A220136 -id:A220136 - cp's OEIS frontend

A326709 Squares of composites such that beta(m) = (tau(m) - 3)/2 where beta(m) = A220136(m) is the number of Brazilian representations of m and tau(m) = A000005(m) is the number of divisors of m.

16, 36, 64, 81, 100, 144, 196, 225, 256, 324, 441, 484, 576, 625, 676, 729, 784, 900, 1024, 1089, 1156, 1225, 1296, 1444, 1764, 1936, 2025, 2116, 2304, 2500, 2601, 2704, 2916, 3025, 3136, 3249, 3364, 3600, 3844, 3969, 4096, 4225, 4356, 4624, 4761, 4900, 5184, 5476, 5625

Offset: 1

Bernard Schott, Aug 29 2019

This sequence is the second subsequence of A326707: squares of composites which have no Brazilian representation with three digits or more.
As tau(m) = 2 * beta(m) + 3, the number of divisors of these squares of composites m is odd with tau(m) >= 5.
The corresponding composites are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 42, ...

			a(1) = 16: tau(16) = 5 and beta(16) = 1 with 16 = 4^2 = 22_7.
a(3) = 64: tau(64) = 7 and beta(64) = 2 with 64 = 8^2 = 44_15 = 22_31.
a(5) = 100: tau(100) = 9 and beta(100) = 3 with 100 = 10^2 = 55_19 = 44_24 = 22_49.

Index entries for sequences related to Brazilian numbers

Subsequence of A000290.
Intersection of A062312 and A326707.
Cf. A326707 = A326708 Union {this sequence} with empty intersection.
Cf. A048691 (number of divisors of n^2).
Cf. A000005 (tau), A220136 (beta).

brazQ[n_, b_] := Length@Union@IntegerDigits[n, b] == 1; beta[n_] := Sum[Boole @ brazQ[n, b], {b, 2, n - 2}]; aQ[n_] := beta[n] == (DivisorSigma[0, n] - 3)/2; Select[Select[Range[75], CompositeQ]^2, aQ] (* Amiram Eldar, Sep 06 2019 *)

A326380 Numbers m such that beta(m) = tau(m)/2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

7, 13, 15, 21, 26, 40, 43, 57, 62, 73, 80, 85, 86, 91, 93, 111, 114, 124, 127, 129, 133, 146, 157, 170, 171, 172, 183, 211, 215, 219, 222, 228, 241, 242, 259, 266, 285, 292, 307, 312, 314, 333, 341, 343, 365, 366, 381, 399, 421, 422, 438, 444, 455, 463, 468, 471, 482, 507, 518, 532, 549, 553, 555, 585, 601, 614, 624

Offset: 1

Bernard Schott, Jul 03 2019

As tau(m) = 2 * beta(m), the terms of this sequence are not squares. Indeed, there are 3 subsequences which realize a partition of this sequence (see examples):
1) Non-oblong composites which have only one Brazilian representation with three digits or more, they form A326387.
2) Oblong numbers that have exactly two Brazilian representations with three digits or more; these oblong integers are a subsequence of A167783 and form A326385.
3) Brazilian primes for which beta(p) = tau(p)/2 = 1, they are in A085104 \ {31, 8191}.

			One example for each type:
15 = 1111_2 = 33_4 with tau(15) = 4 and beta(15) = 2.
3906 = 62 * 63 = 111111_5 = 666_25 = (42,42)_86 = (31,31)_125 = (21,21)_185 = (18,18)_216 = (14,14)_278 = 99_433 = 77_557 = 66_650 = 33_1301 = 22_1952, so tau(3906) = 24 with beta(3906) = 12.
43 = 111_6 is Brazilian prime, so tau(43) = 2 and beta(43) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Brazilian numbers.

Cf. A000005 (tau), A220136 (beta).
Cf. A085104 (Brazilian primes).
Subsequence of A167782.
Cf. A326378 (tau(m)/2 - 2), A326379 (tau(m)/2 - 1), A326381 (tau(m)/2 + 1), A326382 (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).

beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
isok(n) = beta(n) == numdiv(n)/2; \\ Michel Marcus, Jul 03 2019

A326378 Numbers m such that beta(m) = tau(m)/2 - 2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

6, 12, 20, 30, 56, 72, 90, 110, 132, 210, 240, 272, 306, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1482, 1560, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660, 3782

Offset: 1

Bernard Schott, Jul 02 2019

As tau(m) = 2 * (2 + beta(m)), the terms of this sequence are not squares. Indeed, there exists only one family that satisfies this relation and these integers are exactly the oblong numbers that have no Brazilian representation with three digits or more.

There are no integers such as beta(m) = tau(m)/2 - q with q >= 3.

			1) tau(m) = 4 and beta(m) = 0: m = 6 which is not Brazilian.
2) tau(m) = 6 and beta(m) = 1: m = 12, 20.
   12 = 3 * 4 = 22_5, 20 = 4 * 5 = 22_9.
3) tau(m) = 8 and beta(m) = 2: m = 30, 56, 110, 506, 2162, 3422, ...
   30 = 5 * 6 = 33_9 = 22_14, 56 = 7 * 8 = 44_13 = 22_27.
4) tau(m) = 10 and beta(m) = 3: m = 272, ...
   272 = 16 * 17 = 88_32 = 44_67 = 22_135.
5) tau(m) = 12 and beta(m) = 4: m = 72, 90, 132, 306, 380, 650, 812, 992, ...
   72 = 8 * 9 = 66_11 = 44_17 = 33_23 = 22_35.

Amiram Eldar, Table of n, a(n) for n = 1..800
Bernard Schott, Relation beta = f(tau).
Index entries for sequences related to Brazilian numbers.

Cf. A000005 (tau), A220136 (beta).
Subsequence of A002378 (oblong numbers).
Cf. A326379 (tau(m)/2 - 1), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), A326382 (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).
Cf. A326384 (oblongs with tau(m)/2 - 1), A326385 (oblongs with tau(m)/2).

beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
isok(n) = beta(n) == numdiv(n)/2 - 2; \\ Michel Marcus, Jul 08 2019

A326379 Numbers m such that beta(m) = tau(m)/2 - 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

2, 3, 5, 8, 10, 11, 14, 17, 18, 19, 22, 23, 24, 27, 28, 29, 32, 33, 34, 35, 37, 38, 39, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 65, 66, 67, 68, 69, 70, 71, 74, 75, 76, 77, 78, 79, 82, 83, 84, 87, 88, 89, 92, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 112, 113, 115, 116

Offset: 1

Bernard Schott, Jul 03 2019

As tau(m) = 2 * (1 + beta(m)), the terms of this sequence are not squares. Indeed, there are 3 subsequences which realize a partition of this sequence (see examples):
1) Non-oblong composites which have no Brazilian representation with three digits or more, they form A326386.
2) Oblong numbers that have only one Brazilian representation with three digits or more. These oblong integers are a subsequence of A167782 and form A326384.
3) Non Brazilian primes, then beta(p) = tau(p)/2 - 1 = 0.

			One example for each type:
10 = 22_4 and tau(10) = 4 with beta(10) = 1.
42 = 6 * 7 = 222_4 = 33_13 = 22_20 and tau(42) = 8 with beta(42) = 3.
17 is no Brazilian prime with tau(17) = 2 and beta(17) = 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Brazilian numbers.

Cf. A000005 (tau), A220136 (beta).
Cf. A220627 (subsequence of non Brazilian primes).
Cf. A326378 (tau(m)/2 - 2), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), A326382 (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).

beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
isok(n) = beta(n) == numdiv(n)/2 - 1; \\ Michel Marcus, Jul 03 2019

A326382 Numbers m such that beta(m) = tau(m)/2 + 2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

32767, 65535, 67053, 2097151, 4381419, 7174453, 9808617, 13938267, 14348906, 19617234, 21523360, 29425851, 39234468, 43046720, 48686547, 49043085, 58851702, 61035156, 68660319, 71270178, 78468936, 88277553, 98086170, 107894787, 115174101, 117703404, 134217727, 142540356, 175965517

Offset: 1

Bernard Schott, Jul 08 2019

As tau(m) = 2 * (beta(m) - 2) , the terms of this sequence are not squares.
There are 2 subsequences which realize a partition of this sequence (see array in link and examples):
1) Non-oblong composites which have exactly three Brazilian representations with three digits or more, they are in A326389.
2) Oblong numbers that have exactly four Brazilian representations with three digits or more. These integers have been found through b-file of Rémy Sigrist in A290869. These oblong integers are a subsequence of A309062.
There are no primes that satisfy this relation.

			One example for each type:
1) The divisors of 32767 are {1, 7, 31, 151, 217, 1057, 4681, 32767} and tau(32767) = 8; also, 32767 = M_15 = R(15)_2 = 77777_8 = (31,31,31)_32 = (151,151)_216 = (31,31)_1056 = 77_4680 so beta(32767) = 6 with beta'(32767) = 3 and beta"(32767)= 3. The relation is beta(32767) = tau(32767)/2 + 2 = 6.
2) 61035156 = 7812 * 7813 is oblong with tau(61035156) = 144. The four Brazilian representations with three digits or more are 61035156 = R(12)_5 = 666666_25 = (31,31,31,31)_125 = (156,156,156)_625, so beta"(61035156) = 4 and beta(61035156) = tau(61035156)/2 + 2 = 74.

Bernard Schott, Array of relations beta = f(tau)
Index entries for sequences related to Brazilian numbers

Cf. A000005 (tau), A220136 (beta).
Subsequence of A167782, A167783 and A290869.
Cf. A326378 (tau(m)/2 - 2), A326379 (tau(m)/2 - 1), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), this sequence (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).
Cf. A309062, A326389.

Missing a(18) inserted by Bernard Schott, Jul 20 2019

A326386 Non-oblong composites m such that beta(m) = tau(m)/2 - 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

8, 10, 14, 18, 22, 24, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 48, 50, 51, 52, 54, 55, 58, 60, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 82, 84, 87, 88, 92, 94, 95, 96, 98, 99, 102, 104, 105, 106, 108, 112, 115, 116, 117, 118, 119, 120, 122, 123, 125, 126, 128, 130, 134, 135, 136

Offset: 1

Bernard Schott, Jul 12 2019

As tau(m) = 2 * (1 + beta(m)), the terms of this sequence are not squares.
The number of Brazilian representations of a non-oblong number m with repdigits of length = 2 is beta'(n) = tau(n)/2 - 1.
This sequence is the first subsequence of A326379: non-oblong composites which have no Brazilian representation with three digits or more.

			tau(m) = 4 and beta(m)=1 for m = 8, 10, 14, 22, 27, 33, 34, 35, 38, ... 8 = 22_3,
tau(m) = 6 and beta(m)=2 for m = 18, 28, 32, 44, 45, 50, ... 18 = 33_5 = 22_8,
tau(m) = 8 and beta(m)=3 for m = 24, 54, 66, 70, ... 24 = 44_5 = 33_7 = 22_11,
tau(m) = 10 and beta(m) = 4: 48, 112, ... 48 = 66_7 = 44_11 = 33_15 = 22_23.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bernard Schott, beta = f(tau) - Abstract.
Index entries for sequences related to Brazilian numbers.

Cf. A000005 (tau), A220136 (beta).
Subsequence of A308874 and of A326379.
Cf. A326387 (non-oblongs with tau(m)/2), A326388 (non-oblongs with tau(m)/2 + 1), A326389 (non-oblongs with tau(m)/2 + 2).

isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
isok(m) = !isprime(m) && !isoblong(m) && (beta(m) == numdiv(m)/2 - 1); \\ Michel Marcus, Jul 15 2019

A326387 Non-oblong composites m such that beta(m) = tau(m)/2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

15, 21, 26, 40, 57, 62, 80, 85, 86, 91, 93, 111, 114, 124, 129, 133, 146, 170, 171, 172, 183, 215, 219, 222, 228, 242, 259, 266, 285, 292, 312, 314, 333, 341, 343, 365, 366, 381, 399, 422, 438, 444, 455, 468, 471, 482, 507, 518, 532, 549, 553

Offset: 1

Bernard Schott, Jul 14 2019

As tau(m) = 2 * beta(m), the terms of this sequence are not squares.
The number of Brazilian representations of a non-oblong number m with repdigits of length = 2 is beta'(n) = tau(n)/2 - 1.
This sequence is the first subsequence of A326380: non-oblong composites which have only one Brazilian representation with three digits or more.

			tau(m) = 4 and beta(m) = 2 for m = 15, 21, 26, 57, 62, 85, 86, ... with 15 = 1111_2 = 33_4.
tau(m) = 8 and beta(m) = 4 for m = 40 = 1111_3 = 55_7 = 44_9 = 22_19.
tau(m) = 10 and beta(m) = 5 for m = 80 = 2222_3 = 88_9 = 55_15 = 44_19 = 22_39.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Brazilian numbers.

Cf. A000005 (tau), A220136 (beta).
Subsequence of A167782, A308874 and A326380.
Cf. A326386 (non-oblongs with tau(m)/2 - 1), A326388 (non-oblongs with tau(m)/2 + 1), A326389 (non-oblongs with tau(m)/2 + 2).

isoblong(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
isok(m) = !isprime(m) && !isoblong(m) && (beta(m) == numdiv(m)/2); \\ Michel Marcus, Jul 15 2019

A326381 Numbers m such that beta(m) = tau(m)/2 + 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

31, 63, 255, 273, 364, 511, 546, 728, 777, 931, 1023, 1365, 1464, 2730, 3280, 3549, 3783, 4557, 6560, 7566, 7812, 8191, 9114, 9331, 9841, 10507, 11349, 11718, 13671, 14043, 14763, 15132, 15624, 16383, 18291, 18662, 18915, 19608, 19682, 21845, 22351, 22698

Offset: 1

Bernard Schott, Jul 07 2019

As tau(m) = 2 * (beta(m) - 1), the terms of this sequence are not squares.
There are 3 subsequences which realize a partition of this sequence (see examples):
1) Non-oblong composites which have exactly two Brazilian representations with three digits or more, they form A326388.
2) Oblong numbers that have exactly three Brazilian representations with three digits or more; thanks to Michel Marcus, who found the smallest, 641431602. These oblong integers are a subsequence of A290869 and A309062.
3) The two Brazilian primes 31 and 8191 of the Goormaghtigh conjecture (A119598) for which beta(p) = tau(p)/2 + 1 = 2.

			One example for each type:
1) 63 = 111111_2 = 333_4 = 77_8 = 33_20 with tau(63) = 6 and beta(63) = 4.
2) 641431602 = 25326 * 25327 is oblong with tau(641431602) = 256. The three Brazilian representations with three digits or more of 641431602 are 999999_37 = (342,342,342)_1369 = (54,54,54)_3446, so beta"(641431602) = 3 and beta(641431602) = tau(641431602)/2 + 1 = 129.
3) 31 = 11111_2 = 111_5 and 8191 = 1111111111111_2 = 11_90 with beta(p) = tau(p)/2 + 1 = 2.

Wikipedia, Goormaghtigh conjecture.
Index entries for sequences related to Brazilian numbers.

Cf. A000005 (tau), A220136 (beta).
Cf. A119598 (Goormaghtigh conjecture).
Subsequence of A167783.
Cf. A326378 (tau(m)/2 - 2), A326379 (tau(m)/2 - 1), A326380 (tau(m)/2), A326382 (tau(m)/2 + 2), A326383 (tau(m)/2 + 3).

beta(n) = sum(i=2, n-2, #vecsort(digits(n, i), , 8)==1); \\ A220136
isok(n) = beta(n) == numdiv(n)/2 + 1; \\ Michel Marcus, Jul 08 2019

A326383 Numbers m such that beta(m) = tau(m)/2 + 3 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

4095, 262143, 265720, 531440, 1048575, 5592405, 11184810, 122070312, 183105468, 193710244, 244140624, 268435455, 387420488

Offset: 1

Bernard Schott, Jul 08 2019

As tau(m) = 2 * (beta(m) - 3), the terms of this sequence are not squares.

The current known terms are non-oblong composites that have exactly four Brazilian representations with three digits or more; but, maybe, there exist oblong integers that have exactly five Brazilian representations with three digits or more.

			The 24 divisors of 4095 = M_12 are {1, 3, 5, 7, 9, 13, 15, 21, 35, 39, 45, 63, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095} and tau(4095) = 24; also, 4095 = R(12)_2 = 333333_4 = 7777_8 = (15,15,15)_16, so, beta(4095) = 15 with beta'(4095)= 11 and beta''(4095) = 4. The relation is beta(4095) = tau(4095)/2 + 3 = 15 and 4095 is a term.

Index entries for sequences related to Brazilian numbers

Cf. A000005 (tau), A220136 (beta).
Subsequence of A167782, A167783 and A290869.
Cf. A326378 (tau(m)/2 - 2), A326379 (tau(m)/2 - 1), A326380 (tau(m)/2), A326381 (tau(m)/2 + 1), A326382 (tau(m)/2 + 2).

A326385 Oblong numbers m such that beta(m) = tau(m)/2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

Original entry on oeis.org

3906, 37830, 97656, 132860, 1206702, 2441406, 6034392, 10761680, 21441530, 96855122, 148705830, 203932680, 322866992, 747612306, 871696100, 1187526060, 1525878906, 1743939360, 2075941406, 3460321800, 5541090282, 8574111812, 9455714840, 12880093590, 18854722656

Offset: 1

Bernard Schott, Jul 10 2019

The number of Brazilian representations of an oblong number m with repdigits of length = 2 is beta'(n) = tau(n)/2 - 2.

This sequence is the second subsequence of A326380: oblong numbers that have exactly two Brazilian representations with three digits or more.

			3906 = 62 * 63 is oblong, tau(3906) = 24, beta(3906) = 12 with beta'(3906) = 10 and beta"(3906) = 2: 3906 = 111111_5 = 666_25 = (42,42)_92 = (31,31)_125 = (21,21)_185 = (18,18)_216 = (14,14)_278 = 99_433 = 77_557 = 66_650 = 33_130 = 22_1952.

Cf. A000005 (tau), A220136 (beta).
Subsequence of A002378 (oblong numbers) and of A167783.
Cf. A326378 (oblongs with tau(m)/2 - 2), A326384 (oblongs with tau(m)/2 - 1), A309062 (oblongs with tau(m)/2 + k, k >= 1).

a(6)-a(25) from Giovanni Resta, Jul 11 2019

cp's OEIS Frontend

A326709 Squares of composites such that beta(m) = (tau(m) - 3)/2 where beta(m) = A220136(m) is the number of Brazilian representations of m and tau(m) = A000005(m) is the number of divisors of m.

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A326380 Numbers m such that beta(m) = tau(m)/2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

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A326378 Numbers m such that beta(m) = tau(m)/2 - 2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

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A326379 Numbers m such that beta(m) = tau(m)/2 - 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

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A326382 Numbers m such that beta(m) = tau(m)/2 + 2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

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A326386 Non-oblong composites m such that beta(m) = tau(m)/2 - 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

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A326387 Non-oblong composites m such that beta(m) = tau(m)/2 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

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A326381 Numbers m such that beta(m) = tau(m)/2 + 1 where beta(m) is the number of Brazilian representations of m and tau(m) is the number of divisors of m.

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