A004022 -id:A004022 - cp's OEIS frontend

A372034 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nonincreasing order.

4, 8, 9, 22, 32, 33, 44, 55, 64, 77, 88, 93, 99, 422, 633, 775, 844, 933, 993, 4222, 4442, 6333, 6655, 6663, 7533, 7744, 7775, 8444, 8884, 9663, 9993, 44222, 66333, 88444, 99633, 99933, 99993, 933333, 966333, 996663, 999993, 4442222, 6663333, 7777775, 8884444, 9663333, 9666633, 9666663

Offset: 1

Scott R. Shannon, Apr 16 2024

Is it true that no terms end with 1? A separate search on those shows none with < 70 digits. Michael S. Branicky, Apr 23 2024

Testing all products of repunit primes (A004022, A004023), there are no terms ending in 1 less than 10^3000. - Michael S. Branicky, Apr 24 2024

			The initial terms and their factorizations are:
4 = [2, 2]
8 = [2, 2, 2]
9 = [3, 3]
22 = [2, 11]
32 = [2, 2, 2, 2, 2]
33 = [3, 11]
44 = [2, 2, 11]
55 = [5, 11]
64 = [2, 2, 2, 2, 2, 2]
77 = [7, 11]
88 = [2, 2, 2, 11]
93 = [3, 31]
99 = [3, 3, 11]
422 = [2, 211]
633 = [3, 211]
775 = [5, 5, 31]
844 = [2, 2, 211]
933 = [3, 311]
993 = [3, 331]
4222 = [2, 2111]
4442 = [2, 2221]
6333 = [3, 2111]
6655 = [5, 11, 11, 11]
6663 = [3, 2221]
7533 = [3, 3, 3, 3, 3, 31]
7744 = [2, 2, 2, 2, 2, 2, 11, 11]
...

Michael S. Branicky, Table of n, a(n) for n = 1..197 (all terms with <= 21 digits)

Cf. A004022, A004023, A009996, A028867, A372053, A372029, A027746, A372055.

from sympy import factorint, isprime
def ni(s): return sorted(s, reverse=True) == list(s)
def ok(n):
    if n < 4 or isprime(n): return False
    s, f = str(n), "".join(str(p)*e for p, e in factorint(n).items())
    return ni(s+f)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Apr 23 2024

# faster for initial segment of sequence
from sympy import factorint, isprime
from itertools import islice, combinations_with_replacement as mc
def ni(s): return s == "".join(sorted(s, reverse=True))
def bgen(d):
    yield from ("".join(m) for m in mc("987654321", d))
def agen(): # generator of terms
    for d in range(1, 70):
        out = set()
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if ni(s+"".join(str(p)*e for p, e in factorint(t).items())):
                out.add(t)
        yield from sorted(out)
print(list(islice(agen(), 50))) # Michael S. Branicky, Apr 23 2024

A034093 Number of near-repunit primes that can be formed from (10^k - 1)/9 by changing just one digit from 1 to 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 5, 0, 0, 0, 0, 2, 5, 0, 4, 0, 0, 0, 3, 0, 1, 0, 0, 1, 2, 0, 4, 1, 0, 1, 2, 0, 2, 1, 0, 0, 7, 0, 4, 0, 0, 0, 2, 0, 2, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 1, 3, 0, 1, 0, 0, 1, 3, 0, 3, 0, 0, 1, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 0

Offset: 1

Felice Russo

			a(12) = 5 because from (10^12 - 1)/9 = 111111111111, by changing just one digit from 1 to 0, out of the eleven candidates, 111111111101, 111111110111, 111111011111, 111011111111 and 101111111111 are primes.

C. K. Caldwell and H. Dubner, The near repunits primes, J. Rec. Math., Vol. 27(1), 1995, pp. 35-41.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2000
Chris Caldwell, Below are all of the 12-digit Near-Repunit primes.
Chris Caldwell, Repunits.

Cf. A004022, A065074, A065083.

a = {}; Do[ p = IntegerDigits[ (10^n - 1)/9 ]; c = 0; Do[ If[ q = FromDigits[ ReplacePart[p, 0, i]]; PrimeQ[q], c++ ], {i, 2, n} ]; a = Append[a, c], {n, 1, 100} ]; a (* Robert G. Wilson v, Nov 19 2001 *)

a(n)=sum(i=1,n-2,ispseudoprime(10^n\9-10^i)) \\ Charles R Greathouse IV, May 01 2012

from sympy import isprime
def a(n):
    Rn = (10**n-1)//9
    return sum(1 for i in range(n-1) if isprime(Rn-10**i))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Nov 04 2023

More terms from Robert G. Wilson v, Nov 19 2001

Edited by N. J. A. Sloane, Oct 02 2008 at the suggestion of R. J. Mathar

A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors.

Original entry on oeis.org

4, 6, 9, 22, 33, 55, 77, 121, 202, 262, 303, 393, 505, 626, 707, 939, 1111, 1441, 1661, 1991, 3443, 3883, 7997, 10201, 13231, 15251, 18281, 19291, 20602, 22622, 22822, 24842, 26662, 28682, 30903, 31613, 33933, 35653, 37673, 38683, 39993, 60206, 60406, 60806

Offset: 1

Patrick De Geest, Jun 15 1998

Equivalently, semiprime palindromes where both prime factors are palindromes. - Franklin T. Adams-Watters, Apr 11 2011
The sequence "trivially" includes products of palindromic primes p*q where
  a) p = 2 or 3 and q has only digits < 4, as q = 11, 101, 131, 10301, 30103, ...
  b) p <= 11 and q has only digits 0 and 1, as q = 101 and repunit primes A004022
  c) p = 11 and q has only digits spaced out by zeros, as q = 101, 10301, 10501, 10601, 30103, 30203, 30403, 30703, 30803, ... - M. F. Hasler, Jan 04 2022

			The palindrome 35653 is a term since it has 2 factors, 101 and 353, both palindromic.

Lars Blomberg, Table of n, a(n) for n = 1..1000

Cf. A001358 (semiprimes), A002113 (palindromes), A002385 (palindromic primes).
Cf. A046328, A046408.
Subsequence of A188650.

Take[Select[Times@@@Tuples[Select[Prime[Range[5000]],PalindromeQ],2], PalindromeQ]// Union,50] (* Harvey P. Dale, Aug 25 2019 *)

{first(N=50, p=1) = vector(N, i, until( bigomega( p=nxt_A002113(p))==2 && vecmin( apply( is_A002113, factor(p)[,1])),); p)} \\ M. F. Hasler, Jan 04 2022

from sympy import factorint
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal):
    f = factorint(pal)
    return sum(f.values()) == 2 and all(ispal(p) for p in f)
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Aug 14 2022

Intersection of A002113 and A046368; A188649(a(n)) = a(n). - Reinhard Zumkeller, Apr 11 2011

Definition clarified by Franklin T. Adams-Watters, Apr 11 2011

More terms from Lars Blomberg, Nov 06 2015

A116692 Primes with only one distinct decimal digit. Also called repunit primes or repdigit primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 1111111111111111111, 11111111111111111111111

Offset: 1

Rick L. Shepherd, Feb 22 2006

Primes in A010785 (repdigit numbers). Union of single-digit primes and A004022 (repunit primes). A004023 shows that the next term has 317 1's. The Mersenne primes (A000668) are the binary analog (i.e., bits are all 1's).

Clifford A. Pickover, A Passion for Mathematics (2005) at 60, 297.

A004022 is a subsequence.
Cf. A004023, A010785, A000668.
A subsequence of A055387, but not of A225053.

Reference provided by Harvey P. Dale, Apr 19 2014

Definition expanded by N. J. A. Sloane, Jan 22 2023

A187614 Primes p such that the decimal representation of 1/p does not contain every digit 0-9.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 31, 37, 41, 43, 67, 73, 79, 101, 137, 239, 271, 353, 449, 757, 859, 1933, 4649, 8779, 9091, 9901, 21401, 21649, 25601, 27961, 52579, 62003, 123551, 333667, 513239, 538987, 909091, 1676321, 2071723, 2906161, 5882353, 10838689, 35121409, 52986961, 99990001, 265371653, 1056689261, 1058313049, 1360682471

Offset: 1

Michel Lagneau, Mar 12 2011

Every repunit prime (A004022) is here. There are 113 terms of A046107, having periods of up to 256, that are here. The only known unique-period prime (A007615) not here is the one having period 92092. Is this sequence finite? - T. D. Noe, Mar 13 2011

			4649 is in the sequence because 1/4649 = 0.00021510002151000215.... contain
  only the digits 0, 1, 2 and 5.

Cf. A187372.

Cf. A352023 (does not contain digit 9)

Join[{2, 3, 5}, Select[Prime[Range[4, 10000]], Length[Union[RealDigits[1/#][[1, 1]]]] < 10 &]]

from sympy import n_order, nextprime
from itertools import islice
def A187614_gen(): # generator of terms
    yield from (2,3,5)
    p = 7
    while True:
        if len(set('0'+str(10**(n_order(10, p))//p))) < 10:
            yield p
        p = nextprime(p)
A187614_list = list(islice(A187614_gen(),20)) # Chai Wah Wu, Mar 03 2022

Extended by T. D. Noe, Mar 12 2011

A262988 Number of distinct primes, including n if prime, obtained by cyclically shifting the digits of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Felix Fröhlich, Oct 06 2015

First differs from A039999 at n = 103.
Differs from A061264 iff n is a term of A004022.
a(n) = A055642(n) iff n is a term of A068652, except when n is also in A004022.

			a(1013) = 2, because of the four cyclic permutations of the digits of 1013 (1013, 131, 1310, 3101) two, namely 1013 and 131, are prime and those two primes are distinct.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A016114, A039999, A045978, A061264, A068652, A068653, A068654, A069219, A247153.

f[n_] := Block[{len = IntegerLength@ n, s = {n}}, Do[AppendTo[s, FromDigits@ RotateRight@ IntegerDigits@ s[[k - 1]]], {k, 2, len}]; DeleteDuplicates@ Select[s, PrimeQ]]; Array[ Length@ f@ # &, {87}] (* Michael De Vlieger, Oct 07 2015 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
eva(n) = x=0; for(k=1, #n, x=x+(n[k]*10^(#n-k))); x
a(n) = i=0; r=rot(digits(n)); while(r!=digits(n), if(ispseudoprime(eva(r)), i++); r=rot(r)); if(ispseudoprime(eva(r)), i++); i

A372055 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives k such that the digits of L(k) are in nondecreasing order.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 25, 35, 111, 112, 125, 222, 245, 333, 335, 445, 1225, 2225, 11125, 33445, 334445, 3333335, 3334445, 3444445, 33333445, 333333335, 334444445, 3333333335, 33333334445, 333333333335, 33333333334445, 33333333444445, 444444444444445, 2222222222222225

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Apr 22 2024

This is the union of {1,2,3,5,7}, A004022, and A372029.

			L(1) = [1], L(2) = [2,2], L(11) = [11,11], L(12) = [12,2,2,3].
See A372029 for further examples.
The prime k = 1111111111111111111 = A004022(2) will eventually be a term, since L(1111111111111111111) = [1111111111111111111,1111111111111111111], which certainly has digits in nondecreasing order.

Cf. A004022, A372029.

More terms from Michael S. Branicky, Apr 23 2024 using A004022 and A372029.

A046412 Lengths of nonsquarefree repunits.

Original entry on oeis.org

9, 18, 22, 27, 36, 42, 44, 45, 54, 63, 66, 72, 78, 81, 84, 88, 90, 99, 108, 110, 111, 117, 126, 132, 135, 144, 153, 154, 156, 162, 168, 171, 176, 180, 189, 198, 205, 207, 210, 216, 220, 222, 225, 234, 242, 243, 252, 261, 264, 270, 272, 279, 286, 288, 294, 297, 306, 308, 312, 315, 324, 330, 333, 336, 342, 351, 352

Offset: 1

Patrick De Geest, Jul 15 1998

This is the set of all positive multiples of all positive members of A087094. What is the asymptotic density of this set? - Jeppe Stig Nielsen, Dec 28 2015

Patrick De Geest, Repunits prime factors
Shyam Sunder Gupta, Repunit Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 11, 327-352.
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A004022, A046053, A084006, A087094.

remove(t -> numtheory:-issqrfree((10^t-1)/9), [$1..90]); # Robert Israel, Dec 30 2015

Select[Range[300],!SquareFreeQ[(10^#-1)/9]&] (* Harvey P. Dale, Jun 26 2013 *)

isok(n) = ! issquarefree((10^n-1)/9); \\ Michel Marcus, Dec 31 2015

a(n)=k where (10^k-1)/9 is not squarefree. - Ray Chandler, Aug 10 2003

Terms to a(60) from Ray Chandler, Aug 10 2003

a(61)-a(67) from Max Alekseyev, Apr 29 2022

A069869 Largest prime that is a concatenation of the parts of a partition of n, or 0 if no such prime exists.

Original entry on oeis.org

0, 11, 3, 211, 2111, 0, 112111, 1111211, 0, 11131111, 1121111111, 0, 111211111111, 2111111111111, 0, 31111111111111, 212111111111111, 0, 1111111111111111111, 2111111111111111111, 0, 111111111111111121111, 11111111111111111111111, 0, 211111111111111111111111

Offset: 1

Amarnath Murthy, Apr 21 2002

Conjecture: a(n) = 0 only for n = 1 or n = 3k with k>1.

			a(4) = 211 as the partitions of 4 are (4), (3,1), (2,2), (2,1,1) (1,1,1,1). The primes that can be formed are 13, 31, 211 and 211 is the largest prime using a partition.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A069870, A004022.

with(combinat):
a:= proc(n) local k, w;
      if n=1 or n>3 and irem(n, 3)=0 then return 0 fi;
      for k from 0 do w:= max(select(isprime,
        map(x-> parse(cat(x[])), [seq(permute(i)[],
          i=map(x->[x[], 1$(n-k)], partition(k)))]))[]);
        if w>0 then return w fi
      od
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, May 25 2013

f[n_] := If[ PrimeQ@n, n, If[n > 5 && Mod[n, 3] == 0, 0, Block[{len = PartitionsP[n], p = IntegerPartitions[n], t = {}}, Do[ AppendTo[t, Select[FromDigits /@ Join @@@ IntegerDigits /@ Permutations@p[[i]], PrimeQ@# &]], {i, len}]; t = Union@Flatten@t; If[Length@t > 0, Max@t, 0]] ]]; Array[f, 29]

from collections import Counter
from operator import itemgetter
from sympy.utilities.iterables import partitions, multiset_permutations
from sympy import isprime
def A069869(n):
    smax, m = 0, 0
    if n==3 or n%3:
        for s, p in sorted(partitions(n,size=True),key=itemgetter(0),reverse=True):
            if s0:
                smax=s
    return m # Chai Wah Wu, Feb 21 2024

More terms from David Wasserman, Apr 30 2003
a(8) corrected and a(16)-a(24) added by Robert G. Wilson v, Feb 06 2006
a(25) from Alois P. Heinz, May 25 2013

A165210 Primes of the form (6^m - 1)/5.

Original entry on oeis.org

7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371

Offset: 1

Rick L. Shepherd, Sep 07 2009

nonn

Prime repunits in base 6 whose representation consists of m 1's. The exponents m are in A004062. a(5) and a(6) have 55 and 99 decimal digits, respectively.

			a(2) = (6^A004062(2) - 1)/5 = (6^3 - 1)/5 = 215/5 = 43, which is 111_6.

Vincenzo Librandi, Table of n, a(n) for n = 1..9

Cf. A004062, A003464, A085104, A000668, A076481, A086122, A102170, A004022.

[a: n in [1..200] | IsPrime(a) where a is  (6^n-1) div 5 ]; // Vincenzo Librandi, Dec 09 2011

Select[Table[(6^n-1)/5, {n,0,2000}], PrimeQ] (* Vincenzo Librandi, Dec 09 2011 *)

a(n) = (6^A004062(n) - 1)/5.

cp's OEIS Frontend

A372034 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nonincreasing order.

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A034093 Number of near-repunit primes that can be formed from (10^k - 1)/9 by changing just one digit from 1 to 0.

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A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors.

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A116692 Primes with only one distinct decimal digit. Also called repunit primes or repdigit primes.

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A187614 Primes p such that the decimal representation of 1/p does not contain every digit 0-9.

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A262988 Number of distinct primes, including n if prime, obtained by cyclically shifting the digits of n.

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PARI

A372055 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives k such that the digits of L(k) are in nondecreasing order.

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A046412 Lengths of nonsquarefree repunits.

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