A135779 -id:A135779 - cp's OEIS frontend

A095862 Numbers whose number of decimal digits and number of divisors are equal.

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 121, 169, 289, 361, 529, 841, 961, 1003, 1006, 1007, 1011, 1018, 1027, 1037, 1041, 1042, 1043, 1046, 1047, 1055, 1057, 1059, 1067, 1073, 1077, 1079, 1081, 1082, 1094, 1099

Offset: 1

Ray Chandler, Jun 28 2004

An element of this sequence is prime iff it has 2 digits, which is the case for a(2)=11 through a(22); sequence A096489 lists exactly these and the leading term a(1)=1 (the only noncomposite number which is not prime). - M. F. Hasler, Nov 29 2007

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A096489 (= a(1)..a(22) = noncomposite elements of this sequence).

Cf. A135772-A135779 (analog for bases 2...9).

q:= n-> is(length(n)=numtheory[tau](n)):
select(q, [$1..1100])[];  # Alois P. Heinz, Jun 18 2021

Select[Range[1100],IntegerLength[#]==DivisorSigma[0,#]&] (* Harvey P. Dale, Oct 19 2015 *)

Edited by M. F. Hasler, Nov 29 2007

A135772 Numbers having equal number of divisors and binary digits.

Original entry on oeis.org

1, 2, 3, 4, 8, 10, 14, 15, 16, 32, 44, 45, 50, 52, 63, 64, 128, 130, 135, 136, 138, 152, 154, 165, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 256, 441, 484, 512, 567, 592, 656, 688, 752, 848, 891, 944, 976

Offset: 1

M. F. Hasler, Nov 28 2007

			a(1) = 1 since 1 has 1 divisor and 1 binary digit.
a(2), a(3) = 2, 3 since 2 = 10_2 and 3 = 11_2 have 2 divisors and 2 binary digits.
a(4) = 4 = 100_2 is the only number with 3 binary digits having 3 divisors.
8, 10, 14, 15 have 4 binary digits and 4 divisors.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000005, A070939.

Cf. A135773, A135774, A135775, A135776, A135777, A135778, A135779, A095862.

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 2] &] (* G. C. Greubel, Nov 08 2016 *)

for(d=1,10,for(n=2^(d-1),2^d-1,d==numdiv(n)&print1(n", ")))

from sympy import divisor_count
def ok(n): return divisor_count(n) == n.bit_length()
print(list(filter(ok, range(1, 977)))) # Michael S. Branicky, Jul 29 2021

A135778 Numbers having number of divisors equal to number of digits in base 8.

Original entry on oeis.org

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 121, 169, 289, 361, 514, 515, 517, 519, 526, 527, 533, 535, 537, 538, 542, 543, 545, 551, 553, 554, 559, 562, 565, 566, 573, 579, 581, 583, 586, 589, 591, 597, 611, 614, 622, 623, 626, 629, 633, 634

Offset: 1

M. F. Hasler, Nov 28 2007

Since 8 is not a prime, no element > 1 of the sequence A001018(k)=8^k (having k+1 digits in base 8, but much more divisors) can be member of this sequence. Also, no power of a prime less than 8 can be in the sequence, since it will always have fewer divisors than digits in base 8. However all powers of 11 up to 11^6 are in this sequence, having the same number of digits (in base 8) than the same power of 8 (since 6 = floor(log(11/8)/log(8))) and also that number of divisors (since 11 is prime).

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 8 as in any other base).
They are followed by the primes (having 2 divisors {1,p}) between 8 and 8^2 - 1 (to have 2 digits in base 8).
Then come the squares of primes (3 divisors) between 8^2 = 100_8 and 8^3 - 1 = 777_8.
These are followed by all semiprimes and cubes of primes (4 divisors) between 8^3 and 8^4 - 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A135772-A135779, A095862.

Select[Range[1000],IntegerLength[#,8]==DivisorSigma[0,#]&] (* Harvey P. Dale, Mar 04 2016 *)

for(d=1,4,for(n=8^(d-1),8^d-1,d==numdiv(n)&print1(n", ")))

More terms from Harvey P. Dale, Mar 04 2016

A135773 Numbers having number of divisors equal to number of digits in base 3.

Original entry on oeis.org

1, 3, 5, 7, 9, 25, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 81, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428, 436, 452, 475, 477, 507, 508, 524, 531, 539, 548, 549

Offset: 1

M. F. Hasler, Nov 28 2007

Since 3 is a prime, any power 3^k has k+1 divisors { 3^i ; i=0..k } and the same number of digits in base 3; thus the sequence A000244(k) = 3^k is a subsequence of this one. Note that no number in between 3^4 and 3^5, neither in between 3^6 and 3^7, is in this sequence.

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 3).
2 has 2 divisors but only 1 digit in base 3, so it is not member of the sequence.
a(2)..a(4) = 3, 5, 7 all have 2 divisors and 2 digits in base 3.
81 = 3^4 = 10000_3 is the only number with 5 divisors and 5 digits in base 3, so it is followed by 243 = 3^5 = 100000_3.

G. C. Greubel, Table of n, a(n) for n = 1..1250

Cf. A135772 - A135779, A095862.

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 3] &] (* G. C. Greubel, Nov 08 2016 *)

for(d=1,6,for(n=3^(d-1),3^d-1,d==numdiv(n)&print1(n", ")))

A135774 Numbers having number of divisors equal to number of digits in base 4.

Original entry on oeis.org

1, 5, 7, 11, 13, 25, 49, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218

Offset: 1

M. F. Hasler, Nov 28 2007

Since 4 is not a prime, no element > 1 of the sequence A000302(k)=4^k (having k+1 digits in base 4 but 2k+1 divisors) can be member of this sequence. However all powers of 5 up to 5^6 are in this sequence, having the same number of digits (in base 4) than the same power of 4 (since (5/4)^6 < 4 < (5/4)^7) and also that number of divisors.

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 4 as in any other base).
a(2)..a(5) = 5, 7, 11, 13 are the primes (to have 2 divisors {1,p}) between 4 and 4^2 - 1 (to have 2 digits in base 4).
a(6), a(7) = 25, 49 are the squares of primes (3 divisors) between 4^2 = 100[4] and 4^3 - 1 = 333_4.
They are followed by all semiprimes and cubes of primes (4 divisors) between 4^3 and 4^4 - 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A135772-A135779, A095862.

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 4] &] (* G. C. Greubel, Nov 08 2016 *)

for(d=1,4,for(n=4^(d-1),4^d-1,d==numdiv(n)&print1(n", ")))

A135775 Numbers having number of divisors equal to number of digits in base 5.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 25, 49, 121, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213, 214, 215, 217, 218, 219, 221, 226, 235, 237, 247, 249, 253, 254, 259, 262, 265, 267

Offset: 1

M. F. Hasler, Nov 28 2007

Since 5 is a prime, any power 5^k has k+1 divisors { 5^i ; i=0..k } and the same number of digits in base 5; thus the sequence A000351(k)=5^k is a subsequence of this one. It also includes the powers of 7 up to 7^4, since (7/5)^4 < 5 < (7/5)^5.

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 5).
2,3,4 have 2 resp. 3 divisors but only 1 digit in base 5, so they are not members of the sequence.
a(2) = 5 = 10_5 has 2 divisors { 1, 5 } and 2 digits in base 5, so it is (the second term) in this sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A135772-A135779, A095862.

Select[Range[300],DivisorSigma[0,#]==IntegerLength[#,5]&] (* Harvey P. Dale, Mar 14 2013 *)

for(d=1,4,for(n=5^(d-1),5^d-1,d==numdiv(n)&print1(n", ")))

A135776 Numbers having number of divisors equal to number of digits in base 6.

Original entry on oeis.org

1, 7, 11, 13, 17, 19, 23, 29, 31, 49, 121, 169, 217, 218, 219, 221, 226, 235, 237, 247, 249, 253, 254, 259, 262, 265, 267, 274, 278, 287, 291, 295, 298, 299, 301, 302, 303, 305, 309, 314, 319, 321, 323, 326, 327, 329, 334, 335, 339, 341, 343, 346, 355, 358

Offset: 1

M. F. Hasler, Nov 28 2007

Since 6 is not a prime, no element > 1 of the sequence A000400(k)=6^k (having k+1 digits in base 6, but much more divisors) can be a member of this sequence. However, all powers of 7 up to 7^11 are in this sequence, having the same number of digits (in base 6) as the same power of 6 (since 11 = floor(log(7/6)/log(6))) and also that number of divisors (since 7 is prime).

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 6 as in any other base).
They are followed by the primes (having 2 divisors {1,p}) between 6 and 6^2 - 1 (to have 2 digits in base 6).
Then come the squares of primes (3 divisors) between 6^2 = 100_6 and 6^3 - 1 = 555_6.
These are followed by all semiprimes and cubes of primes (4 divisors) between 6^3 and 6^4 - 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Abel Jansma, E_8 Symmetry Structures in the Ising model, Master's thesis, University of Amsterdam, 2018.

Cf. A135772-A135779, A095862.

Select[Range[500], DivisorSigma[0, #] == IntegerLength[#, 6] &] (* G. C. Greubel, Nov 08 2016 *)

for(d=1,4,for(n=6^(d-1),6^d-1,d==numdiv(n)&print1(n", ")))

A135777 Numbers having number of divisors equal to number of digits in base 7.

Original entry on oeis.org

1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 121, 169, 289, 343, 346, 355, 358, 362, 365, 371, 377, 381, 382, 386, 391, 393, 394, 395, 398, 403, 407, 411, 413, 415, 417, 422, 427, 437, 445, 446, 447, 451, 453, 454, 458, 466, 469, 471, 473, 478, 481

Offset: 1

M. F. Hasler, Nov 28 2007

Since 7 is a prime, any power 7^k has k+1 divisors { 7^i ; i=0..k } and the same number of digits in base 7; thus the sequence A000420(k)=7^k is a subsequence of this one.

			a(1) = 1 since 1 has 1 divisor and 1 digit (in base 7 as in any other base).
All other numbers have at least 2 divisors so there are no other members of the sequence below a(2) = 7 = 10_7 having 2 divisors { 1, 7 } and 2 digits in base 7.

G. C. Greubel, Table of n, a(n) for n = 1..1250
Abel Jansma, E_8 Symmetry Structures in the Ising model, Master's thesis, University of Amsterdam, 2018.

Cf. A135772-A135779, A095862.

Select[Range[500],DivisorSigma[0,#]==IntegerLength[#,7]&] (* Harvey P. Dale, Feb 14 2015 *)

for(d=1,4,for(n=7^(d-1),7^d-1,d==numdiv(n)&print1(n", ")))

cp's OEIS Frontend

A095862 Numbers whose number of decimal digits and number of divisors are equal.

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A135772 Numbers having equal number of divisors and binary digits.

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A135778 Numbers having number of divisors equal to number of digits in base 8.

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A135773 Numbers having number of divisors equal to number of digits in base 3.

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A135774 Numbers having number of divisors equal to number of digits in base 4.

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A135775 Numbers having number of divisors equal to number of digits in base 5.

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A135776 Numbers having number of divisors equal to number of digits in base 6.

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A135777 Numbers having number of divisors equal to number of digits in base 7.