Daniel Lignon - cp's OEIS frontend

A329383 Positive integers that have more Brazilian representations than any smaller positive integer.

1, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, 840, 1260, 1440, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880

Offset: 1

Daniel Lignon, Dec 30 2019

By analogy with highly composite numbers (A002182), these numbers could be called highly Brazilian numbers.
Also, records in A284758.
The representation n = 11_(n-1) is allowed in A066044, but it is not allowed for Brazilian numbers. Hence 3 = 11_2 = A066044(2) is not Brazilian and therefore not highly Brazilian. However, except for 3, the sequences A066044 and this one are the same.
The first time the name "highly Brazilian number" was used is in Daniel Lignon's book in reference. - Bernard Schott, Jul 27 2020

			40 is a term since 40 = 1111_3 = 55_7 = 44_9 = 22_19 and it's the smallest number with 4 representations as a Brazilian number.

D. Lignon, Dictionnaire de (presque) tous les nombres entiers, Editions Ellipses, 2012, see p. 420. [In French.]

Bernard Schott, Table of n, a(n) for n = 1..91.
Wikipédia, Nombre brésilien (last paragraph).

Cf. A066044, A279930, A284758, A309039, A309493, A371812.

A307624 Least number whose digits can be used to form exactly n distinct composite numbers (not necessarily using all digits).

Original entry on oeis.org

1, 4, 12, 18, 46, 103, 122, 104, 102, 108, 124, 128, 126, 148, 246, 468, 1002, 1008, 1137, 1077, 1014, 1055, 1044, 1022, 1124, 1126, 1079, 1145, 1037, 1224, 1266, 1448, 1379, 1039, 1367, 1036, 1057, 1034, 1027, 1047, 1024, 1023, 1025, 1029, 1026, 1068, 1247, 1235, 3579, 1234, 1257, 1289, 1239, 1236, 1278, 1245

Offset: 0

Daniel Lignon, Apr 19 2019

a(n) always exists because with 10^n, you can form exactly n composite numbers... but, in general, it's not the least.

			The digits of 103 can be used to form the numbers 1, 3, 10, 13, 30, 31, 103, 130, 301, and 310. Of these, exactly 5 are composite (10, 30, 130, 301 = 7*43, and 310). Since 103 is the smallest such number, a(5) = 103.

Cf. A002808 (composite numbers).

Cf. A076449 (the same with primes instead of composite numbers) and A307623 (the sequence of corresponding records).

f[n_] := Length[Union[ Select[FromDigits /@ Flatten[Permutations /@ Subsets[IntegerDigits[n]],  1], CompositeQ]]];
t = Table[0, {100}]; Do[ a = f[n]; If[a < 100 && t[[a + 1]] == 0, t[[a + 1]] = n], {n, 100000}]; t

A307623 Numbers that set a record for the number of distinct composite numbers that can be obtained by permuting some subset of their digits.

Original entry on oeis.org

1, 4, 12, 18, 46, 102, 108, 124, 126, 148, 246, 468, 1002, 1008, 1014, 1022, 1023, 1025, 1026, 1068, 1234, 1236, 1245, 1248, 1268, 1458, 2456, 2468, 10023, 10025, 10026, 10068, 10124, 10125, 10146, 10224, 10234, 10236, 10245, 10248, 10458, 12345, 12348, 12369

Offset: 1

Daniel Lignon, Apr 19 2019

			108 is in this sequence because the number of composite numbers which can be obtained by permuting some or all of digits of 108 is larger than the number of composite numbers obtainable in the same way for any smaller integer. With 108, you can form 9 composite numbers: 8, 10, 18, 80, 81, 108, 180, 801, 810. It's impossible to form n >= 9 composite numbers in the same way with any integer < 108.

Daniel Lignon, Table of n, a(n) for n = 1..74

Cf. A072857 (the same with primes instead of composite numbers) and A307624.

f[n_] := Length[Union[ Select[FromDigits /@ Flatten[Permutations /@ Subsets[IntegerDigits[n]],  1], CompositeQ]]];
d=-1; res={};Do[b=f[n];If[b>d,AppendTo[res,n];d=b],{n,10000}];res

A320726 Composite numbers such that all other numbers obtained from all permutations of all subsets of the digits are noncomposite.

Original entry on oeis.org

4, 6, 8, 9, 10, 20, 22, 30, 32, 33, 35, 50, 55, 70, 77, 111

Offset: 1

Daniel Lignon, Oct 19 2018

Sequence is finite since it is a subsequence of a finite sequence (A071070).

This is complete: there are only 16 terms in the sequence.

			371 is in this sequence because it's composite and none of the numbers 1, 3, 7, 13, 17, 31, 37, 137, 173, 317, 713 and 731 is composite.

Subsequence of A071070. Cf. A320725 (the same for prime numbers).

Select[Range[4, 10^3], Function[n, And[CompositeQ@ n, NoneTrue[DeleteCases[Flatten@ Map[If[Length@ # > 1, FromDigits /@ Permutations@ #, #] &, Rest@ Subsets@ IntegerDigits@ n], ?(# == n &)], CompositeQ]]]] (* _Michael De Vlieger, Nov 13 2018 *)

A320725 Prime numbers such that all other numbers obtained from all permutations of all subsets of the digits are nonprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 6469

Offset: 1

Daniel Lignon, Oct 19 2018

Sequence is finite since it is a subsequence of a finite sequence (A071062).

This is complete: there are only 14 terms in the sequence.

			449 is in this sequence because it's prime and none of the numbers 4, 9, 44, 49, 94, 494 and 944 is prime.

Subsequence of A071062.

Cf. A320726 (the same for composite numbers).

Select[Prime@ Range[10^3], NoneTrue[DeleteCases[FromDigits /@ Rest@ Union@ Apply[Join, Permutations /@ Subsets@ IntegerDigits@ #], #], PrimeQ] &] (* Michael De Vlieger, Oct 22 2018 *)

A262043 Carmichael numbers (A002997) that are not absolute Euler pseudoprimes (A033181).

Original entry on oeis.org

561, 1105, 2821, 6601, 8911, 10585, 29341, 52633, 62745, 63973, 101101, 115921, 126217, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 410041, 512461, 552721, 658801, 748657, 825265, 852841, 1024651, 1033669, 1082809, 1152271, 1193221, 1461241, 1569457, 1909001

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

These are composite numbers n such that b^(n-1) == 1 (mod n) and abs(b^((n-1)/2) mod n) <> 1 for every b coprime to n.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..131 from Daniel Lignon)

Cf. A002997 (Carmichael numbers), A033181 (absolute Euler pseudoprimes).

A263239 Euler pseudoprimes to base 9: composite integers such that abs(9^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

4, 28, 91, 121, 286, 532, 671, 703, 949, 1036, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4636, 4961, 5551, 6364, 6601, 7381, 8401, 8911, 10585, 11011, 11476, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 19684, 23521, 24046, 24661, 24727

Offset: 1

Daniel Lignon, Oct 12 2015

nonn

Even numbers are permitted since 9 is an integer square. - Charles R Greathouse IV, Oct 12 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..116 from Daniel Lignon)

Cf. A020138 (pseudoprimes to base 9).

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), A262053 (base 6), A262054 (base 7), A262055 (base 8).

eulerPseudo9Q[n_]:=(Mod[9^((n-1)/2)+1,n]==0 ||Mod[9^((n-1)/2)-1,n]==0) && Not[PrimeQ[n]];
Select[Range[2,200000],eulerPseudo9Q]

is(n) = abs(centerlift(Mod(3, n)^(n-1)))==1 && !isprime(n) && n>1 \\ Charles R Greathouse IV, Oct 12 2015

A262053 Euler pseudoprimes to base 6: composite integers such that abs(6^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

185, 217, 301, 481, 1111, 1261, 1333, 1729, 2465, 2701, 3421, 3565, 3589, 3913, 5713, 6533, 8365, 10585, 11041, 11137, 12209, 14701, 15841, 17329, 18361, 20017, 21049, 22049, 29341, 31021, 31621, 34441, 36301, 38081, 39305, 39493, 41041, 43621, 44801, 46657

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..86 from Daniel Lignon)

Cf. A006970 (base 2), A262051 (base 3), A262052 (base 5), this sequence (base 6), A262054 (base 7), A262055 (base 8).

eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[25000] + 1, eulerPseudoQ[#, 6] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(6, (2*n+1))^n == 1 ||  Mod(6, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A262052 Euler pseudoprimes to base 5: composite integers such that abs(5^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

217, 781, 1541, 1729, 5461, 5611, 6601, 7449, 7813, 11041, 12801, 13021, 13333, 14981, 15751, 15841, 16297, 21361, 23653, 24211, 25351, 29539, 30673, 38081, 40501, 41041, 44173, 44801, 46657, 47641, 48133, 53971, 56033, 67921, 75361, 79381, 90241, 98173, 100651, 102311

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (term 1..65 from Daniel Lignon)

Cf. A006970 (base 2), A262051 (base 3), this sequence (base 5), A262053 (base 6), A262054 (base 7), A262055 (base 8).

eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[27000] + 1, eulerPseudoQ[#, 5] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(5, (2*n+1))^n == 1 ||  Mod(5, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

A262051 Euler pseudoprimes to base 3: composite integers such that abs(3^((n - 1)/2)) == 1 mod n.

Original entry on oeis.org

121, 703, 1541, 1729, 1891, 2465, 2821, 3281, 4961, 7381, 8401, 8911, 10585, 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009, 29341, 30857, 31621, 31697, 41041, 44287, 46657, 47197, 49141, 50881, 52633, 55969, 63139, 63973, 72041, 74593, 75361

Offset: 1

Daniel Lignon, Sep 09 2015

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..64 from Daniel Lignon)

Cf. A006970 (base 2), this sequence (base 3), A001567 (base 4), A262052 (base 5), A262053 (base 6), A262054 (base 7), A262055 (base 8).

eulerPseudoQ[n_?PrimeQ, b_] = False; eulerPseudoQ[n_, b_] := Block[{p = PowerMod[b, (n - 1)/2, n]}, p == Mod[1, n] || p == Mod[-1, n]]; Select[2 Range[26000] + 1, eulerPseudoQ[#, 3] &] (* Michael De Vlieger, Sep 09 2015, after Jean-François Alcover at A006970 *)

for(n=1, 1e5, if( Mod(3, (2*n+1))^n == 1 ||  Mod(3, (2*n+1))^n == 2*n && bigomega(2*n+1) != 1 , print1(2*n+1", "))); \\ Altug Alkan, Oct 11 2015

cp's OEIS Frontend

User: Daniel Lignon

A329383 Positive integers that have more Brazilian representations than any smaller positive integer.

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A307624 Least number whose digits can be used to form exactly n distinct composite numbers (not necessarily using all digits).

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A307623 Numbers that set a record for the number of distinct composite numbers that can be obtained by permuting some subset of their digits.

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A320726 Composite numbers such that all other numbers obtained from all permutations of all subsets of the digits are noncomposite.

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A320725 Prime numbers such that all other numbers obtained from all permutations of all subsets of the digits are nonprime.

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A262043 Carmichael numbers (A002997) that are not absolute Euler pseudoprimes (A033181).

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A263239 Euler pseudoprimes to base 9: composite integers such that abs(9^((n - 1)/2)) == 1 mod n.

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A262053 Euler pseudoprimes to base 6: composite integers such that abs(6^((n - 1)/2)) == 1 mod n.

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A262052 Euler pseudoprimes to base 5: composite integers such that abs(5^((n - 1)/2)) == 1 mod n.

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