A092405 -id:A092405 - cp's OEIS frontend

A137179 a(n) = the smallest positive integer m such that d(m) + d(m+1) = n, where d(m) is the number of positive divisors of m. (a(n) is the smallest m where A092405(m) = n.)

1, 2, 3, 5, 8, 11, 15, 20, 24, 39, 35, 59, 80, 84, 195, 167, 120, 119, 224, 239, 399, 335, 440, 359, 360, 480, 1520, 539, 899, 719, 1224, 720, 840, 1079, 3135, 1259, 5183, 1260, 2400, 2160, 1680, 1679, 9408, 2880, 7056, 2639, 3024, 2520, 6240, 2519, 7055, 6929

Offset: 3

Leroy Quet, May 11 2008

nonn

Robert Israel, Table of n, a(n) for n = 3..250

Cf. A092405.

N:= 100: # for a(3)..a(N)
V:= Array(3..N):
count:= 0: dp:= 1:
for m from 1 while count < N-2 do
  d:= dp; dp:= numtheory:-tau(m+1);
  v:= d+dp;
  if v <= N and V[v] = 0 then
    V[v]:= m;
    count:= count+1;
  fi
od:
convert(V,list); # Robert Israel, Mar 31 2021

a = {}; For[n = 3, n < 60, n++, i = 1; While[ ! DivisorSigma[0, i] + DivisorSigma[0, i + 1] == n, i++ ]; AppendTo[a, i]]; a (* Stefan Steinerberger, May 18 2008 *)

More terms from Stefan Steinerberger, May 18 2008

A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

Original entry on oeis.org

-1, 1, -1, 2, 0, 4, 2, 6, 3, 7, 5, 11, 9, 13, 9, 14, 12, 18, 16, 22, 18, 22, 20, 28, 25, 29, 25, 31, 29, 37, 35, 41, 37, 41, 37, 46, 44, 48, 44, 52, 50, 58, 56, 62, 56, 60, 58, 68, 65, 71, 67, 73, 71, 79, 75, 83, 79, 83, 81, 93, 91, 95, 89, 96, 92, 100, 98, 104, 100, 108

Offset: 1

Ilya Gutkovskiy, Apr 22 2019

sign

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000005, A006218, A051950, A059851, A068762, A068773, A092405, A263084 (bisection).

Cf. A001620 (gamma), A002162.

nmax = 70; Rest[CoefficientList[Series[1/(1 - x) Sum[(-x)^k/(1 - (-x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^k DivisorSigma[0, k], {k, 1, n}], {n, 1, 70}]
Accumulate[Array[(-1)^#*DivisorSigma[0, #] &, 70]] (* Amiram Eldar, Oct 14 2022 *)

a(n) = Sum_{k=1..n} (-1)^k*A000005(k).

a(n) = n*log(n)/2 + (gamma - log(2) - 1/2)*n + O(n^(131/416 + eps)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022

A069904 Number of prime factors of n-th triangular number (with multiplicity).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 3, 5, 4, 3, 3, 3, 4, 3, 2, 4, 5, 3, 4, 5, 3, 3, 3, 5, 6, 3, 3, 5, 4, 2, 3, 5, 4, 3, 3, 3, 5, 4, 2, 5, 6, 4, 4, 4, 3, 4, 5, 5, 5, 3, 2, 4, 4, 2, 4, 8, 7, 4, 3, 3, 4, 4, 3, 5, 5, 2, 4, 5, 4, 4, 3, 5, 8, 5, 2, 4, 5, 3, 3, 5, 4, 4, 5

Offset: 1

Reinhard Zumkeller, Apr 10 2002

nonn

			A000217(8) = 8*(8+1)/2 = 36 = 2*2*3*3, therefore a(8) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A069901, A069902, A069903, A092405, A101745, A108815, A114435, A114436, A114437, A164977, A240527, A240528, A240529, A278253.

Array[Plus@@Last/@FactorInteger[ #*(#+1)/2]&,33] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)

A069904(n) = bigomega((n*(n+1))/2); \\ Antti Karttunen, Oct 07 2017

a(n) = A001222(A000217(n)).
From Antti Karttunen, Oct 07 2017: (Start)
a(n) = (A001222(n)+A001222(n+1))-1.
a(n) = A001222(A278253(n)). (End)
From Alois P. Heinz, Aug 05 2019: (Start)
a(n) = 2 <=> n in { A164977 }.
a(n) = 3 <=> n in { A108815 }.
a(n) = 4 <=> n in { A114435 }.
a(n) = 5 <=> n in { A114436 }.
a(n) = 6 <=> n in { A114437 }.
a(n) = 7 <=> n in { A240527 }.
a(n) = 8 <=> n in { A240528 }.
a(n) = 9 <=> n in { A240529 }.
a(n) = 10 <=> n im { A101745 }. (End)

A092403 a(n) = sigma(n) + sigma(n+1).

Original entry on oeis.org

4, 7, 11, 13, 18, 20, 23, 28, 31, 30, 40, 42, 38, 48, 55, 49, 57, 59, 62, 74, 68, 60, 84, 91, 73, 82, 96, 86, 102, 104, 95, 111, 102, 102, 139, 129, 98, 116, 146, 132, 138, 140, 128, 162, 150, 120, 172, 181, 150, 165, 170, 152, 174, 192, 192, 200, 170, 150, 228, 230

Offset: 1

Jon Perry, Mar 22 2004

nonn

a(n) is the area of the terrace in the level 1 of the polycube called "tower" described in A221529 where the longest side of its base is equal to n + 1, thus a(n) has a symmetric representation. - Omar E. Pol, Jul 22 2021

Cf. A000203, A092405, A221529, A237593, A346533.

Total/@Partition[DivisorSigma[1,Range[70]],2,1] (* Harvey P. Dale, Feb 19 2018 *)

for(i=1,60,print1(","sigma(i)+sigma(i+1)))

a(n) = A346533(n+1,n). - Omar E. Pol, Jul 22 2021

A350593 Numbers k such that tau(k) + tau(k+1) = 6, where tau is the number of divisors function A000005.

Original entry on oeis.org

5, 6, 7, 10, 13, 22, 37, 46, 58, 61, 73, 82, 106, 157, 166, 178, 193, 226, 262, 277, 313, 346, 358, 382, 397, 421, 457, 466, 478, 502, 541, 562, 586, 613, 661, 673, 718, 733, 757, 838, 862, 877, 886, 982, 997, 1018, 1093, 1153, 1186, 1201, 1213, 1237, 1282

Offset: 1

Jon E. Schoenfield, Jan 08 2022

nonn

Since tau(k) + tau(k+1) = 6, (tau(k), tau(k+1)) must be (1,5), (2,4), (3,3), (4,2), or (5,1); of these, (1,5) and (5,1) are impossible (tau(m) = 1 only for m=1, but then neither m+1 nor m-1 would have 5 divisors), and (3,3) is also impossible (both k and k+1 would have to be squares of primes), so (tau(k), tau(k+1)) must be either (2,4) or (4,2).
For every prime p, tau(p) = 2. For every semiprime s, tau(s) = 4, with the exception of the squares of primes; for p prime, tau(p^2) = 3, since the divisors of p^2 are 1, p, and p^2.
The only numbers that have exactly 4 divisors but are not semiprimes are the cubes of primes; for prime p, the divisors of p^3 are 1, p, p^2, and p^3.
As a result, this sequence consists of:
(1) the primes p such that (p+1)/2 is prime (A005383), with the exception of p=3 (since p+1 = 4 has 3 divisors, not 4),
(2) semiprimes of the form prime - 1 (A077065), with the exception of the semiprime 4 (since it does not have 4 divisors), and
(3) the special case k = 7, since it is the unique prime p such that p+1 has 4 divisors but is not a semiprime.
For all k > 4, tau(k) + tau(k+1) >= 6; for k = 1..4, tau(k) + tau(k+1) = 3, 4, 5, 5.

			   k  tau(k)  tau(k+1)  tau(k) + tau(k+1)
  --  ------  --------  -----------------
   1     1        2         1 + 2 = 3
   2     2        2         2 + 2 = 4
   3     2        3         2 + 3 = 5
   4     3        2         3 + 2 = 5
   5     2        4         2 + 4 = 6   so   5 = a(1)
   6     4        2         4 + 2 = 6   so   6 = a(2)
   7     2        4         2 + 4 = 6   so   7 = a(3)
   8     4        3         4 + 3 = 7
   9     3        4         3 + 4 = 7
  10     4        2         4 + 2 = 6   so  10 = a(4)
  11     2        6         2 + 6 = 8
  12     6        2         6 + 2 = 8
  13     2        4         2 + 4 = 6   so  13 = a(5)

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000005, A005383, A077065, A092405, A164977.

Numbers k such that Sum_{j=0..N-1} tau(k+j) = 2*Sum_{k=1..N} tau(k): A000040 (N=1), (this sequence) (N=2), A350675 (N=3), A350686 (N=4), A350699 (N=5), A350769 (N=6), A350773 (N=7), A350854 (N=8).

Select[Range[1300], Plus @@ DivisorSigma[0, # + {0, 1}] == 6 &] (* Amiram Eldar, Jan 08 2022 *)
Position[Total/@Partition[DivisorSigma[0,Range[1300]],2,1],6]//Flatten (* Harvey P. Dale, Sep 03 2022 *)

isok(k) = numdiv(k) + numdiv(k+1) == 6; \\ Michel Marcus, Jan 08 2022

from itertools import count, islice
from sympy import divisor_count
def A350093_gen(): # generator of terms
    a, b = divisor_count(1), divisor_count(2)
    for k in count(1):
        if a + b == 6:
            yield k
        a, b = b, divisor_count(k+2)
A350093_list = list(islice(A350093_gen(),12)) # Chai Wah Wu, Jan 11 2022

{ k : tau(k) + tau(k+1) = 6 }.
UNION(A005383 \ {3}, A077065 \ {4}, {7}).
a(n) = A164977(n+1) for n>=4. - Hugo Pfoertner, Jan 08 2022

A175143 a(1)=1. a(n) = the smallest integer > a(n-1) such that d(a(n))+d(a(n)+1) > d(a(n-1))+d(a(n-1)+1), where d(m) = the number of divisors of m.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 15, 20, 24, 35, 59, 80, 84, 119, 224, 239, 335, 359, 360, 480, 539, 719, 720, 840, 1079, 1259, 1260, 1679, 2519, 4199, 5039, 5040, 6720, 7559, 9360, 10079, 10080, 15119, 20159, 25199, 25200, 27719, 32759, 43680, 50399, 55439, 75599

Offset: 1

Leroy Quet, Feb 24 2010

nonn

Those n where A092405(n) sets records.

Nicolas proved that: (1) Except for a finite number of terms, if k is in this sequence either k or k+1 is a largely composite number (A067128). (2) Except for a finite number of terms if k is a highly composite number (A002182) then k-1 is a term of this sequence. Apparently the only exceptions of (1) are 15, 80, 224, 6720, and 9360, and the only exceptions of (2) are 1, 24, 48, 180, 840, and 45360. - Amiram Eldar, Aug 24 2019

Amiram Eldar, Table of n, a(n) for n = 1..145
Jean-Louis Nicolas, Nombres hautement composés, Acta Arithmetica, Vol. 49 (1988), pp. 395-412, alternative link. See p. 398.

Cf. A000005, A002182, A067128, A092405.

A092405 := proc(n) numtheory[tau](n)+numtheory[tau](n+1) ; end proc: read("transforms") ; a092405 :=[seq(A092405(n),n=1..90000)] ; RECORDS(a092405)[2] ; # R. J. Mathar, Mar 05 2010

d1 = 1; dm = 0; s = {}; Do[d2 = DivisorSigma[0, n]; d = d1 + d2; If[d > dm, dm = d; AppendTo[s, n - 1]]; d1 = d2, {n, 2, 80000}]; s (* Amiram Eldar, Aug 24 2019 *)
smi[n_]:=Module[{k=n+1,ds=DivisorSigma[0,n]+DivisorSigma[0,n+1]},While[ DivisorSigma[ 0,k]+DivisorSigma[0,k+1]<=ds,k++];k]; NestList[smi,1,50] (* Harvey P. Dale, Apr 25 2020 *)

Extended by Ray Chandler, Mar 05 2010

Terms beyond 80 from R. J. Mathar, Mar 05 2010

A346562 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000005 together with the sum of A000005(n-1) and A000005(n), with a(1) = 1.

Original entry on oeis.org

1, 3, 1, 4, 1, 2, 5, 1, 2, 2, 5, 1, 2, 2, 3, 6, 1, 2, 2, 3, 2, 6, 1, 2, 2, 3, 2, 4, 6, 1, 2, 2, 3, 2, 4, 2, 7, 1, 2, 2, 3, 2, 4, 2, 4, 7, 1, 2, 2, 3, 2, 4, 2, 4, 3, 6, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 8, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 6

Offset: 1

Omar E. Pol, Jul 23 2021

T(n,k) is the total number of divisors related to the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower described in A221529.

			Triangle begins:
1;
3;
1, 4;
1, 2, 5;
1, 2, 2, 5;
1, 2, 2, 3, 6;
1, 2, 2, 3, 2, 6;
1, 2, 2, 3, 2, 4, 6;
1, 2, 2, 3, 2, 4, 2, 7;
1, 2, 2, 3, 2, 4, 2, 4, 7;
1, 2, 2, 3, 2, 4, 2, 4, 3, 6;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 6;
...

The length of row n is A028310(n-1).
Row sums give A006218, n >= 1.
Leading diagonal gives A092405.
Other diagonals give A000005.
Column 1 gives the absolute values of A260196.
Companion of A346533.
Cf. A221529, A237270, A237593, A336811, A338156, A340035.

cp's OEIS Frontend

A137179 a(n) = the smallest positive integer m such that d(m) + d(m+1) = n, where d(m) is the number of positive divisors of m. (a(n) is the smallest m where A092405(m) = n.)

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A307704 Expansion of (1/(1 - x)) * Sum_{k>=1} (-x)^k/(1 - (-x)^k).

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A069904 Number of prime factors of n-th triangular number (with multiplicity).

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A092403 a(n) = sigma(n) + sigma(n+1).

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A350593 Numbers k such that tau(k) + tau(k+1) = 6, where tau is the number of divisors function A000005.

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A175143 a(1)=1. a(n) = the smallest integer > a(n-1) such that d(a(n))+d(a(n)+1) > d(a(n-1))+d(a(n-1)+1), where d(m) = the number of divisors of m.

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A346562 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000005 together with the sum of A000005(n-1) and A000005(n), with a(1) = 1.

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