A092517 -id:A092517 - cp's OEIS frontend

A123000 a(n) is the smallest positive integer such that d(a(n))d(a(n)+1) > d(a(n-1))d(a(n-1)+1), where d(m) is the number of divisors of m and n > 1; a(1) = 1.

1, 2, 3, 5, 8, 14, 15, 20, 35, 63, 80, 99, 104, 195, 224, 384, 440, 560, 935, 1224, 1539, 2015, 2079, 5264, 5984, 12375, 21735, 41040, 78624, 123200, 126224, 156519, 176175, 201824, 313599, 338624, 395199, 453375, 638000, 1154439, 1890944

Offset: 1

Leroy Quet, Jul 06 2008

nonn

a(n) also equals the smallest positive integer such that d(a(n)(a(n)+1)) > d(a(n-1)(a(n-1)+1)).

That is, this is the sequence of indices of oblong numbers that have more divisors than the preceding oblong numbers. - Michel Marcus, Jul 13 2019

			Since a(7) = 15, we want for a(8) the smallest positive integer m such that d(m)*d(m+1) > d(15)d(16) = 4*5=20. Checking: d(16)*d(17)=10, d(17)*d(18)=12, d(18)*d(19)=12, d(19)*d(20)=12. All of these are <= 20. But d(20)*d(21) = 6*4=24, which is > 20. So a(8) = 20.

Cf. A002378 (oblong numbers), A092517.

lista(nn) = {my(m = 0, nm); for (n=1, nn, if ((nm = numdiv(n*(n+1))) > m, m = nm; print1(n, ", ")););} \\ Michel Marcus, Jul 13 2019

More terms from Max Alekseyev, Apr 26 2010

A192488 Numbers that set records for number of divisors of n(n-1).

Original entry on oeis.org

2, 3, 4, 6, 9, 15, 16, 21, 36, 64, 81, 100, 105, 196, 225, 385, 441, 561, 936, 1225, 1540, 2016, 2080, 5265, 5985, 12376, 21736, 41041, 78625, 123201, 126225, 156520, 176176, 201825, 313600, 338625, 395200, 453376, 638001, 1154440, 1890945, 2203201, 2697696, 2756160, 4921840, 6969600

Offset: 1

J. Lowell, Jul 02 2011

nonn

Places of records in A092517.

Bases for which it is easy to find divisibility rules for many numbers in those bases; in base 64 the final digit rule works for 1,2,4,8,16,32,64 and the add the digits rule works for 1,3,7,9,21,63.

			6 qualifies because 6*5=30 has 8 divisors, more than any smaller number of the form n(n-1).

Charles R Greathouse IV, Table of n, a(n) for n = 1..61

DeleteDuplicates[Table[{n,DivisorSigma[0,n(n-1)]},{n,2,7*10^6}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, May 25 2024 *)

r=0;t1=1;for(n=2,1e8,t2=numdiv(n);if(t1*t2>r,r=t1*t2;print1(n", "));t1=t2) \\ Charles R Greathouse IV, Jul 03 2011

A229949 Number of divisors of the n-th positive quarter-square.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 5, 6, 3, 8, 9, 8, 3, 8, 7, 12, 5, 12, 9, 8, 3, 12, 15, 12, 3, 8, 9, 16, 9, 20, 9, 10, 3, 12, 15, 12, 3, 12, 15, 24, 9, 16, 9, 8, 3, 16, 21, 24, 5, 12, 9, 16, 7, 24, 15, 12, 3, 16, 27, 16, 3, 12, 11, 24, 9, 16, 9, 16, 9, 36, 25, 18, 3, 8

Offset: 1

Omar E. Pol, Oct 24 2013

Also A048691 and A092517 interleaved.
The first bisection gives A048691, the number of divisors of the squares. The second bisection gives A092517, the number of divisors of the oblong numbers.
a(n) has the same parity of n.

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

Cf. A000005, A000290, A002378, A002620, A048691, A092517, A220492.

a[n_] := DivisorSigma[0, Floor[(n+1)^2/4]]; Array[a, 100] (* Amiram Eldar, Dec 03 2023 *)

a(n) = numdiv((n+1)^2\4); \\ Amiram Eldar, Dec 03 2023

a(n) = A000005(A002620(n+1)).

A330321 a(n) = Sum_{i=1..n} tau(i)*tau(i+1)/2, where tau(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

1, 3, 6, 9, 13, 17, 21, 27, 33, 37, 43, 49, 53, 61, 71, 76, 82, 88, 94, 106, 114, 118, 126, 138, 144, 152, 164, 170, 178, 186, 192, 204, 212, 220, 238, 247, 251, 259, 275, 283, 291, 299, 305, 323, 335, 339, 349, 364, 373, 385, 397, 403, 411, 427, 443, 459, 467, 471, 483, 495, 499, 511, 532, 546, 562, 570, 576, 588

Offset: 1

N. J. A. Sloane, Dec 11 2019

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000005, A092517.

Partial sums of A063123.

Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, n+1] / 2 &]; Array[a, 68]] (* Vincenzo Librandi, Jan 11 2020 *)

lista(nmax) = {my(d1 = 1, d2, s = 0); for(k = 2, nmax, d2 = numdiv(k); s += (d1 * d2 / 2); print1(s, ", "); d1 = d2);} \\ Amiram Eldar, Apr 19 2024

From Amiram Eldar, Apr 19 2024: (Start)
a(n) = A330320(n)/2.
a(n) ~ (3/Pi^2) * n * log(n)^2. (End)

A350693 Number of b > 0 which permit n^3 to be written as a sum of powers of b in n parts. Each exponent c is an integer >= 0, n^3 = b^c_1 + b^c_2 + ... + b^c_n.

Original entry on oeis.org

3, 5, 8, 7, 10, 13, 17, 19, 12, 20, 16, 18, 18, 25, 25, 21, 14, 28, 31, 34, 19, 22, 29, 34, 28, 33, 29, 38, 19, 33, 30, 31, 34, 51, 44, 30, 20, 41, 38, 44, 18, 37, 42, 52, 27, 30, 37, 59, 39, 50, 28, 35, 37, 82, 64, 44, 19, 36, 27, 36, 27, 52, 85, 65, 35, 40, 29

Offset: 2

Thomas Scheuerle, Jan 12 2022

nonn

If n^3 is written in different number bases, a(n) is an upper limit for the count of number bases which allow n^3 to be written as a base-b number with a digit sum of n (generalized Dudeney numbers).

a(n) has an upper limit in the number of divisors of n^3-n. Let d be one of these divisors, then it appears that a lower limit can be found by excluding all divisors d where d+1 does not share all its prime divisors with binomial(n^3, n) (A107444).

			a(2) = 3 because 2^3 = 2^2 + 2^2 = 4^1 + 4^1 = 7^1 + 7^0.

Cf. A000005, A046459, A092517, A107444.

a(n) = sum(d=2, n^3, s=sumdigits(n^3, d); s<=n&&(n-s)%(d-1)==0); \\ Jinyuan Wang, Jan 15 2022

a(n) <= A000005(n^3-n). Conjectured to become a(n) = A000005(n^3-n), if the definition would permit negative values for b and only the absolute value of the sum needs to be equal to n^3.

More terms from Jinyuan Wang, Jan 15 2022

cp's OEIS Frontend

A123000 a(n) is the smallest positive integer such that d(a(n))*d(a(n)+1) > d(a(n-1))*d(a(n-1)+1), where d(m) is the number of divisors of m and n > 1; a(1) = 1.

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A192488 Numbers that set records for number of divisors of n(n-1).

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A229949 Number of divisors of the n-th positive quarter-square.

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A330321 a(n) = Sum_{i=1..n} tau(i)*tau(i+1)/2, where tau(n) = A000005(n) is the number of divisors of n.

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A350693 Number of b > 0 which permit n^3 to be written as a sum of powers of b in n parts. Each exponent c is an integer >= 0, n^3 = b^c_1 + b^c_2 + ... + b^c_n.

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A123000 a(n) is the smallest positive integer such that d(a(n))d(a(n)+1) > d(a(n-1))d(a(n-1)+1), where d(m) is the number of divisors of m and n > 1; a(1) = 1.