

A340539 a(n) is the least prime that is the concatenation of two n-digit primes, and such that the concatenation of the same primes in the other order is also prime or 0 if no such prime exists.

37, 1123, 101107, 10091789, 1000710709, 100003100363, 10000031000303, 1000001910002521, 100000007100010321, 10000000071000000349, 1000000001910000000799, 100000000003100000009939, 10000000000391000000012387, 1000000000003710000000034573, 100000000000031100000000014113

Offset: 1

J. M. Bergot and Robert Israel, Jan 10 2021

Conjecture: a(n) > 0 and for n > 1 the first n digits of a(n) = A003617(n). - Chai Wah Wu, Jan 13 2021

			For n=4, 1009, 1789, 10091789 and 17891009 are all prime.

Robert Israel, Table of n, a(n) for n = 1..171

Cf. A003617, A340487.

f:= proc(d) local P,a,b;
  a:= prevprime(10^(d-1));
  do
    a:= nextprime(a);
    if a > 10^d then return FAIL fi;
    b:= prevprime(10^(d-1));
    do
      b:= nextprime(b);
      if b > 10^d then break fi;
      if isprime(10^d*a+b) and isprime(10^d*b+a) then return 10^d*a+b fi;
  od od:
  FAIL
end proc:
f(1):= 37:
map(f, [$1..20]);

A379140 Numbers k such that the greatest prime < 10^k and the least prime > 10^k share no decimal digits.

Original entry on oeis.org

1, 2, 8, 11, 15, 16, 17, 18, 21, 25, 26, 30, 40, 44, 46, 47, 50, 51, 53, 55, 60, 63, 64, 74, 77, 81, 86, 88, 89, 93, 95, 101, 123, 130, 131, 133, 134, 140, 152, 154, 158, 161, 164, 166, 176, 181, 189, 192, 198, 209, 214, 215, 233, 245, 264, 268, 274, 291, 293, 295, 297, 324, 326, 334, 352, 357

Offset: 1

Robert Israel, Dec 16 2024

Charles R Greathouse IV conjectures that A107801(n) = prime(n) for n sufficiently large (and similarly for other related sequences). If that is the case, this sequence must be finite.

			a(3) = 8 is a term because the greatest prime < 10^8 and the least prime > 10^8 are 99999989 and 100000007 respectively, and these have no digits in common.
5 is not a term because the greatest prime < 10^5 and the least prime > 10^5 are 99991 and 100003 respectively, and these have digit 1 in common.

Robert Israel, Table of n, a(n) for n = 1..250

Cf. A003617, A003618, A107801.

filter:= t -> convert(convert(prevprime(10^t),base,10),set) intersect convert(convert(nextprime(10^t),base,10),set) = {}:
select(filter, [$1..400]);

A038694 Smallest odd number with n prime factors all of different number of decimal digits.

Original entry on oeis.org

3, 33, 3333, 3362997, 33653510979, 3365452058432937, 3365462154789112298811, 33654685491672063981243677409, 3365468784750004839828815609605741863, 3365468808308286333078849488407451130240193041, 33654688147026770688645935212572651582143501884563667779

Offset: 1

Jeff Burch

Cf. A003617.

for(n=0,10,print1(3*prod(k=1,n,nextprime(10^k)),", ")) \\ Hugo Pfoertner, Sep 23 2020

Title changed and a(11) from Hugo Pfoertner, Sep 23 2020

A046445 Smallest composite with n prime factors that are distinct in length.

Original entry on oeis.org

1, 22, 2222, 2241998, 22435673986, 2243634705621958, 2243641436526074865874, 22436456994448042654162451606, 2243645856500003226552543739737161242, 2243645872205524222052566325604967420160128694

Offset: 0

Patrick De Geest, Jul 15 1998

Cf. A003617, A033873. Initial terms of A046442, A046443, A046444.

p = 2; Join[{1}, Table[p = p*Prime[PrimePi[10^n] + 1], {n, 9}]] (* Jayanta Basu, Jun 24 2013 *)

Corrected by Jayanta Basu, Jun 24 2013

A078728 a(n) is the smallest m such that m < 10^n, 10^n + m is prime and if the natural number k is such that 1 < k < 10 and 3 doesn't divide k10^n + m then k10^n+m is prime.

Original entry on oeis.org

3, 57, 297, 177, 237, 25111, 231339, 67419, 273817, 345111, 2001753, 912277, 5236153, 9228627, 10599391, 2835261, 60120003, 14054037, 27923629, 41783347, 24590943, 112161513, 230484021, 11446969, 205242589, 583389307, 873650007

Offset: 1

Farideh Firoozbakht, Dec 26 2003

nonn

a(n) is the smallest m such that m < 10^n and all six numbers 10^n + m, (Mod[m, 3]+2)*10^n + m, 4*10^n + m, (Mod[m, 3]+5)*10^n + m, 7*10^n + m & (Mod[m, 3]+8)*10^n + m are primes.

Carlos Rivera in Puzzle 245 of www.primepuzzles.net wrote "if the Faride's results ( a(n) for n=1,...,24 ) are plotted in Excel and a trend 'potential' function is asked, we obtain that a(n) is approximately equal to 0.5*n^6; this means that for n=999 a(n)=5*10^17, approximately." Since 10^n+a(n) is prime, for each n a(n)=0 (mod 3) or a(n)=1 (mod 3).

			a(6)=25111 because all the six numbers 1025111, 3025111, 4025111, 6025111, 7025111, 9025111 are primes and 25111 is the smallest number with this property.

Carlos Rivera, Puzzle 245. As 13

Cf. A003617, A033873.

Cf. A091199.

a[n_] := (For[m=1, !PrimeQ[10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+2)10^n+2m-1]||! PrimeQ[4*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+5)10^n+2m-1]||!PrimeQ [7*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+8)10^n+2m-1], m++ ];2m-1); Do[Print[a[n]], {n, 32}]

a[n_] := (For[m=1, !PrimeQ[10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+2)10^n+2m-1]||! PrimeQ[4*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+5)10^n+2m-1]||!PrimeQ [7*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+8)10^n+2m-1], m++ ];2m-1)

A097515 a(n) = (largest prime < 10^n) + (smallest prime > 10^n).

Original entry on oeis.org

18, 198, 2006, 19980, 199994, 1999986, 20000010, 199999996, 1999999944, 19999999986, 199999999980, 2000000000028, 20000000000008, 200000000000004, 2000000000000026, 19999999999999998, 200000000000000000, 1999999999999999992, 20000000000000000012, 200000000000000000028

Offset: 1

Cino Hilliard, Aug 27 2004

nonn

Cf. A003617, A003618.

Table[NextPrime[10^n,-1]+NextPrime[10^n],{n,20}] (* Harvey P. Dale, May 03 2018 *)

a(n) = A003618(n) + A003617(n+1). - Amiram Eldar, Jul 02 2024

More terms from Amiram Eldar, Jul 02 2024

A098963 Position of the smallest n-digit prime in the decimal expansion of e.

Original entry on oeis.org

1, 201, 196, 6226, 151208, 2778287, 21544761, 46847251, 172814008, 1976183900

Offset: 1

Dominic C. Milioto (dmilioto(AT)sw.rr.com), Oct 22 2004 (who also used PiFast to calculate e)

a(10) was found by Shigeru Kondo, using the program PiFast of Xavier Gourdon to calculate e.

			n...Prime...................Location.....String around value
1...2............................1.............2718281828459
2...11.........................201.............1019011573834
3...101........................196.............1952510190115
4...1009......................6226..............424341009057
5...10007...................151208.............4361910007991
6...100003.................2778287.............8243510000312
7...1000003...............21544761...........279031000003794
8...10000019..............46847251...........299541000001984
9...100000007............172814008.......2861310000000776457
10..1000000007..........1976183900.......3649610000000077111

X. Gourdon, PiFast

Cf. A001113, A003617.

A099657 a(n) is the least prime following A002277(n) repdigits.

Original entry on oeis.org

2, 5, 37, 337, 3343, 33343, 333337, 3333373, 33333347, 333333349, 3333333403, 33333333343, 333333333367, 3333333333347, 33333333333437, 333333333333389, 3333333333333343, 33333333333333391, 333333333333333391

Offset: 0

Labos Elemer, Nov 17 2004

			n=3: 33 is followed by 37.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[3*(10^n-1)/9], {n, 0, 35}]

A099661 a(n) is the least prime following A002281(n) repdigits.

Original entry on oeis.org

2, 11, 79, 787, 7789, 77783, 777781, 7777801, 77777803, 777777799, 7777777781, 77777777827, 777777777841, 7777777777859, 77777777777837, 777777777777787, 7777777777777867, 77777777777777797, 777777777777777817

Offset: 0

Labos Elemer, Nov 17 2004

			n=6: 77 is followed by 79.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Table[NextPrime[7*(10^n-1)/9], {n, 0, 35}]
NextPrime/@LinearRecurrence[{11,-10},{0,7},35] (* Harvey P. Dale, Dec 12 2021 *)

A099663 a(n) is the largest prime before A002276(n).

Original entry on oeis.org

19, 211, 2221, 22193, 222199, 2222219, 22222199, 222222193, 2222222137, 22222222189, 222222222169, 2222222222197, 22222222222201, 222222222222151, 2222222222222203, 22222222222222153, 222222222222222221, 2222222222222222177, 22222222222222222169, 222222222222222222149, 2222222222222222222161

Offset: 2

Labos Elemer, Nov 17 2004

			n=2: 19 is before 22.

Cf. A007917, A002276.

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099669.

Table[NextPrime[2(10^n-1)/9, -1], {n, 2, 35}]
Drop[NextPrime[#,-1]&/@LinearRecurrence[{11,-10},{0,2},20],2] (* Harvey P. Dale, Dec 19 2020 *)

a(n) = A007917(A002276(n)). - Michel Marcus, Jun 29 2025

More terms from Michel Marcus, Jun 29 2025

cp's OEIS Frontend

A340539 a(n) is the least prime that is the concatenation of two n-digit primes, and such that the concatenation of the same primes in the other order is also prime or 0 if no such prime exists.

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A379140 Numbers k such that the greatest prime < 10^k and the least prime > 10^k share no decimal digits.

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Maple

A038694 Smallest odd number with n prime factors all of different number of decimal digits.

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PARI

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A046445 Smallest composite with n prime factors that are distinct in length.

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A078728 a(n) is the smallest m such that m < 10^n, 10^n + m is prime and if the natural number k is such that 1 < k < 10 and 3 doesn't divide k*10^n + m then k*10^n+m is prime.

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A097515 a(n) = (largest prime < 10^n) + (smallest prime > 10^n).

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Mathematica

Formula

Extensions

A098963 Position of the smallest n-digit prime in the decimal expansion of e.

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A099657 a(n) is the least prime following A002277(n) repdigits.

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Examples

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Mathematica

A099661 a(n) is the least prime following A002281(n) repdigits.

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Examples

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Programs

A078728 a(n) is the smallest m such that m < 10^n, 10^n + m is prime and if the natural number k is such that 1 < k < 10 and 3 doesn't divide k10^n + m then k10^n+m is prime.