A075036 -id:A075036 - cp's OEIS frontend

A035004 Number of divisors of the n-th nonprime.

1, 3, 4, 4, 3, 4, 6, 4, 4, 5, 6, 6, 4, 4, 8, 3, 4, 4, 6, 8, 6, 4, 4, 4, 9, 4, 4, 8, 8, 6, 6, 4, 10, 3, 6, 4, 6, 8, 4, 8, 4, 4, 12, 4, 6, 7, 4, 8, 6, 4, 8, 12, 4, 6, 6, 4, 8, 10, 5, 4, 12, 4, 4, 4, 8, 12, 4, 6, 4, 4, 4, 12, 6, 6, 9, 8, 8, 8, 4, 12, 8, 4, 10, 8, 4, 6, 6, 4, 4, 16, 3, 4, 4, 6, 4, 12, 8, 4

Offset: 1

Jason Earls, Jul 05 2001

a(n+1) = a(n) only when a(n) is even. Will these repeat terms include each even integer >= 4? (See also A075036.) - Bill McEachen, Nov 16 2021

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

Cf. A000005, A002808, A018252, A075036.

DivisorSigma[0,Select[Range[200],!PrimeQ[#]&]] (* Harvey P. Dale, Mar 07 2016 *)

for(n=1,33, if(!isprime(n), print1(numdiv(n), ", "))) \\ edited by Michel Marcus, Jan 04 2016

a(n) = A000005(A018252(n)). - Omar E. Pol, Jan 31 2013

More terms from Lior Manor, Aug 16 2001

A190646 Least number k such that d(k-1) = d(k+1) = 2n or 0 if no such k exists, where d(n)=A000005(n).

Original entry on oeis.org

4, 7, 19, 41, 127252, 199, 26890624, 919, 17299, 6641, 25269208984376, 3401, 3900566650390624, 640063, 8418574, 18089, 1164385682220458984374, 41651, 69528379848480224609374, 128465, 34084859374, 12164095, 150509919493198394775390626, 90271

Offset: 1

Juri-Stepan Gerasimov, May 15 2011, May 20 2011

nonn

a(28) = 2319679. a(30) = 3568049.
From Chai Wah Wu, Mar 13 2019: (Start)
a(26) = 64505245697, a(27) = 3959299, a(29) = 237828698392557762563228607177734374, a(31) = 26711406049549496732652187347412109374, a(32) = 441559, a(34) = 12535291248641, a(36) = 352351, a(37) = 1749348542212388688829378224909305572509765626, a(38) = 193405731995647.
Conjecture: if p is an odd prime, then a(p) is even.
(End)

			a(16)=18089 because d(18088)=d(18090)=2*16.

Hugo van der Sanden, Table of n, a(n) for n = 1..40

Cf. A067888, A075036, A190355.

a(7), a(11), a(13), and a(15) from T. D. Noe, May 25 2011
a(17), a(19), a(21)-a(23) from Chai Wah Wu, Mar 13 2019
b-file extending to a(40) from Hugo van der Sanden, Mar 04 2022

A292580 T(n,k) is the start of the first run of exactly k consecutive integers having exactly 2n divisors. Table read by rows.

Original entry on oeis.org

5, 2, 6, 14, 33, 12, 44, 603, 242, 10093613546512321, 24, 104, 230, 3655, 11605, 28374, 171893, 48, 2511, 7939375, 60, 735, 1274, 19940, 204323, 368431323, 155385466971, 18652995711772, 15724736975643, 2973879756088065948, 9887353188984012120346

Offset: 1

Jon E. Schoenfield, Sep 19 2017

The number of terms in row n is A119479(2n).
Düntsch and Eggleton (1989) has typos for T(3,5) and T(10,3) (called D(6,5) and D(20,3) in their notation). Letsko (2015) and Letsko (2017) both have a wrong value for T(7,3).
The first value required to extend the data is T(6,13) <= 586683019466361719763403545; the first unknown value that may exist is T(12,19). See the a-file for other known values and upper bounds up to T(50,7).

			T(1,1) = 5 because 5 is the start of the first "run" of exactly 1 integer having exactly 2*1=2 divisors (5 is the first prime p such that both p-1 and p+1 are nonprime);
T(1,2) = 2 because 2 is the start of the first run of exactly 2 consecutive integers having exactly 2*1=2 divisors (2 and 3 are the only consecutive integers that are prime);
T(3,4) = 242 because the first run of exactly 4 consecutive integers having exactly 2*3=6 divisors is 242 = 2*11^2, 243 = 3^5, 244 = 2^2*61, 245 = 5*7^2.
Table begins:
   n  T(n,1), T(n,2), ...
  ==  ========================================================
   1  5, 2;
   2  6, 14, 33;
   3  12, 44, 603, 242, 10093613546512321;
   4  24, 104, 230, 3655, 11605, 28374, 171893;
   5  48, 2511, 7939375;
   6  60, 735, 1274, 19940, 204323, 368431323, 155385466971, 18652995711772, 15724736975643, 2973879756088065948, 9887353188984012120346, 120402988681658048433948, T(6,13), ...;
   7  192, 29888, 76571890623;
   8  120, 2295, 8294, 153543, 178086, 5852870, 17476613;
   9  180, 6075, 959075, 66251139635486389922, T(9,5);
  10  240, 5264, 248750, 31805261872, 1428502133048749, 8384279951009420621, 189725682777797295066519373;
  11  3072, 2200933376, 104228508212890623;
  12  360, 5984, 72224, 2919123, 15537948, 973277147, 33815574876, 1043710445721, 2197379769820, 2642166652554075, 17707503256664346, T(12,12), ...;
  13  12288, 689278976, 1489106237081787109375;
  14  960, 156735, 23513890624, 4094170438109373, 55644509293039461218749, 4230767238315793911295500109374, 273404501868270838132985214432619890621;
  15  720, 180224, 145705879375, 10868740069638250502059754282498, T(15,5);
  16  840, 21735, 318680, 6800934, 57645182, 1194435205, 14492398389;
  ...

Hugo van der Sanden, Table of n, a(n) for n = 1..32
Ivo Düntsch and Roger B. Eggleton, Equidivisible consecutive integers, 1989.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.
Vladimir A. Letsko, Table of a(n) for all even n such that exact value of a(n) is proved, 2017.
Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors, The Prime Puzzles and Problems Connection.
Hugo van der Sanden, calculation of T(6,11).
Hugo van der Sanden, calculation of T(6,12).
Hugo van der Sanden, A-file with known values and bounds up to T(50,7)

Cf. A000005, A006558, A072507, A119479.
Cf. A005237, A005238, A006601, A049051, A049052, A049053.
Cf. A003680, A075036, A075040.

T(n,2) = A075036(n). - Jon E. Schoenfield, Sep 23 2017

a(1)-a(25) from Düntsch and Eggleton (1989) with corrections by Jon E. Schoenfield, Sep 19 2017
a(26)-a(27) from Giovanni Resta, Sep 20 2017
a(28)-a(29) from Hugo van der Sanden, Jan 12 2022
a(30) from Hugo van der Sanden, Sep 03 2022
a(31) added by Hugo van der Sanden, Dec 05 2022; see "calculation of T(6,11)" link for a list of the people involved.
a(32) added by Hugo van der Sanden, Dec 18 2022; see "calculation of T(6,12)" link for a list of the people involved.

A343818 a(n) is the least number k such that k and k+1 both have n Fermi-Dirac factors (A064547).

Original entry on oeis.org

2, 14, 104, 2079, 21735, 3341624, 103488384, 6110171144

Offset: 1

Amiram Eldar, Apr 30 2021

Since the number of infinitary divisors of k is A037445(k) = 2^A064547(k), a(n) is also the least number k such that k and k+1 both have 2^n infinitary divisors.

a(9) > 2*10^11, if it exists.

			a(1) = 2 since A064547(2) = A064547(3) = 1.
a(2) = 14 since A064547(14) = A064547(15) = 2.

Cf. A064547, A343819.

Similar sequences: A045920, A052215, A075036, A093548, A115186.

fd[1] = 0; fd[n_] := Plus @@ DigitCount[FactorInteger[n][[;;,2]], 2, 1]; seq[m_] := Module[{s = Table[0, {m}], c = 0, n = 1, fd1, fd2}, fd1=fd[n]; While[c < m, fd2 = fd[++n]; If[fd1 == fd2 && fd1 <= m && s[[fd1]] == 0, s[[fd1]] = n-1; c++]; fd1=fd2]; s]; seq[5]

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

A349261 a(n) is the least number k such that A349258(k) = A349258(k+1) = n.

Original entry on oeis.org

2, 14, 125, 135, 2079, 21735, 2730375, 916352, 5955200, 4122495, 444741759, 7391633535, 98228219264

Offset: 1

Amiram Eldar, Nov 12 2021

			2 is a term since A349258(2) = A349258(3) = 1.
14 is a term since A349258(14) = A349258(15) = 2.

Cf. A349258.

Similar sequences: A075036, A093548, A115186, A343818.

f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; c[1] = 0; c[n_] := Plus @@ f @@@ FactorInteger[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], k = 0, n = 1, i}, While[n < nmax && k < len, i = c[n]; If[c[n + 1] == i && i <= len && s[[i]] == 0, k++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[8, 3*10^6]

A356766 Least number k such that k and k+2 both have exactly 2n divisors, or -1 if no such number exists.

Original entry on oeis.org

3, 6, 18, 40, 127251, 198, 26890623, 918, 17298, 6640, 25269208984375, 3400, 3900566650390623, 640062, 8418573, 18088, 1164385682220458984373, 41650, 69528379848480224609373, 128464, 34084859373, 12164094, 150509919493198394775390625, 90270, 418514293125, 64505245696

Offset: 1

Jean-Marc Rebert, Aug 26 2022

nonn

			For n=1, numdiv(3) = numdiv(5) = 2 = 2*1, and no number < 3 satisfies this, hence a(1) = 3.

Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), A356743 (m=6), A356744 (m=8).

Cf. A000005 (d(n)), A003680, A005238, A006558, A006601, A062832, A067888, A067889, A075036.

a={}; n=1; nmax=10; For[k=1, n<=nmax, k++, If[DivisorSigma[0, k] == DivisorSigma[0, k+2] == 2n, AppendTo[a, k]; k=1; n++]]; a (* Stefano Spezia, Aug 26 2022 *)
Flatten[Table[SequencePosition[DivisorSigma[0,Range[27*10^6]],{2n,,2n},1],{n,10}],1][[;;,1]] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant but the program will take a long time to run. *) (* _Harvey P. Dale, Jul 01 2023 *)

a(n)=for(k=1,+oo,if(numdiv(k)==2*n&&numdiv(k+2)==2*n,return(k)))

More terms from Jinyuan Wang, Aug 28 2022

cp's OEIS Frontend

A035004 Number of divisors of the n-th nonprime.

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A190646 Least number k such that d(k-1) = d(k+1) = 2n or 0 if no such k exists, where d(n)=A000005(n).

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A292580 T(n,k) is the start of the first run of exactly k consecutive integers having exactly 2n divisors. Table read by rows.

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A343818 a(n) is the least number k such that k and k+1 both have n Fermi-Dirac factors (A064547).

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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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A349261 a(n) is the least number k such that A349258(k) = A349258(k+1) = n.

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A356766 Least number k such that k and k+2 both have exactly 2n divisors, or -1 if no such number exists.

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