A085104 -id:A085104 - cp's OEIS frontend

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A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

1, 6, 8, 12, 14, 15, 20, 21, 24, 27, 33, 38, 50, 54, 57, 62, 66, 69, 75, 77, 78, 80, 90, 99, 105, 110, 111, 117, 119, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 168, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278

Offset: 1

Bernard Schott, Dec 18 2012

nonn

All these numbers are in A002384 but not in A053182.
The generated prime numbers n^2 + n + 1 are in A185632.
All the generated numbers n^2 + n + 1 = 111_n are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A085104, A125134, A185632.

Select[Range@ 280, And[! PrimeQ@ #, PrimeQ[#^2 + # + 1]] &] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = ! isprime(n) && isprime(n^2 + n + 1); \\ Michel Marcus, Sep 04 2013

A288783 Brazilian numbers which have only one Brazilian representation.

Original entry on oeis.org

7, 8, 10, 12, 13, 14, 16, 20, 22, 27, 33, 34, 35, 38, 39, 43, 46, 51, 55, 58, 65, 69, 73, 74, 77, 81, 82, 87, 94, 95, 106, 115, 118, 119, 121, 122, 123, 125, 127, 134, 141, 142, 143, 145

Offset: 1

Bernard Schott, Jun 15 2017

These numbers could be called 1-Brazilian numbers.
The smallest number of this sequence is 7 = 111_2 which is also the smallest Brazilian number (A125134) and the smallest Brazilian prime (A085104), and as such belongs to A329383.
a(2) = 8 is the smallest composite Brazilian number and so the smallest even composite Brazilian with 8 = 22_3 (A220571).
a(10) = 27 is the smallest odd composite Brazilian in this sequence because 27 = 33_8 but 15 is the smallest odd composite Brazilian with 15 = 1111_2 = 33_4 so with two representations.
121 is the only square of prime which is Brazilian with 121 = 11111_3.
In this sequence, there are:
1) The Brazilian primes (except for 31 and 8191) and the only square of prime 121 which are all repunits in a base >= 2 with a string of at least three 1's.
2) The composite numbers which are such that n = a * b = (aa)_(b-1) with 1 < a < b-1 with only one such product a * b.

			13 = 111_3; 127 = 1111111_2.
20 = 2 * 10 = 22_9; 55 = 5 * 11 = 55_10; 69 = 3 * 23 = 33_22.
31 = 11111_2 = 111_5 so 31 is not a term.

D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, 2012, page 420.

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.

Cf. A085104, A125134, A220570, A220571, A284758, A290015, A290016, A290017, A290018, A329383.

Select[Range@ 145, Function[n, Count[Range[2, n - 2], b_ /; SameQ @@ IntegerDigits[n, b]] == 1]] (* Michael De Vlieger, Jun 16 2017 *)

A340795 a(n) is the number of divisors of n that are Brazilian.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 0, 1, 0, 2, 2, 1, 0, 3, 0, 2, 1, 3, 0, 3, 1, 3, 1, 1, 2, 3, 0, 1, 2, 4, 0, 4, 1, 2, 2, 1, 0, 5, 1, 2, 1, 3, 0, 3, 1, 5, 1, 1, 0, 6, 0, 2, 3, 4, 2, 3, 0, 2, 1, 5, 0, 6, 1, 1, 2, 2, 2, 4, 0, 6, 2, 1, 0, 7, 1, 2, 1, 4, 0, 6

Offset: 1

Bernard Schott, Jan 21 2021

The cases a(n) = 0 and a(n) = 1 are respectively detailed in A341057 and A341058.

			For n = 16, the divisors are 1, 2, 4, 8 and 16. Only 8 = 22_3 and 16 = 22_7 are Brazilian numbers, so a(16) = 2.
For n = 30, the divisors are 1, 2, 3, 5, 6, 10, 15 and 30. Only 10 = 22_4, 15 = 33_4 and 30 = 33_9 are Brazilian numbers, so a(30) = 3.
For n = 49, the divisors are 1, 7 and 49. Only 7 = 111_2 is Brazilian, so a(49) = 1 although 49 that is square of prime <> 121 is not Brazilian.

David A. Corneth, Table of n, a(n) for n = 1..10000
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A085104, A125134, A220627, A308851, A340796, A340797, A341057, A341058.

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; a[n_] := DivisorSum[n, 1 &, brazQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 21 2021 *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
a(n) = sumdiv(n, d, isb(d)); \\ Michel Marcus, Jan 24 2021

A225148 Primes of the form (k^p-1)/(k-1) not having representation in the form (m^q+1)/(m+1), where k,m > 1 and p,q > 2.

Original entry on oeis.org

127, 1093, 2801, 19531, 22621, 30941, 55987, 88741, 131071, 245411, 292561, 346201, 524287, 637421, 732541, 797161, 837931, 2625641, 3500201, 3835261, 5229043, 6377551, 8108731, 12207031, 15018571, 16007041, 21700501, 25646167, 28792661, 30397351, 35615581

Offset: 1

Thomas Ordowski, Apr 30 2013

nonn

The exponent p must be a prime p > 3. If p=3 then (k^p-1)/(k-1) = (m^q+1)/(m+1) for m=k+1 and q=3.
Are almost all primes of the form (k^p-1)/(k-1), where k > 1 and p > 3, in the sequence? Except 31 and 8191. See:
  31 = (2^5-1)/(2-1) = (5^3-1)/(5-1) = (6^3+1)/(6+1),
  8191 = (2^13-1)/(2-1) = (90^3-1)/(90-1) = (91^3+1)/(91+1).

Cf. A085104, A059055.

Numbers in A085104 but not in A059055.

Extended by T. D. Noe, Apr 30 2013

A290015 Brazilian numbers which have exactly two Brazilian representations.

Original entry on oeis.org

15, 18, 21, 26, 28, 30, 31, 32, 44, 45, 50, 52, 56, 57, 62, 64, 68, 75, 76, 85, 86, 91, 92, 93, 98, 99, 110, 111, 116, 117, 129, 133, 146, 147, 148, 153, 164, 175, 183, 188, 207, 212, 215, 219, 236, 243, 244, 245, 259, 261, 268, 275, 279, 284, 314, 316, 325, 332, 338, 341, 343, 356, 363, 365, 369, 381, 387, 388

Offset: 1

Bernard Schott, Jul 17 2017

These numbers could be called 2-Brazilian numbers.
The smallest number of this sequence is 15 which is also the smallest odd composite Brazilian in A257521 with 15 = 11111_2 = 33_4. The number 15 is highly Brazilian in A329383.
Following the Goormaghtigh conjecture, only two primes, 31 and 8191, which are both Mersenne numbers, are Brazilian in two different bases (A119598).

			18 = 2 * 9 = 22_8 = 3 * 6 = 33_5.
26 = 2 * 13 = 2 * 111_3 = 222_3 = 22_12.
31 = 11111_2 = 111_5;
8191 = 1111111111111_2 = 111_90.

Wikipedia, Goormaghtigh conjecture

Cf. A085104, A125134, A220570, A220571, A257521, A284758, A288783, A290016, A290017, A290018, A329383.

bresilienbaseb:=proc(n,b)
local r,q,coupleq:
if n0 then
return [couple[1]+1,r]
else
return [0,0]
end if
end if
end proc;
bresil:=proc(n)
local b,L,k,t:
k:=0:
for b from 2 to (n-2) do
t:=bresilienbase(n,b):
if t[1]>0 then
k:=k+1
L[k]:=[b,t[1],t[2]]:
end if:
end do:
seq(L[i],i=1..k);
end proc;
nbbresil:=n->nops([bresil(n)]);
#Numbers 2 times Brazilian
for n from 1 to 100 do if nbbresil(n)=2 then print(n,bresil(n)) else fi; od:

Mathematica

Flatten@ Position[#, 2] &@ Table[Count[Range[2, n - 2], ?(And[Length@ # != 1, Length@ Union@ # == 1] &@ IntegerDigits[n, #] &)], {n, 400}] (* _Michael De Vlieger, Jul 18 2017 *)

A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.

Original entry on oeis.org

6, 14, 39, 62, 155, 254, 3279, 5219, 16382, 19607, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67977559, 150508643, 310747739, 344964203

Offset: 1

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Author

Walter Kehowski, Jan 08 2020

Keywords

nonn
easy

Comments

Also numbers with power-spectral basis {q,p^k}. The equation q=(p^k-1)/(p-1) is equivalent to the decomposition of the identity q + p^k = pq + 1 in Z/pqZ, and it is now easily verified that {q,p^k} is the spectral basis of p*q, consisting of primes and powers.

The numbers p^(r^e)*q, where p, q, r are primes, and q=(p^(r^e)-1)/(p^(r^(e-1))-1), e>0, have power-spectral basis {q,p^(r^e)}. However, the primes q for e>1 are usually quite large, while e=1 is accessible. For example, the table in A003424 has 4738 entries with all primes q<10^12, but only 8 have y>1.

Examples

a(5) = 5*(5^3-1)/(5-1) = 5*31 = 155. The number 155 has spectral basis {31,125}.

Links

Walter Kehowski, Table of n, a(n) for n = 1..4731

Crossrefs

Cf. A003424, A023194, A023195, A085104, A330833, A330834, A330835.

Formula

a(n) = A330833(n) * A330835(n).

A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

Original entry on oeis.org

2, 2, 3, 2, 5, 2, 3, 17, 2, 7, 41, 5, 59, 2, 71, 13, 89, 101, 2, 17, 131, 3, 167, 173, 23, 29, 293, 383, 5, 13, 43, 677, 701, 743, 17, 761, 773, 827, 839, 857, 911, 1091, 1097, 5, 1163, 1181, 1193, 1217, 73, 1373, 1427, 79, 1487, 1559, 1583, 83, 2, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset: 1

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Walter Kehowski, Jan 08 2020

Keywords

nonn
easy

Examples

a(5) = 5 and, since A330834(5) = 3, then A330835(5) = (5^3-1)/(5-1) = 31 is prime.

Links

Walter Kehowski, Table of n, a(n) for n = 1..4731

Crossrefs

Cf. A003424, A023194, A023195, A085104, A330832, A330834, A330835.

A330834 The exponents k of A330832, that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=k.

Original entry on oeis.org

2, 3, 3, 5, 3, 7, 7, 3, 13, 5, 3, 7, 3, 17, 3, 5, 3, 3, 19, 5, 3, 13, 3, 3, 5, 5, 3, 3, 11, 7, 5, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 13, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 3, 5, 31, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

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Author

Walter Kehowski, Jan 08 2020

Keywords

nonn

Examples

a(5) = 3, and, since A330833(5)=5, then A330835(5)=(5^3-1)/(5-1) = 31 is prime.

Links

Walter Kehowski, Table of n, a(n) for n = 1..4731

G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.

Crossrefs

Cf. A003424, A023194, A023195, A085104, A330832, A330833, A330835.

A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

Original entry on oeis.org

3, 7, 13, 31, 31, 127, 1093, 307, 8191, 2801, 1723, 19531, 3541, 131071, 5113, 30941, 8011, 10303, 524287, 88741, 17293, 797161, 28057, 30103, 292561, 732541, 86143, 147073, 12207031, 5229043, 3500201, 459007, 492103, 552793, 25646167, 579883, 598303, 684757

Offset: 1

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Author

Walter Kehowski, Jan 08 2020

Keywords

nonn

Comments

The terms in the b-file are the same as those of A003424 with y=1, but with an ordering based on that of A330832. The ordering allows the inclusion of the only duplicate 2^5-1=31 and (5^3-1)/(5-1)=31.

Examples

a(5)=31 since A330833(5)=5, A330834(5)=3, and (5^3-1)/(5-1) = 31 is prime.

Links

Walter Kehowski, Table of n, a(n) for n = 1..4731

G. Sobczyk, The Missing Spectral Basis in Algebra and Number Theory, The American Mathematical Monthly 108(4), April 2001.

Crossrefs

Cf. A003424, A023194, A023195, A085104, A330832, A330833, A330834.

Formula

a(n) = (A330833(n) ^ A330834(n) - 1) / (A330833(n) - 1).

A125598 a(n) = ((n+1)^(n-1) - 1)/n.

Original entry on oeis.org

0, 1, 5, 31, 259, 2801, 37449, 597871, 11111111, 235794769, 5628851293, 149346699503, 4361070182715, 139013933454241, 4803839602528529, 178901440719363487, 7143501829211426575, 304465936543600121441

Offset: 1

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Author

Alexander Adamchuk, Nov 26 2006

Keywords

nonn

Comments

Odd prime p divides a(p-2).

a(n) is prime for n = {3,4,6,74, ...}; prime terms are {5, 31, 2801, ...}.

a(n) is the (n-1)-th generalized repunit in base (n+1). For example, a(5) = 259 which is 1111 in base 6. - Mathew Englander, Oct 20 2020

Links

G. C. Greubel, Table of n, a(n) for n = 1..350

Crossrefs

Cf. A000272 (n^(n-2)), A125599.

Cf. other sequences of generalized repunits, such as A125118, A053696, A055129, A060072, A031973, A173468, A023037, A119598, A085104, and A162861.

Programs

Magma

[((n+1)^(n-1) -1)/n: n in [1..25]]; // G. C. Greubel, Aug 15 2022

Mathematica

Table[((n+1)^(n-1)-1)/n, {n,25}]

Sage

[gaussian_binomial(n,1,n+2) for n in range(0,18)] # Zerinvary Lajos, May 31 2009

Formula

a(n) = ((n+1)^(n-1) - 1)/n.

a(n) = (A000272(n+1) - 1)/n.

a(2k-1)/(2k+1) = A125599(k) for k>0.

From Mathew Englander, Dec 17 2020: (Start)

a(n) = (A060072(n+1) - A083069(n-1))/2.

For n > 1, a(n) = Sum_{k=0..n-2} (n+1)^k.

For n > 1, a(n) = Sum_{j=0..n-2} n^j*C(n-1,j+1). (End)

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A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

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PARI

A288783 Brazilian numbers which have only one Brazilian representation.

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Mathematica

A340795 a(n) is the number of divisors of n that are Brazilian.

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A225148 Primes of the form (k^p-1)/(k-1) not having representation in the form (m^q+1)/(m+1), where k,m > 1 and p,q > 2.

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A290015 Brazilian numbers which have exactly two Brazilian representations.

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Maple

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A330832 Numbers of the form p*q, where p is prime and q=(p^k-1)/(p-1) is also prime for some integer k>1.

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A330833 a(n) = first prime factor p of the term A330832(n) = p*q.

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A330834 The exponents k of A330832, that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=k.

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A330835 Primes q appearing in A330832: that is, if A330832(n)=p*q, where p is prime and q=(p^k-1)/(p-1) is prime, then a(n)=q.

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