A005238 -id:A005238 - cp's OEIS frontend

A171666 Numbers n such that n, n+1 and n+2 have the same number of divisors, and that number of divisors is larger than 4.

230, 242, 243, 374, 603, 663, 902, 1105, 1274, 1309, 1334, 1832, 1885, 1924, 2013, 2054, 2133, 2264, 2343, 2504, 2523, 2665, 2696, 2936, 3110, 3655, 3656, 3729, 4203, 4401, 4503, 4504, 4614, 4669, 4695, 4807, 4923, 5133, 5862, 5943, 5944, 6054, 6061

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2009

nonn

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

Cf. A005237, A005238, A171665

f[n_]:=Length[Divisors[n]]; lst={};Do[If[f[n]==f[n+1]==f[n+2]&&f[n]>4,AppendTo[lst,n]],{n,8!}];lst
Flatten[Position[Partition[DivisorSigma[0,Range[7000]],3,1],?(Length[ Union[#]]==1&&#[[1]]>4&),{1},Heads->False]] (* _Harvey P. Dale, Jul 20 2014 *)
SequencePosition[Table[If[DivisorSigma[0,n]>4,DivisorSigma[0,n],n],{n,6100}],{x_,x_,x_}][[All,1]] (* Harvey P. Dale, Nov 15 2021 *)

A347603 Numbers k such that tau(k) = 2*tau(k-1) and sigma(k) = sigma(k-1), where tau(k) and sigma(k) are respectively the number and sum functions of the divisors of k.

Original entry on oeis.org

4365, 74919, 79827, 111507, 347739, 445875, 739557, 2168907, 4481986, 7263945, 7845387, 9309465, 10838247, 12290055, 12673095, 18151479, 22083215, 25645707, 39175955, 62634519, 69076995, 72794967, 80889207, 81166839, 87215967, 94682133, 107522943, 110768835, 119192283

Offset: 1

Claude H. R. Dequatre, Sep 08 2021

Conjecture: the asymptotic density of terms is equal to 0 and this sequence is infinite.

			a(1) = 4365 because the divisors of 4365 are: 1, 3, 5, 9, 15, 45, 97, 291, 485, 873, 1455, 4365; so, tau(4365) = 12 and sigma(4365) = 7644. The divisors of 4364 are: 1, 2, 4, 1091, 2182, 4364; so, tau(4364) = 6 and sigma(4364) = 7644. Thus tau(4365) = 2*tau(4364), sigma(4365) = sigma(4364) and so 4365 is a term.

Kevin P. Thompson, Table of n, a(n) for n = 1..288

Cf. A027750, A000005, A000203.

Similar sequences: A002961, A005237, A005238, A006601, A049051, A347076.

Select[Range[2, 10^6], DivisorSigma[0, #] == 2*DivisorSigma[0, # - 1] && DivisorSigma[1, #] == DivisorSigma[1, # - 1] &] (* Amiram Eldar, Sep 08 2021 *)

for(k=2,100000000,if(numdiv(k)==2*numdiv(k-1) && sigma(k)==sigma(k-1),print1(k", ")))

from sympy import divisor_count as tau, divisor_sigma as sigma
print([k for k in range(2, 10**6) if tau(k) == 2*tau(k-1) and sigma(k) == sigma(k-1)]) # Karl-Heinz Hofmann, Jan 15 2022

A355711 Starts of runs of 3 consecutive numbers with the same number of 5-smooth divisors.

Original entry on oeis.org

33, 85, 93, 145, 213, 265, 393, 445, 453, 475, 505, 633, 685, 753, 805, 813, 865, 933, 985, 993, 1045, 1113, 1165, 1293, 1345, 1353, 1405, 1430, 1533, 1585, 1624, 1653, 1705, 1713, 1765, 1833, 1885, 1893, 1945, 2013, 2065, 2193, 2245, 2253, 2275, 2305, 2433, 2485

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355583(k) = A355583(k+1) = A355583(k+2).

			33 is a term since A355583(33) = A355583(34) = A355583(35) = 2.

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

Cf. A355583.
Subsequence of A355710.
A355712 is a subsequence.
Similar sequences: A005238, A006073, A045939, A332313, A332387.

f[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); s = {}; m = 3; fs = f /@ Range[m]; Do[If[Equal @@ fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 2500}]; s

s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);
s1 = s(1); s2 = s(2); for(k = 3, 2500, s3 = s(k); if(s1 == s2 && s2 == s3, print1(k-2,", ")); s1 = s2; s2 = s3);

A171668 Fibonacci numbers sandwiched between two numbers having same number of divisors.

Original entry on oeis.org

34, 55, 144, 10946, 46368, 196418, 9227465, 1134903170, 4052739537881, 117669030460994, 420196140727489673, 12200160415121876738, 3928413764606871165730, 22698374052006863956975682, 68330027629092351019822533679447, 13598018856492162040239554477268290

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..70

Cf. A000005, A000045, A005237, A005238, A067889, A171666, A171667.

f[n_]:=Length[Divisors[n]]; lst={};Do[fi=Fibonacci[n];If[f[fi-1]==f[fi+1],AppendTo[lst,fi]],{n,170}];lst

a(15)-a(16) from Amiram Eldar, Aug 08 2024

A333056 Numbers k such that k, k+1 and k+2 have different prime signatures and d(k) = d(k+1) = d(k+2), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

59318, 72063, 72224, 184190, 185192, 215648, 300320, 355454, 362624, 384128, 548936, 550016, 640790, 682624, 707966, 723896, 758888, 828872, 828873, 858494, 860030, 888704, 901503, 963486, 963710, 993375, 1039742, 1039743, 1081214, 1248776, 1261897, 1340630

Offset: 1

Amiram Eldar, Mar 06 2020

nonn

Apparently most of the numbers k such that d(k) = d(k+1) = d(k+2) (A005238) are terms of A052214, i.e., k, k+1 and k+2 have the same prime signature.

Of the first 10000 terms of A005238, 6406 are also in A052214, 3578 have a pair (k and k+1, k and k+2, or k+1 and k+2) with the same prime signature, and only 16 are in this sequence.

			59318 is a term since d(59318) = d(59319) = d(59320) = 16, and the prime signatures of these 3 numbers are different: 59318 = 2 * 7 * 19 * 223, 59319 = 3^3 * 13^3, and 59320 = 2^3 * 5 * 1483 have 3 different ordered prime signatures (A124010): [1, 1, 1, 1], [3, 3], and [1, 1, 3].

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

Subsequence of A005238.

Cf. A000005, A052214, A124010, A333055.

psig[n_] := Sort @ FactorInteger[n][[;; , 2]]; d[sig_] := Times @@ (sig + 1); vsig = psig /@ Range[2, 4]; seqQ[v_] := Length@Union[v] == 3 && Length @ Union[d /@ v] == 1; seq = {}; Do[If[seqQ[vsig], AppendTo[seq, n - 3]]; vsig = Join[Rest[vsig], {psig[n]}], {n, 5, 10^6}]; seq

A171669 Pell numbers sandwiched between two numbers having same number of divisors.

Original entry on oeis.org

12, 5741, 2744210, 15994428, 21300003689580, 723573111879672, 1111984844349868137938112, 293199986221627877463941823267862, 9960168529794442859224531878561050, 27749033099085295754434173207717704165, 66992092050551637663438906713182313772

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 14 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..19

Cf. A000005, A000129, A005237, A005238, A067889, A171666, A171667, A171668.

f[n_]:=Length[Divisors[n]]; a=1;b=0;c=0;lst={};Do[c=a+b+c;If[f[c-1]==f[c+1],AppendTo[lst,c]];a=b;b=c,{n,80}];lst

a(8)-a(11) from Amiram Eldar, Aug 08 2024

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

A171666 Numbers n such that n, n+1 and n+2 have the same number of divisors, and that number of divisors is larger than 4.

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A347603 Numbers k such that tau(k) = 2*tau(k-1) and sigma(k) = sigma(k-1), where tau(k) and sigma(k) are respectively the number and sum functions of the divisors of k.

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A355711 Starts of runs of 3 consecutive numbers with the same number of 5-smooth divisors.

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A171668 Fibonacci numbers sandwiched between two numbers having same number of divisors.

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A333056 Numbers k such that k, k+1 and k+2 have different prime signatures and d(k) = d(k+1) = d(k+2), where d(k) is the number of divisors of k (A000005).

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A171669 Pell numbers sandwiched between two numbers having same number of divisors.

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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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