A102782 -id:A102782 - cp's OEIS frontend

A105992 Near-repunit primes.

101, 113, 131, 151, 181, 191, 211, 311, 811, 911, 1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111, 10111, 11113, 11117, 11119, 11131, 11161, 11171, 11311, 11411, 16111, 101111, 111119, 111121, 111191, 111211, 111611, 112111, 113111, 131111, 311111, 511111

Offset: 1

Shyam Sunder Gupta, Apr 29 2005

According to the prime glossary "a near-repunit prime is a prime all but one of whose digits are 1." This would also include {2, 3, 5, 7, 13, 17, 19, 31, 41, 61 and 71}, but this sequence only lists terms with more than two digits. - M. F. Hasler, Feb 10 2020

			a(2)=113 is a term because 113 is a prime and all digits are 1 except one.

C. Caldwell and H. Dubner, "The near repunit primes 1(n-k-1)01(1k)," J. Recreational Math., 27 (1995) 35-41.
Heleen, J. P., "More near-repunit primes 1(n-k-1)D(1)1(k), D=2,3, ..., 9," J. Recreational Math., 29:3 (1998) 190-195.

T. D. Noe, Table of n, a(n) for n = 1..1000
Chris Caldwell, The Top 20 Near-repdigit Primes
Chris Caldwell, The Prime Glossary, Near-repunit prime

Cf. A000042, A002275, A004022, A046053, A046413, A065074, A034093, A065083, A102782, A118505.

lst = {}; Do[r = (10^n - 1)/9; Do[AppendTo[lst, DeleteCases[Select[FromDigits[Permutations[Append[IntegerDigits[r], d]]], PrimeQ], r]], {d, 0, 9}], {n, 2, 14}]; Sort[Flatten[lst]] (* Arkadiusz Wesolowski, Sep 20 2011 *)

A046413 Numbers k such that the repunit of length k (11...11, with k 1's) has exactly 2 prime factors.

Original entry on oeis.org

3, 4, 5, 7, 11, 17, 47, 59, 71, 139, 211, 251, 311, 347, 457, 461

Offset: 1

Patrick De Geest, Jul 15 1998

347, 457, 461 and 701 are also terms. The only other possible terms up to 1000 are 263, 311, 509, 557, 617, 647 and 991; repunits of these lengths are known to be composite but the linked sources do not provide their factors. - Rick L. Shepherd, Mar 11 2003
The Yousuke Koide reference now shows the repunit of length 263 partially factored; 263 is no longer a possible candidate for this sequence. - Ray Chandler, Sep 06 2005
The repunit of length 263 has 3 prime factors; the repunit of length 617 has one known prime factor and a large composite. Possible terms > 1000 are 1117, 1213, 1259, 1291, 1373, 1447, 1607, 1637, 1663, 1669, 1759, 1823, 1949, 1987, 2063 & 2087. - Robert G. Wilson v, Apr 26 2010
All terms are either primes or squares of primes in A004023. In particular, the only composite below a million is 4. - Charles R Greathouse IV, Nov 21 2014
a(17) >= 509. The only confirmed term below 2500 is 701. As of July 2019, no factorization is known for the potential terms 509, 557, 647, 991, 1117, 1259, 1447, 1607, 1637, 1663, 1669, 1759, 1823, 1949, 1987, 2063, 2087, 2111, 2203, 2269, 2309, 2341, 2467, 2503, 2521, ... Unless the least prime factors of the respective composites have fewer than ~80 decimal digits and are thus accessible by massive ECM computations, there is no chance for an extension using current publicly available factorization methods. See links to factordb.com for the status of the factorization of the smallest unknown terms. - Hugo Pfoertner, Jul 30 2019

			7 is a term because 1111111 = 239*4649.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

Patrick De Geest, Repunits prime factors.
Shyam Sunder Gupta, Repunit Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 11, 327-352.
Makoto Kamada, Factorizations of 11...11 (Repunit).
Yousuke Kiode, Factorizations of Repunit Numbers.
Eric Weisstein's World of Mathematics, Repunit.
Status of (10^509-1)/9 in factordb.com.
Status of (10^557-1)/9 in factordb.com.
Status of (10^647-1)/9 in factordb.com.

Cf. A000042, A004022 (repunit primes), A046053, A102782.

Select[Range[60],PrimeOmega[FromDigits[PadRight[{},#,1]]]==2&] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Aug 26 2024 *)

More terms from Rick L. Shepherd, Mar 11 2003

a(13)-a(16) from Robert G. Wilson v, Apr 26 2010

A196104 Repdigit semiprimes (semiprimes composed of identical digits).

Original entry on oeis.org

4, 6, 9, 22, 33, 55, 77, 111, 1111, 11111, 1111111, 11111111111, 11111111111111111, 2222222222222222222, 3333333333333333333, 5555555555555555555, 7777777777777777777, 22222222222222222222222, 33333333333333333333333, 55555555555555555555555

Offset: 1

Michel Lagneau, Oct 27 2011

A semiprime can be repdigit (base 10) in only three ways. It can be a single-digit semiprime, a repunit semiprime (A102782), or a repunit prime times a prime digit {2, 3, 5, 7}. Occurs in proof that the sequence is infinite in which a(n) is the least semiprime > a(n-1) such that a(n) has no digit in common with a(n-1). - Jonathan Vos Post; corrected by Max Alekseyev, Sep 14 2022

			a(12) = 11111111111 = 21649 * 513239 is semiprime.

Max Alekseyev, Table of n, a(n) for n = 1..35

Subsequence of A046328.
Except for initial terms, subsequence of A116063.
Cf. A000042, A001358, A004023, A046413, A102782.

with(numtheory):for n from 1 to 23 do:for b from 1 to 9 do:x:=(((10^n)- 1)/9)*b:if bigomega(x)=2 then printf(`%d, `,x):else fi:od:od:

Select[FromDigits/@Flatten[Table[PadRight[{},i,n],{i,25},{n,9}],1], PrimeOmega[ #] ==2&] (* Harvey P. Dale, Mar 11 2019 *)

print1("4, 6, 9");for(n=1,20,t=10^n\9;if(bigomega(t)==2,print1(", "t)); if(isprime(t),forprime(p=2,7,print1(", "p*t)))) \\ Charles R Greathouse IV, Oct 27 2011

Union of {4, 6, 9}, A102782, 2*A004022, 3*A004022, 5*A004022, and 7*A004022. - Jonathan Vos Post and R. J. Mathar, Oct 27 2011

Edited by Max Alekseyev, Sep 14 2022

A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

Original entry on oeis.org

4, 6, 14, 15, 55, 95, 247, 447, 511, 1535, 2047, 7167, 12287, 32255, 49151, 98303, 196607, 393215, 983039, 1572863, 3145727, 6291455, 8388607, 33423359, 50331647, 117440511, 201326591, 528482303, 805306367, 1879048191, 3221225471

Offset: 1

Jonathan Vos Post, Jun 23 2007

Semiprime analog of A061712. Extended by Stefan Steinerberger. Includes the subset Mersenne semiprimes A092561.

			a(1) = 4 because the first semiprime A001358(1) is 4 (base 10) which is written 100 in binary, the latter representation having exactly 1 one.
a(2) = 6 since A001358(2) = 6 = 110 (base 2) has exactly 2 ones.
a(4) = 15 since A001358(6) = 15 = 1111 (base 2) has exactly 4 ones and, as it also has no zeros, is the smallest of the Mersenne semiprimes.

Donovan Johnson, Table of n, a(n) for n = 1..250

Cf. A000043, A000120, A000337, A000668, A001358, A007088, A061712, A085724, A089226, A089998, A089999, A091991, A092558, A092559, A092561, A092562, A081093, A102782, A110472, A110699, A110700.

Join[{4},Table[SelectFirst[Sort[FromDigits[#,2]&/@Permutations[ Join[ PadRight[{}, n,1],{0}]]],PrimeOmega[#]==2&],{n,2,40}]] (* Harvey P. Dale, Feb 06 2015 *)

A118694 Semiprimes which are divisible by the product of their digits.

Original entry on oeis.org

4, 6, 9, 15, 111, 115, 1111, 1115, 11111, 1111111, 1111117, 111111115, 1111113111, 1111711111, 11111111111, 111111111115, 1111111111113, 1111117111111, 11171111111111, 1111111111711111, 1111711111111111, 11111111111111111

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 20 2006

The Mathematica coding is only good for multidigital, nonrepunits numbers. Obviously 4, 6 and 9 are members and so are A102782: Repunit semiprimes. - Robert G. Wilson v, Jun 10 2006

			115 is in the sequence because (1) it is a semiprime, (2) the product of its digits is 1*1*5=5 and (3) 115 is divisible by 5.

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A001358, A102782, A046413, A118693.

sp:= proc(n) evalb(2=add (i[2], i=ifactors(n) [2])) end: dp:= proc(n) local m; m:=n; 1; while m<>0 do %*irem(m, 10, 'm') od; % end: select(x-> irem(x, dp(x))=0 and sp(x), sort([{4, 6, 9, seq(seq(seq(parse(cat(1$(k-j), t, 1$j)), j=0..k), t=[1, 3, 5, 7]), k=1..20)} []]))[]; # Alois P. Heinz, Nov 17 2009

lst = {}; Do[ p = Times @@ IntegerDigits@n; If[ PrimeQ@p && PrimeQ[n/p], AppendTo[lst, n]; Print[n]], {n, 275*10^6}]; lst (* Robert G. Wilson v, Jun 10 2006 *)

A007954(n)= { local(resul,ncpy); if(n<10, return(n) ); ncpy=n; resul = ncpy % 10; ncpy = (ncpy - ncpy%10)/10; while( ncpy > 0, resul *= ncpy %10; ncpy = (ncpy - ncpy%10)/10; ); return(resul); } { for(n=4,50000000, if( bigomega(n)==2, dr=A007954(n); if(dr !=0 && n % dr == 0, print1(n,","); ); ); ); } \\ R. J. Mathar, May 23 2006

a(n) = A001358(k): A007954(a(n)) | a(n). - R. J. Mathar, May 23 2006

More terms from R. J. Mathar, May 23 2006
a(12) from Robert G. Wilson v, Jun 10 2006
Further terms from Alois P. Heinz, Nov 17 2009

A361628 Sphenic numbers (products of 3 distinct primes) whose digits are primes.

Original entry on oeis.org

222, 255, 273, 322, 357, 555, 777, 2222, 2233, 2235, 2255, 2337, 2355, 2373, 2522, 2553, 2555, 2737, 2755, 3237, 3322, 3333, 3335, 3355, 3522, 3535, 3553, 3723, 5222, 5235, 5253, 5322, 5335, 5522, 5523, 5555, 5727, 5735, 5757, 7222, 7257, 7322, 7337, 7527, 7535, 7553, 7557

Offset: 1

Massimo Kofler, Mar 18 2023

Intersection of A046034 and A007304.

Cf. A102782.

Select[Flatten@ Table[FromDigits /@ Tuples[Prime[Range[4]], n], {n, 3, 4}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Mar 18 2023 *)

isok(k) = (omega(k)==3) && (bigomega(k)==3) && !#select(x->!isprime(x), digits(k)); \\ Michel Marcus, Mar 21 2023

cp's OEIS Frontend

A105992 Near-repunit primes.

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A046413 Numbers k such that the repunit of length k (11...11, with k 1's) has exactly 2 prime factors.

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A196104 Repdigit semiprimes (semiprimes composed of identical digits).

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A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

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Mathematica

A118694 Semiprimes which are divisible by the product of their digits.

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Maple

Mathematica

PARI

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Extensions

A361628 Sphenic numbers (products of 3 distinct primes) whose digits are primes.

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Mathematica

PARI