A004022 -id:A004022 - cp's OEIS frontend

A182051 Primes with a majority of one digit.

2, 3, 5, 7, 11, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811, 877, 881, 883, 887, 911, 919, 929, 977, 991, 997, 1117, 1151

Offset: 1

Arkadiusz Wesolowski, Apr 08 2012

a(n+5) = A164937(n) for n <= 89.

			1151 is prime and the number of ones is greater than the number of all other digits, so this number is in the sequence.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Supersequence of A164937 and of A164968.

Cf. A004022, A105992.

lst = {}; Do[i = IntegerDigits[n]; If[PrimeQ[n] && Count[i, First[Commonest@i]] > IntegerLength[n]/2, AppendTo[lst, n]], {n, 10^4}]; lst

A242541 Undulating primes: prime numbers whose digits follow the pattern A, B, A, B, A, B, A, B, ...

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, 1212121, 1616161, 323232323

Offset: 1

J. Lowell, May 17 2014

All numbers in this sequence with three or more digits must have an odd number of digits. Any number with an even number of digits that follow this pattern is divisible by a number of the form 1010101...1010101 where the number of digits is one less than the number of digits in the original number.
Union of A004022 and A032758. - Arkadiusz Wesolowski, May 17 2014
Because A may equal B, 11 (and other prime repunits) are terms in this sequence (but not of A032758). - Harvey P. Dale, May 26 2015

			121 = 11*11 is not prime and thus is not a term of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..135

Cf. A004022, A032758, A033619.

select(isprime,[$0..99,seq(seq(seq(a*(10^(d+1)-10^(d+1 mod 2))/99 + b*(10^d - 10^(d mod 2))/99, b=0..9),a=1..9,2),d=3..9,2)]); # Robert Israel, Jul 08 2016

Select[Union[Flatten[Table[FromDigits[PadRight[{},n,#]],{n,9}]&/@ Tuples[ Range[0,9],2]]],PrimeQ] (* Harvey P. Dale, May 26 2015 *)

from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    yield from primerange(2, 100)
    yield from (t for t in (int((A+B)*d2+A) for d2 in count(1) for A in "1379" for B in "0123456789") if isprime(t))
print(list(islice(agen(), 51))) # Michael S. Branicky, Jun 09 2022

A092696 Smoothly undulating palindromic primes of the form (12*10^n-21)/99.

Original entry on oeis.org

1212121, 12121212121, 1212121212121212121212121212121212121212121

Offset: 1

Rick L. Shepherd, Mar 04 2004

The De Geest link calls these smoothly undulating palindromic primes. Corresponding n are given in A062209. Equivalently, primes of the form 1212...121: Decimal digits "12" repeated k times with 1 appended (or "21" repeated k times with 1 prefixed). Corresponding k are given in A056803. The next term, a(4), has "12" repeating A056803(4) = 69 times and length A062209(4) = 2*A056803(4) + 1 = 139 decimal digits.

Cf. A056803 (number of 12's (or 21's)), A062209 (corresponding decimal digit lengths).

Cf. A032758, A033619, A030291, A059168, A242541, A004022.

a(n) = (4*10^A062209(n)-7)/33. - M. F. Hasler, Jul 30 2015

Edited by M. F. Hasler, Jul 30 2015

A107845 Transposable-digit primes: Primes such that if any single pair of adjacent digits is transposed the result is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 137, 179, 199, 277, 311, 331, 337, 373, 379, 397, 419, 491, 577, 613, 733, 811, 877, 911, 919, 977, 991, 1013, 1031, 1091, 1117, 1213, 1231, 1579, 1777, 1811, 1999, 2113, 2131, 2399, 2411, 2677, 2699, 2719

Offset: 1

Rick L. Shepherd, May 24 2005

A single transposition error (of decimal digits) when recording or communicating such a prime still results in a prime (possibly the same prime). A003459 and A004022 (its subsequence) are subsequences. A003459 is also a subsequence of A068652.

			137 is a term because it is prime and 173 and 317 are also prime.
173 is not a term because 713 is not prime (even though 173 and 137 are prime). {Hence none of 137,173,317,371,713,731 is a term of A003459.}
3119 is a term because it is prime and 1319 and 3191 are primes.
As 3119, 1193, 1931 and 9311 are all prime, 3119 is also a term of A068652.
Finally, although 1913 is also prime, neither 1139, 1391, 3911, 9113, nor 9131 is prime so 3119's twelve total permutations are not terms of A003459.

Giovanni Resta, Table of n, a(n) for n = 1..5000 (first 3565 terms from Felix Fröhlich)

Cf. A003459 (absolute primes), A004022 (repunit primes), A068652 (every cyclic permutation is prime).

swap[lst_List, i_Integer] := Block[{lsu = lst}, ReplacePart[ lsu, {i -> lsu[[i + 1]], i + 1 -> lsu[[i]]}]]; fQ[n_] := Block[{id = IntegerDigits@ n, l = Floor@ Log10@ n}, And @@ Table[ PrimeQ@ FromDigits@ swap[id, j], {j, l}] == True]; Select[ Prime@ Range@ 500, fQ] (* Robert G. Wilson v, Nov 29 2014 *)

eva(n) = subst(Pol(n), x, 10)
switchdigits(v, pos) = my(vt=v[pos]); v[pos]=v[pos+1]; v[pos+1]=vt; v
is(n) = my(d=digits(n)); for(k=1, #d-1, if(!ispseudoprime(eva(switchdigits(d, k))), return(0))); 1
forprime(p=1, , if(is(p), print1(p, ", "))) \\ Felix Fröhlich, Sep 21 2019

Offset changed from 0 to 1 by Felix Fröhlich, Sep 21 2019

A108387 Doubly-transmutable primes: primes such that simultaneously exchanging pairwise all occurrences of any two disjoint pairs of distinct digits results in a prime.

Original entry on oeis.org

113719, 131797, 139177, 139397, 193937, 313979, 317179, 317399, 331937, 371719, 739391, 779173, 793711, 793931, 797131, 917173, 971713, 971933, 979313, 997391, 1111793, 3333971, 7777139, 9999317, 13973731, 31791913, 79319197, 97137379

Offset: 1

Rick L. Shepherd, Jun 02 2005

By my definition of (a nontrivial) transmutable prime, each digit of each term must be capable of being an ending digit of a prime, so this sequence is a subsequence of A108387, primes p such that p's set of distinct digits is {1,3,7,9}. The repunit primes (A004022), which would otherwise trivially be (doubly-)transmutable and primes whose distinct digits are other proper subsets of {1,3,7,9} are excluded here by the two-disjoint-pair condition.

			a(0) = 113719 as this is the first prime having four distinct digits and such that all three simultaneous pairwise exchanges of all distinct digits as shown below 'transmutate' the original prime into other primes:
(1,3) and (7,9): 113719 ==> 331937 (prime),
(1,7) and (3,9): 113719 ==> 779173 (prime),
(1,9) and (3,7): 113719 ==> 997391 (prime).

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A108387, A108388 (transmutable primes), A108389 (transmutable primes with four distinct digits), A107845 (transposable-digit primes), A003459 (absolute primes).

N:= 100: # to get a(1) to a(N)
R:= NULL: count:= 0:
S[1] := [0=1,1=3,2=7,3=9]:
S[2] := [0=3,1=1,2=9,3=7]:
S[3] := [0=7,1=9,2=1,3=3]:
S[4] := [0=9,1=7,2=3,3=1]:
g:= L -> add(L[i]*10^(i-1),i=1..nops(L)):
for d from 6 while count < N do
for n from 4^d to 2*4^d-1 while count < N do
  L:= convert(n,base,4)[1..-2];
  if nops(convert(L,set)) < 4 then next fi;
  if andmap(isprime,[seq(g(subs(S[i],L)),i=1..4)]) then
    R:= R, g(subs(S[1],L)); count:= count+1;
  fi
od od:
R; # Robert Israel, Jul 27 2020

Offset changed by Robert Israel, Jul 27 2020

A108388 Transmutable primes: Primes with distinct digits d_i, i=1,m (2<=m<=4) such that simultaneously exchanging all occurrences of any one pair (d_i,d_j), i<>j results in a prime.

Original entry on oeis.org

13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 179, 191, 199, 313, 331, 337, 773, 911, 919, 1171, 1933, 3391, 7717, 9311, 11113, 11119, 11177, 11717, 11933, 33199, 33331, 77171, 77711, 77713, 79999, 97777, 99991, 113111, 131111, 131113, 131171, 131311

Offset: 1

Rick L. Shepherd, Jun 02 2005

a(n) is a term iff a(n) is prime and binomial(m,2) 'transmutations' (see example) of a(n) are different primes. A083983 is the subsequence for m=2: one transmutation (The author of A083983, Amarnath Murthy, calls the result of such a digit-exchange a self-complement. {Because I didn't know until afterwards that this sequence was a generalization of A083983 and as this generalization always leaves some digits unchanged for m>2, I've chosen different terminology.}). A108389 ({1,3,7,9}) is the subsequence for m=4: six transmutations. Each a(n) corresponding to m=3 (depending upon its set of distinct digits) and having three transmutations is also a member of A108382 ({1,3,7}), A108383 ({1,3,9}), A108384 ({1,7,9}), or A108385 ({3,7,9}). The condition m>=2 only eliminates the repunit (A004022) and single-digit primes. The condition m<=4 is not a restriction because if there were more distinct digits, they would include even digits or the digit 5, in either case transmuting into a composite number. Some terms such as 1933 are reversible primes ("Emirps": A006567) and the reverse is also transmutable. The transmutable prime 3391933 has three distinct digits and is also a palindromic prime (A002385). The smallest transmutable prime having four distinct digits is A108389(0) = 133999337137 (12 digits).

			179 is a term because it is prime and its three transmutations are all prime:
exchanging ('transmuting') 1 and 7: 179 ==> 719 (prime),
exchanging 1 and 9: 179 ==> 971 (prime) and
exchanging 7 and 9: 179 ==> 197 (prime).
(As 791 and 917 are not prime, 179 is not a term of A068652 or A003459 also.).
Similarly, 1317713 is transmutable:
exchanging all 1's and 3s: 1317713 ==> 3137731 (prime),
exchanging all 1's and 7s: 1317713 ==> 7371173 (prime) and
exchanging all 3s and 7s: 1317713 ==> 1713317 (prime).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A108382, A108383, A108384, A108385, A108386, A108389 (transmutable primes with four distinct digits), A083983 (transmutable primes with two distinct digits), A108387 (doubly-transmutable primes), A006567 (reversible primes), A002385 (palindromic primes), A068652 (every cyclic permutation is prime), A003459 (absolute primes).

from gmpy2 import is_prime
from itertools import combinations, count, islice, product
def agen(): # generator of terms
    for d in count(2):
        for p in product("1379", repeat=d):
            p, s = "".join(p), sorted(set(p))
            if len(s) == 1: continue
            if is_prime(t:=int(p)):
                if all(is_prime(int(p.translate({ord(c):ord(d), ord(d):ord(c)}))) for c, d in combinations(s, 2)):
                    yield t
print(list(islice(agen(), 50))) # Michael S. Branicky, Dec 15 2023

A147556 Largest prime factor of prime(n)-th repunit number.

Original entry on oeis.org

11, 37, 271, 4649, 513239, 265371653, 5363222357, 1111111111111111111, 11111111111111111111111, 77843839397, 57336415063790604359, 2212394296770203368013, 201763709900322803748657942361

Offset: 1

Farideh Firoozbakht, Dec 26 2008

nonn

The sequence of repunit primes is a subsequence of this sequence.

			Prime(15)=47 and (10^47-1)/9 = 35121409*316362908763458525001406154038726382279, so a(15)=316362908763458525001406154038726382279.

Table of n, a(n) for n = 1..70
Makoto Kamada, Factorizations of 11...11 (Repunit).

Cf. A000040, A002275, A003020, A004022, A006530, A019328, A147555.

Table[FactorInteger[FromDigits[PadRight[{},n,1]]][[-1,1]],{n,Prime[ Range[15]]}] (* Harvey P. Dale, Feb 23 2016 *)

a(n) = A003020(A000040(n)) = A006530(A002275(A000040(n))) = A006530(A019328(A000040(n))). - Ray Chandler, May 11 2017

Edited by Ray Chandler, Apr 06 2011
terms to a(66) in b-file from Ray Chandler, May 11 2017
a(67)-a(70) in b-file from Max Alekseyev, Apr 26 2022

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

A206159 Numbers needing at most two digits to write all positive divisors in decimal representation.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 22, 31, 33, 41, 55, 61, 71, 77, 101, 113, 121, 131, 151, 181, 191, 199, 211, 311, 313, 331, 661, 811, 881, 911, 919, 991, 1111, 1117, 1151, 1171, 1181, 1511, 1777, 1811, 1999, 2111, 2221, 3313, 3331, 4111, 4441, 6661, 7177, 7717, 8111, 9199, 10111, 11113

Offset: 1

Reinhard Zumkeller, Feb 05 2012

The terms of A203897 having all divisors in A020449 (in particular, the first 1022 terms) are a subsequence. - M. F. Hasler, May 02 2022

Since 1 and the term itself are divisors, one must only check repdigits and those containing only 1 and another digit. - Michael S. Branicky, May 02 2022

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..500 from M. F. Hasler)

Cf. A027750, A031955, A011531, A106101, A004022, A062634.

Cf. A203897 (an "almost subsequence"), A020449 (primes with only digits 0 & 1), A095048 (number of distinct digits in divisors(n)).

Select[Range[12000],Length[Union[Flatten[IntegerDigits/@Divisors[#]]]]<3&] (* Harvey P. Dale, May 03 2022 *)

select( {is_A206159(n)=#Set(concat([digits(d)|d<-divisors(n)]))<3}, [1..10^4]) \\ M. F. Hasler, May 02 2022

from sympy import divisors
def ok(n):
    digits_used = set()
    for d in divisors(n, generator=True):
        digits_used |= set(str(d))
        if len(digits_used) > 2: return False
    return True
print([k for k in range(1, 9000) if ok(k)]) # Michael S. Branicky, May 02 2022

A095048(a(n)) <= 2.

Terms corrected by Harvey P. Dale, May 02 2022

Edited by N. J. A. Sloane, May 02 2022

A337139 Indices m of repunits R_m that are not Colombian (or self) numbers.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

Michel Marcus, Aug 19 2020

Note that 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, 5794777, 8177207 (see A004023) are terms. [Last 2 terms added by Serge Batalov, Aug 24 2021]

While all currently known A004023 terms are in this sequence, there is no clear argument that it would hold for all future values. - Serge Batalov, Aug 24 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A002275 (repunits), A004022 (repunit primes), A004023 (indices of repunit primes), A176995 (not Colombian).

Cf. A337208 (complement).

upto(n)= {my(res = List()); for(i = 1, n, if(is(i), listput(res, i); print1(i", "))); res}
is(n) = {if(n < 8, return(isprime(n))); qd = n; n = 10^n\9; r = 1 + (n-1)%9; h = (r + 9 * (r%2))/2; ld = 10; while(h + 9*qd >= n % ld, ld*=10); vs = qd - valuation(ld, 10); n %= ld; for(i = 0, qd, if(vs + vecsum(digits(n - h - 9*i)) == h + 9*i, return(1))); 0} \\ David A. Corneth, Aug 20 2020

cp's OEIS Frontend

A182051 Primes with a majority of one digit.

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A242541 Undulating primes: prime numbers whose digits follow the pattern A, B, A, B, A, B, A, B, ...

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A092696 Smoothly undulating palindromic primes of the form (12*10^n-21)/99.

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A107845 Transposable-digit primes: Primes such that if any single pair of adjacent digits is transposed the result is a prime.

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A108387 Doubly-transmutable primes: primes such that simultaneously exchanging pairwise all occurrences of any two disjoint pairs of distinct digits results in a prime.

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A108388 Transmutable primes: Primes with distinct digits d_i, i=1,m (2<=m<=4) such that simultaneously exchanging all occurrences of any one pair (d_i,d_j), i<>j results in a prime.

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A147556 Largest prime factor of prime(n)-th repunit number.

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A196104 Repdigit semiprimes (semiprimes composed of identical digits).

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