A215199 -id:A215199 - cp's OEIS frontend

A075036 Smaller of two smallest consecutive numbers with 2n divisors.

2, 14, 44, 104, 2511, 735, 29888, 2295, 6075, 5264, 2200933376, 5984, 689278976, 156735, 180224, 21735, 2035980763136, 223244, 9399153082499072, 458864, 41680575, 701443071, 2503092614937444351, 201824, 2707370000, 29785673727, 46977524, 5475519, 1737797404898095794225152

Offset: 1

Amarnath Murthy, Sep 03 2002

nonn

There cannot be two consecutive numbers with the same odd number of divisors as both cannot be squares.
These numbers have the property that a(n) * (a(n) + 1) has 4*n^2 divisors. - David A. Corneth, Jun 24 2016
Conjecture: if a term k is even, the highest p-adic order of k (the maximum may be attained by several p's) occurs at p=2 and the highest p-adic order of k+1 occurs at p=3. If a term k is odd, the highest p-adic order of k occurs at p=3 and the highest p-adic order of k+1 occurs at p=2. - Chai Wah Wu, Mar 12 2019
a(49) = 378401464109375, a(58) = 79921490583489592950783. - Jon E. Schoenfield, May 07 2022
a(51) = 34210814718574592, a(55) = 2481402804069375, a(57) = 394311388855795712. - Jon E. Schoenfield, Nov 06 2023 - Nov 08 2023

			a(4) = 104 as tau(104) = tau(105) = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..48
Ray Chandler, Known terms and upper limits.

Cf. A000005, A005237, A039832, A049103, A174456, A215197, A215199, A274357, A274358, A274359.

a[n_] := (For[k=1, ! (DivisorSigma[0, k] == 2*n && DivisorSigma[0, k+1] == 2*n), k++]; k); Array[a, 10] (* Giovanni Resta, Jun 24 2016 *)

a(n) = my(k=1); while(numdiv(k)!=2*n || numdiv(k+1)!=2*n, k++); k \\ Felix Fröhlich, Jun 24 2016

a(n) <= A215199(n-1) for n > 1. Conjecture: if p is prime, then a(p) = A215199(p-1). This conjecture is true if the conjecture in A215199 is true. The b-file of A215199 thus shows that a(p) = A215199(p-1) for prime p < 1279. - Chai Wah Wu, Mar 12 2019

a(5)-a(24) from Max Alekseyev, Mar 12 2009
a(25)-a(28) from Giovanni Resta, Jun 24 2016
a(29) from Chai Wah Wu, Mar 12 2019

A344315 a(n) is the least number k such that A048105(k) = A048105(k+1) = 2*n, and 0 if it does not exist.

Original entry on oeis.org

1, 27, 135, 2511, 2295, 6975, 5264, 12393728, 12375, 2200933376, 108224, 257499, 170624, 3684603215871, 4402431, 2035980763136, 126224, 41680575, 701443071, 46977524, 1245375, 2707370000, 4388175, 3129761024, 1890944

Offset: 0

Amiram Eldar, May 14 2021

There are no two consecutive numbers with an odd number of non-unitary divisors, since A048105(k) is odd only if k is a perfect square.

a(25) <= 1965640805422351777791, a(26) <= 3127059999. In general, a(n) <= A215199(n+1). - Daniel Suteu, May 20 2021

			a(0) = 1 since A048105(1) = A048105(2) = 0.
a(1) = 27 since A048105(27) = A048105(28) = 2.

Cf. A048105, A075036, A093548, A309181, A344314.

nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; seq[max_] := Module[{s = Table[0, {max}], k = 2, c = 0, nd1 = 0}, While[c < max, If[(nd2 = nd[k]) == nd1 && nd2 < 2*max && s[[nd2/2 + 1]] == 0, c++; s[[nd2/2 + 1]] = k - 1]; nd1 = nd2; k++]; s]; seq[7]

A048105(n) = numdiv(n) - 2^omega(n);
isok(n,k) = A048105(k) == 2*n && A048105(k+1) == 2*n;
a(n) = for(k=1, oo, if(isok(n, k), return(k))); \\ Daniel Suteu, May 16 2021

a(13)-a(24) confirmed by Martin Ehrenstein, May 20 2021

A085080 Smallest k such that n, k and n+k have the same prime signature (canonical form), or 0 if no such number exists.

Original entry on oeis.org

0, 3, 2, 0, 2, 15, 0, 0, 0, 55, 2, 63, 0, 21, 6, 0, 2, 45, 0, 637, 14, 33, 0, 351, 0, 39, 0, 147, 2, 165, 0, 0, 6, 21, 22, 0, 0, 39, 26, 20237, 2, 231, 0, 325, 18, 39, 0, 4136875, 0, 18, 6, 423, 0, 135, 10, 1375, 34, 33, 2, 90, 0, 15, 12, 0, 21, 165, 0, 207, 22, 385, 2

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 01 2003

nonn

a(n) = 2 if n and n+2 form a twin prime pair.
a(n) = 0 if n is a perfect prime power or an odd prime such that n+2 is composite.
Here is a temporary list of integers <= 1000 for which a(n) is unknown (greater than a(48) or 0): 72, 200, 288, 432, 500, 648, 800, 864, 968, 972. - Michel Marcus, David A. Corneth, Mar 08 2019
a(96) = 1841996779; a(160) = 28521479; a(448) = 184390625; a(608) = 4633767. - Michel Marcus, Mar 08 2019
From David A. Corneth, Mar 08 2019: (Start)
By Fermat's Last Theorem, a(m^e) = 0 for e > 2 and positive integer m. For example, a(216) = a(6^3) = 0.
a(n) = 0 for squares < 1000, see worked example for n = 36 for the method.
a(192) = 30927921875, a(320) = 355182331, a(480) = 7771875, a(640) = 18243947439, a(832) = 194546043, a(896) = 2157109375, a(960) = 157546875. For the values to do, they are > 10^11 if a(n) > 0.
If n is even and a(n) > 0 and the exponent of 2 in the factorization of n is the largest in the prime signature then a(n) isn't necessarily odd. Ray Chandler found n = 392 as an example where a(n) = 108 is even. (End)
a(384) = 1281916327741, a(768) <= 1367088016014857. - Daniel Suteu, Mar 18 2019; confirmed by Michel Marcus, Mar 18 2019
a(768) = 85001950390625. - Ray Chandler, Mar 26 2019

			a(12) = 63 as 12 + 63 = 75, 2^2*3 + 3^2*7 = 5^2*3, all have the prime signature p^2*q.
a(1) = 0, because the only possible value for k is then 1, giving n+k=2, with a different signature.
a(2) = 3, because 2, 3 and 2+3=5 have the same prime signature.
a(36) = 0, because if a(n) exists then k exists such that k^2 + 36 = m^2 where k^2, 36 and m^2 have the same prime signature. Rewriting 36 = m^2 - k^2 = (m - k)*(m + k) and then inspection over divisors of 36 gives no terms. Alternatively checking Pythagorean triples gives the same result. - _David A. Corneth_, Mar 08 2019

Cf. A085072 (only n and n+k have same prime signature), A215199.

a[n_?PrimeQ] := If[PrimeQ[n + 2], 2, 0]; a[2] = 3; a[36] = 0; ps[n_] := Sort[ FactorInteger[n][[;; , 2]] ]; a[n_] := Module[{k = 2, f = FactorInteger[n]}, ps0 = Sort[f[[;; , 2]]]; If[Length[f] == 1, 0, While[ps[k] != ps0 || ps[n + k] != ps0, k++]; k]]; Array[a, 71] (* Amiram Eldar, Mar 07 2019 works for n <= 71 *)

sigt(n) = vecsort(factor(n)[,2]~);
a(n) = {
  if ((n==1) || (isprimepower(n) && !isprime(n)), return(0));
  if (isprimepower(n) && !isprime(n), return(0));
  if ((n!=2) && isprime(n), if (isprime(n+2), return(2), return(0)));
  if (n==36, return(0));
  my(k=2, v = sigt(n));
  while ((sigt(k) != v) || (sigt(n+k) != v), k++);
  k;
} \\ Michel Marcus, Mar 07 2019; works for n <= 71

a(20)-a(47) from Max Alekseyev, Aug 12 2013

a(48)-a(71) from Amiram Eldar, Mar 05 2019

A324204 a(n) is the smallest denominator D of the fraction N/D with 1 <= D < 2^n which is closest to (3/2)^n.

Original entry on oeis.org

1, 1, 3, 1, 5, 23, 35, 97, 203, 409, 205, 1297, 933, 311, 10819, 18239, 37611, 118535, 135251, 394609, 217989, 72663, 24221, 2788129, 4663029, 32000089, 11702925, 3900975, 1300325, 358347383, 596378755, 1232862847, 3274265813, 6818044999, 3453941395

Offset: 1

David A. Corneth, Mar 14 2019

First 10000 terms are odd. Is that the general case?

Cf. A215199.

a[n_] := Denominator[FromContinuedFraction[ContinuedFraction[(3/2)^n][[1 ;; -2]]]]; Array[a, 35] (* Amiram Eldar, Jul 10 2019 *)

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A075036 Smaller of two smallest consecutive numbers with 2n divisors.

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A344315 a(n) is the least number k such that A048105(k) = A048105(k+1) = 2*n, and 0 if it does not exist.

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A085080 Smallest k such that n, k and n+k have the same prime signature (canonical form), or 0 if no such number exists.

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A324204 a(n) is the smallest denominator D of the fraction N/D with 1 <= D < 2^n which is closest to (3/2)^n.

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