A085104 -id:A085104 - cp's OEIS frontend

A258165 Odd non-Brazilian numbers > 1.

3, 5, 9, 11, 17, 19, 23, 25, 29, 37, 41, 47, 49, 53, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293, 311, 313, 317, 331, 337

Offset: 1

Daniel Lignon and Robert G. Wilson v, May 22 2015

nonn

Complement of A257521 in A144396 (odd numbers > 1).
The terms are only odd primes or squares of odd primes.
Most odd primes are present except those in A085104.
All terms which are not primes are squares of odd primes except 121 = 11^2.

			11 is present because there is no base b < 11 - 1 = 10 such that the representation of 11 in base b is a repdigit (all digits are equal). In fact, we have: 11 = 1011_2 = 102_3 = 23_4 = 21_5 = 15_6 = 14_7 = 13_8 = 12_9, and none of these representations are repdigits. - _Bernard Schott_, Jun 21 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1150

Cf. A085104, A125134, A190300, A220570, A220571, A220627, A242397, A253260, A253261, A257521.

fQ[n_] := Block[{b = 2}, While[b < n - 1 && Length@ Union@ IntegerDigits[n, b] > 1, b++]; b+1 == n]; Select[1 + 2 Range@ 170, fQ]

forstep(n=3, 300, 2, c=1; for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), c=0;  break));if(c,print1(n,", "))) \\ Derek Orr, May 27 2015

from sympy.ntheory.factor_ import digits
l=[]
for n in range(3, 301, 2):
    c=1
    for b in range(2, n - 1):
        d=digits(n, b)[1:]
        if max(d)==min(d):
            c=0
            break
    if c: l.append(n)
print(l) # Indranil Ghosh, Jun 22 2017, after PARI program

A288939 Nonprime numbers k such that k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

Original entry on oeis.org

1, 6, 14, 26, 38, 40, 46, 56, 60, 66, 68, 72, 80, 87, 93, 95, 115, 122, 126, 128, 146, 156, 158, 160, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 264, 266, 280, 282, 290, 294, 300, 303, 320, 341, 350, 363, 381, 395, 399, 404, 405, 417, 418, 436, 438, 447, 450

Offset: 1

Bernard Schott, Jun 19 2017

nonn

A163268 Union {This sequence} = A100330.

The corresponding prime numbers k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 = 1111111_k are in A194194; all these Brazilian primes belong to A085104 and A285017.

			6 is in the sequence because 6^6 + 6^5 + 6^4 + 6^3 + 6^2 + 6 + 1 = 1111111_6 = 55987 which is prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A053716, A085104, A088550, A100330, A163268, A194194, A194257, A285017.

for n from 1 to 200 do s(n):= 1+n+n^2+n^3+n^4+n^5+n^6;
if not isprime(n) and isprime(s(n)) then print(n,s(n)) else fi; od:

Select[Range@ 450, And[! PrimeQ@ #, PrimeQ@ Total[#^Range[0, 6]]] &] (* Michael De Vlieger, Jun 19 2017 *)

isok(n) = !isprime(n) && isprime(1+n+n^2+n^3+n^4+n^5+n^6); \\ Michel Marcus, Jun 19 2017

from sympy import isprime
A288939_list = [n for n in range(10**3) if not isprime(n) and isprime(n*(n*(n*(n*(n*(n + 1) + 1) + 1) + 1) + 1) + 1)] # Chai Wah Wu, Jul 13 2017

A325658 Brazilian composites of the form 1 + b + b^2 + b^3 + ... + b^k, b > 1, k > 1.

Original entry on oeis.org

15, 21, 40, 57, 63, 85, 91, 111, 121, 133, 156, 183, 255, 259, 273, 341, 343, 364, 381, 400, 507, 511, 553, 585, 651, 703, 781, 813, 820, 871, 931, 993, 1023, 1057, 1111, 1191, 1261, 1333, 1365, 1407, 1464, 1555, 1561, 1641, 1807, 1885, 1893, 1981, 2047, 2071, 2163, 2257, 2353

Offset: 1

Bernard Schott, May 12 2019

Composites that are repunits in base b >= 2 with three or more digits. If the Goormaghtigh conjecture is true, there are no composite numbers which can be represented as a string of three or more 1's in a base >= 2 in more than one way (A119598).
Only three known perfect powers belong to this sequence: 121, 343 and 400 (A208242).
Except for 121, each term of this sequence have also at least one Brazilian representation with only 2 digits.

			121 = (11111)_3, 133 = (111)_11 = (77)_18.

Robert Israel, Table of n, a(n) for n = 1..10000
Yann Bugeaud and T. N. Shorey, On the Diophantine Equation (x^m - 1)/(x-1) = (y^n - 1)/(y-1), Pacific Journal of Mathematics, Vol. 207, No 1, November 2002.
Sean A. Irvine, Java program (github)
Michel Waldschmidt, Lecture on the abc conjecture and some of its consequences, 6th World Conference on 21st Century Mathematics 2013, Lahore, p. 14 (Goormaghtigh conjecture).
Wikipedia, Goormatigh conjecture.

Complement of A085104 with respect to A053696.
Intersection of A053696 and A220571.
Cf. A119598, A208242, A325659.

N:= 3000:
Res:= NULL:
for m from 2 while 1+m+m^2 <= N do
  for k from 2 do
    v:= (m^(k+1)-1)/(m-1);
    if v > N then break fi;
    if not isprime(v) then  Res:= Res, v fi
od od:
sort(convert({Res},list)); # Robert Israel, May 13 2019

lista(nn) = {forcomposite(n=1, nn, for(b=2, sqrtint(n), my(d=digits(n, b), sd=Set(d)); if ((#d >= 3) && (#sd == 1) && (sd[1] == 1), print1(n, ", "); break);););} \\ Michel Marcus, May 18 2019

A341058 Numbers that have only one divisor that is Brazilian.

Original entry on oeis.org

7, 8, 10, 12, 13, 15, 18, 22, 27, 31, 33, 34, 38, 43, 46, 49, 51, 55, 57, 58, 69, 73, 74, 82, 85, 87, 94, 95, 106, 111, 115, 118, 121, 122, 123, 125, 127, 134, 141, 142, 145, 157, 158, 159, 166, 169, 177, 178, 183, 185, 187, 194, 201, 202, 205, 206, 209, 211, 213, 214, 218

Offset: 1

Bernard Schott, Feb 15 2021

m is a term iff m is a Brazilian prime (A085104), or m is the square of a Brazilian prime, or m = 121, the only square of prime that is Brazilian, or m = p*q >= 10 with p>q are non-Brazilian primes, or m is the cube of a Brazilian prime, or m = 12 or 18 (see corresponding examples).

			One example for each type of terms that has k divisors:
-> k=2: 7 is a Brazilian prime, hence 7 = 111_2  is a term.
-> k=3: 169 has three divisors {1, 13, 169} and 13 = 111_3 is the only divisor of 169 that is Brazilian, hence 169 is a term.
-> k=3: 121 has three divisors {1, 11, 121} and 121 = 11111_3, hence, 121 that is the only square of prime that is Brazilian, is a term.
-> k=4: 34 has four divisors {1, 2, 17, 34} and 34 = 22_16  is the only divisor of 34 that is Brazilian, hence 34 is a term.
-> k=4: 27 has four divisors {1, 3, 9, 27} and 27 = 33_8 is the only divisor of 27 that is Brazilian, hence 27 is a term.
-> k=6: only two cases: 12 and 18, these integers have each 6 divisors and only 12 = 22_5 and 18 = 33_5 are Brazilian.

Cf. A125134, A220627, A308851, A340795, A341057.

Subsequence: A085104.

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length @ Union[IntegerDigits[n, b]] > 1, b++]; b < n - 1]; Select[Range[200], DivisorSum[#, 1 &, brazQ[#1] &] == 1 &] (* Amiram Eldar, Feb 16 2021 *)

A340795(a(n)) = 1.

A341434 a(n) is the number of bases 1 < b < n in which n is divisible by its product of digits.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 3, 2, 2, 1, 5, 2, 3, 4, 6, 1, 5, 1, 5, 4, 4, 1, 9, 2, 2, 4, 5, 1, 7, 3, 9, 4, 2, 3, 12, 1, 2, 3, 10, 1, 7, 2, 7, 7, 2, 1, 15, 2, 5, 3, 6, 1, 10, 3, 10, 4, 3, 1, 14, 1, 2, 7, 14, 3, 8, 1, 6, 3, 6, 1, 20, 2, 3, 8, 7, 3, 7, 1, 16, 7, 2, 1, 14

Offset: 1

Amiram Eldar, Feb 11 2021

			a(3) = 1 since 3 is divisible by its product of digits only in base 2: 3 = 11_2 and 1*1 | 3.
a(6) = 2 since 6 is divisible by its product of digits in 2 bases: in base 4, 6 = 12_4 and 1*2 | 6, and in base 5, 6 = 11_5 and 1*1 | 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007602, A068953, A080221, A085104, A088323, A119598.

q[n_, b_] := (p = Times @@ IntegerDigits[n, b]) > 0 && Divisible[n, p]; a[n_] := Count[Range[2, n], _?(q[n, #] &)]; Array[a, 100]

a(n) = sum(b=2, n-1, my(x=vecprod(digits(n, b))); x && !(n%x)); \\ Michel Marcus, Feb 12 2021

a(n) > 0 for all numbers n > 2 since n in base b = n-1 is 11.
a(n) > 1 for all even numbers > 4 since n in base b = n-2 is 12. Similarly, a(n) > 1 for all composite numbers > 4 since if n = k*m, then n is divisible by its product of digits in bases n-m and n-k.
a(p) > 1 for primes p in A085104.
a(p) > 2 for primes p in A119598 (i.e., 31, 8191, ...).
a(n) >= A088323(n), with equality if n = 4 or if n is a prime.

A343774 Primes of the form (c^k+1)/(c+1) not having a representation in the form (b^q-1)/(b-1), where b, c > 1 and k, q > 2.

Original entry on oeis.org

3, 11, 61, 521, 547, 683, 2731, 9091, 13421, 19141, 43691, 61681, 152381, 174763, 185641, 224071, 398581, 909091, 1151041, 1623931, 1824841, 2031671, 2796203, 3341101, 4778021, 5200081, 7027567, 8987221, 10678711, 15790321, 22796593, 25058741, 31224301, 32222107

Offset: 1

Bernard Schott, Apr 29 2021

The exponents k, q are necessarily primes.
Equivalently: primes of the form (c^k+1)/(c+1) that are not Brazilian: intersection of A059055 and A220627.
Except for 3 where k = 3, all the terms of this sequence are of the form (c^k+1)/(c+1) with k prime >= 5.
The only known prime of this form with k prime >= 5 that is not present is 43 = (2^7+1)/(2+1) because also 43 = (7^3+1)/(7+1) = (6^3-1)/(6-1) = 111_6, so 43 belongs to A002383.

			3 = (2^3+1)/(2+1) is not Brazilian, hence 3 is a term.
11 = (2^5+1)/(2+1) is not Brazilian, hence 11 is a term.
547 = (3^7+1)/(3+1) is not Brazilian, hence 547 is a term.
9091 = (10^5+1)/(10+1) is not Brazilian, hence 9091 is a term.

H. Dubner and T. Granlund, Primes of the form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

Primes of the form (b^k-1)/(b-1) = A085104 (Brazilian primes).
Primes of the form (c^q+1)/(c+1) = A059055.
Primes of the form (b^k-1)/(b-1) and (c^q+1)/(c+1): A002383 \ {3} is a subsequence, but, maybe the intersection (conjecture).
Primes of the form (b^k-1)/(b-1) but not (c^q+1)/(c+1) = A225148.
Primes of the form (c^q+1)/(c+1) but not (b^k-1)/(b-1) = this sequence.
Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = A343775.

isc(p) = for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1); ); );
isnotb(p) = for (b=2, p-1, my(d=digits(p, b), md=vecmin(d)); if ((#d > 2) && (md == 1) && (vecmax(d) == 1), return (0)); ); return (1);
isok(p) = isprime(p) && isc(p) && isnotb(p); \\ Michel Marcus, May 01 2021

More terms from Michel Marcus, Apr 30 2021

A376298 Numbers which are the sum of at least three successive terms of a geometric progression.

Original entry on oeis.org

7, 13, 14, 15, 21, 26, 28, 30, 31, 35, 39, 40, 42, 43, 45, 49, 52, 56, 57, 60, 62, 63, 65, 70, 73, 75, 77, 78, 80, 84, 85, 86, 90, 91, 93, 98, 104, 105, 111, 112, 114, 117, 119, 120, 121, 124, 126, 127, 129, 130, 133, 135, 140, 143, 146, 147, 150, 154, 155, 156, 157

Offset: 1

Andrew Howroyd, Sep 19 2024

nonn

Multiples of terms in A053696.

Numbers of the form m*(b^n-1)/(b-1) for n > 2 and b > 1, m > 0.

			7 is a term because 7 = 1 + 2 + 4.
13 is a term because 13 = 1 + 3 + 9.
14 is a term because 14 = 2 + 4 + 8.
15 is a term because 15 = 1 + 2 + 4 + 8.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Wikipedia, Geometric progression.

Cf. A053696, A084733, A085104.

B(k,lim)={vector(logint(lim*(k-1)+1,k)-2, i, (k^(i+2) - 1)/(k-1))}
upto(lim=200)={my(v=concat(vector(sqrtint(lim)-1, k, B(k+1,lim)))); Set(concat(vector(#v, i, my(t=v[i]); t*[1..lim\t])))}

A179625 Legal generalized repunit prime numbers.

Original entry on oeis.org

5, 7, 13, 31, 43, 73, 157, 211, 241, 1093, 2801, 19531, 22621, 30941, 55987, 88741, 245411, 292561, 346201, 797161, 5229043, 8108731, 12207031, 25646167, 305175781, 321272407, 917087137, 16148168401, 2141993519227, 10778947368421, 17513875027111, 610851724137931, 50544702849929377

Offset: 1

Tim Johannes Ohrtmann, Jan 09 2011

nonn

In Chris Caldwell's sense, a legal generalized repunit prime is a prime number of the form (b^p-1)/(b-1) such that 3 <= b <= 5*p, b != 10, and p prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..640, replacing an earlier file of T. D. Noe
Chris Caldwell, The Top Twenty: Generalized Repunit
Robert Granger and Andrew Moss, Generalised Mersenne numbers revisited, arXiv:1108.3054 [math.NT], 2011-2012.

Cf. A076481, A086122, A165210, A102170 (repunit primes in bases 3, 5, 6, and 7)

This sequence except for the term 5 is subsequence of A085104.

lim=10^17; n=1; Sort[Reap[While[p=Prime[n]; b=3; While[num=Cyclotomic[p,b]; b<=5p && num<=lim, If[b!=10 && PrimeQ[num], Sow[num]]; b++]; b>3, n++]][[2,1]]]

upTo(lim)=my(v=List(),t);forprime(p=2,log(2*lim+1)\log(3),for(b=3,5*p,if(b==10,next);t=(b^p-1)/(b-1);if(t>lim,break);if(isprime(t),listput(v,t))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Aug 21 2011

A308238 Nonprimes k such that k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

Original entry on oeis.org

1, 20, 21, 30, 60, 86, 172, 195, 212, 224, 258, 268, 272, 319, 339, 355, 365, 366, 390, 398, 414, 480, 504, 534, 539, 543, 567, 592, 626, 654, 735, 756, 766, 770, 778, 806, 812, 874, 943, 973, 1003, 1036, 1040, 1065, 1194, 1210, 1239, 1243, 1264, 1309, 1311

Offset: 1

Bernard Schott, May 16 2019

nonn

A240693 Union {this sequence} = A162862.

The corresponding prime numbers, (11111111111)_k, are Brazilian primes and belong to A085104 and A285017 (except 11).

			(11111111111)_20 = (20^11 - 1)/19 = 10778947368421 is prime, thus 20 is a term.

Cf. A162861, A162862, A240693, A286301.
Intersection of A064108 and A285017.
Similar to A182253 for k^2+k+1, A286094 for k^4+k^3+k^2+k+1, A288939 for k^6+k^5+k^4+k^3+k^2+k+1.

[1] cat [n:n in [2..1500]|not IsPrime(n) and IsPrime(Floor((n^11-1)/(n-1)))]; // Marius A. Burtea, May 16 2019

filter:= n -> not isprime(n) and isprime((n^11-1)/(n-1)) : select(filter, [$2..5000]);

Select[Range@ 1320, And[! PrimeQ@ #, PrimeQ@ Total[#^Range[0, 10]]] &] (* Michael De Vlieger, Jun 09 2019 *)

isok(n) = !isprime(n) && isprime(polcyclo(11, n)); \\ Michel Marcus, May 19 2019

A343775 Primes that are neither of the form (c^q+1)/(c+1) and nor of the form (b^k-1)/(b-1) for any b, c > 1 and k, q primes > 2.

Original entry on oeis.org

2, 5, 17, 19, 23, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367

Offset: 1

Bernard Schott, Apr 29 2021

Equivalently, non-Brazilian primes that are not of the form (c^q+1)/(c+1) for some c > 1, q prime > 2.

Equals A220627 \ A059055.

Primes of the form (b^k-1)/(b-1) = A085104 (Brazilian primes).
Primes of the form (c^q+1)/(c+1) = A059055.
Primes of the form (b^k-1)/(b-1) and also (c^q+1)/(c+1): A002383 \ {3} is a subsequence, but, maybe the intersection (conjecture).
Primes of the form (b^k-1)/(b-1) but not (c^q+1)/(c+1) = A225148.
Primes of the form (c^q+1)/(c+1) but not (b^k-1)/(b-1) = A343774.
Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = this sequence.

isc(p) = for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1); ); );
isnotb(p) = for (b=2, p-1, my(d=digits(p, b), md=vecmin(d)); if ((#d > 2) && (md == 1) && (vecmax(d) == 1), return (0)); ); return (1);
isok(p) = isprime(p) && !isc(p) && isnotb(p); \\ Michel Marcus, May 01 2021

cp's OEIS Frontend

A258165 Odd non-Brazilian numbers > 1.

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A288939 Nonprime numbers k such that k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

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A325658 Brazilian composites of the form 1 + b + b^2 + b^3 + ... + b^k, b > 1, k > 1.

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A341058 Numbers that have only one divisor that is Brazilian.

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A341434 a(n) is the number of bases 1 < b < n in which n is divisible by its product of digits.

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A343774 Primes of the form (c^k+1)/(c+1) not having a representation in the form (b^q-1)/(b-1), where b, c > 1 and k, q > 2.

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A376298 Numbers which are the sum of at least three successive terms of a geometric progression.

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A179625 Legal generalized repunit prime numbers.