Today's submission to OEIS:

\[5, 13, 37, 109, 2917, 19131877, 57395629, 16210220612075905069\]

They are the primes of the form \(4\times 3^n+1\), for n up to \(39\) (after which I got lazy and stopped, but it may be possible to find larger ones without specialized algorithms).

To my knowledge T. Venkataraman was the first to consider these primes; he showed in 1975 that multiplying them by 3 produces perfect totient numbers.

Venkataraman, T. (1975). "Perfect totient number". The Mathematics Student 43: 178. MR0447089.

ETA: The 1972 Math. Comp. reference in A005537 is a little older, but less specific, so I think the title of this post is still appropriate.




Comments:

leonardo_m:
2007-02-07T17:23:54Z

Few more for you:

n = 201:
31873678665104972320653758644293555219508014318407339516948995
40869109890815626195932616388528013

n = 249:
25424499748906215411968868182837001026834723513219778324939153
9881181660045297550875174703528321187968562717038040968333

n = 1005:
12850528365353440510575225124790843237183902158350880189703025
97916477396620997162353636521062791036228920303921360150049984
32785380180158233311312895073097115484685889276782136943036055
05538257187074689142961367359773372767856477937958423202085722
47604772005137966894762191736873921806734576337943778484406647
02104399278198610202073915526703580483682477052176735631079139
10346618604120422955984102516420067341591606385759079355485748
50582245188714317163880472255891373575273840973

n = 1254:
81679563799560186762169799209669591262866306458151570875282277
85947940476595919884419054757911426353773514620512704829065688
19411259497688591039268900396618142258185827330949888505013446
62181667331017343119789605225036682812197135207571992847410467
17476539254330325519596024884626002757381624652559858571082378
76804547027281566226983183366064429154001784077696551761897658
46340955513462702308835048918127758370327451934697423307982456
53521438114339448599379120983448616108815465351647527065355354
94386418359437926739062815583130565760771918215557360456274484
80280972493656269384368803681660964024677

n = 1635:
49580563455637426873192095918756690249176904281714767405798856
51095186193546691804386752060397758265482273689253079409890139
79473111207547718217941527901958027446792720942934116615467086
55329708194416829297143999901767064379196840147948721071720824
07750751581719878165388220502626779619906617927369260005710086
97564661083857734088844288785871751559075899263651680829686301
57372612285919711562253497667715931871430001530942771706683346
47569826366862414841661618686170519463396773414177479270001231
38776661020892370226725086643574137979946026970311841842306610
93414702825102803975781320561056415372534241739752806751224522
12521034774524886863560433642319356995806321758738136153384843
02318611744771169152849065672588587011513389469755157273342829
5939702910862954896050565022383454829

Then I got lazy too...

11011110:
2007-02-07T17:56:50Z

Much less lazy than I.

So the sequence of exponents is 0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635...

It turns out to be already in OEIS: A005537. I wonder why the primes themselves aren't listed...

leonardo_m:
2007-02-07T19:28:18Z

>I wonder why the primes themselves aren't listed...

Maybe because they grow too much quickly, so they need too much space in the database and page...