230 
Prof. Sylvester on the Totients §c. 
XXXIII. Table of Totients, of Sum-totients, and of Sir 2 into the 
Squares, of all the Numbers from 501 to 1000 inclusive. 
By J. J. Sylvester*. 
n 
r(n) 
T(») 
£* 
n 
r{n) 
T(«) 
3 o 
- i* 
» 
r{n) TOO 
3 2 
501 
332 
76448 
7629515 
542 
270 
89452 
89293-54 
583 
520 10."556 
103313-87 
502 
250 
76(598 
76600-03 
543 
360 
89812 89623-34 
584 
288 j 103844 1103668-60 
503 
502 
77200 
76905-52 544 
256 
90068! 89953-75 
585 
288 104132 104023-93 
504 
144 
77344 
7721161 545 
432 
90500 90284-77 
586 
292 104424 10437987 
505 
400 
77744 
77518-31 546 1 144 
90644 9061639 
587 
586 105010 104736-42 
506 
220 
77964 
77825-62 ; 547 1 546 
91190 90948 62 
588 
168 105178 105093-58 
507 
312 
78276 
78133-54 | 548 
272 
91462 91281-46 
589 
540 105718 
105451-35 
508 
252 
78528 
7844206 549 
360 
91822 
91614-91 
590 
232 105950 
105809-72 
509 
508 
79036 
78751-19 
550 
200 
92022 
91948-97 
591 
392 10(5342 
106168-70 
510 
128 
79164 
79060-93 
551 
504 
92526 
92283-64 
592 
288 406630 
106528-29 
511 
432 
79596 
79371-28 
552 
176 
92702 
92618-91 
593 
592 KI7222 
106888-49 
512 
256 
79852 
79SS2-:3 
553 
468 
93170 
92954-79 
51 »4 180 107402 
107249-29 
513 
324 
80176 
79993-79 
554 
276 
93446 
93291-28 
595 384 107786 
107610-70 
514 
256 
80432 
80305-96 
555 
288 
93734 
93628-38 
596 
296 108082 
107972-72 
515 
408 
80840 
80618-74' 
556 
276 
94010 
93966 08 
597 
396 
108478 
10833535 
516 
168 
81008 
80932-13! 
557 
556 
94566 
94304-39 
598 
264 
108742 
108698-59 
517 
460 
81468 
8124612 
558 
180 
94746 
94643-31 
599 
598 
109340 
109062-43 
518 
216 
81684 
81560-72, 
559 
504 
95250 
94982-84 
600 
160 
111' 151 in 
109426-88 
519 
344 
82028 
81875-93 
560 
192 
95442 
95322-98 
601 
600 1 110100 
109791-94 
520 
192 
82220 
8219V75 
561 
320 
95762 
95663-72 
602 
252 110352 
110157-61 
521 
520 
82740 
82508-18 
562 
280 
96042 
96005-07 
603 
396 110748 
110523-89 
522 
168 
82908 
82825-21 
563 
562 
96604 
96347 03 
604 
300 111048 
110890-77 
523 
522 
83430 
83142-85 
564 
184 
96788 
96689-60 
605 
440 ! 111488 
111258-26 
524 
260 
83690 
83461-10 
565 
448 
97236 
97032-77 
606 
200111688 
111626-36 
525 
240 
83930 
83779-95 
566 
282 
97518 
97376-55 
607 
606 112294 
11199507 
526 
262 
84192 
8409942! 
567 
324 
97842 
97720-94 
608 
288 112582 112364-39 
527 
480 
84672 
84419-49! 
568 
280 
98122 
98065-94 
609 
336 112918 
112734-31 
528 
160 
84S32 
84740-17 
569 
568 
98690 
98411-55 
610 
240 113158 
113104-84 
529 
506 
85338 
85061-46 
570 
144 
98834 
98757-76 
611 
552 113710 
113475-98 
530 
208 
85546 
85383-36 
571 
570 
99404 
9910458 
612 
192 113902 
11384773 
531 
348 
85894 
85705-87 
572 
240 
99644 
9945201 
j 613 
612 114514 
114220-09 
532 
216 
86110 
86028-98 
573 
380 
100024 
9980005 
614 
306 114820 
114593-05 
533 
480 
86590 
86352-70 
574 
240 
100264 
100148-70 
615 
320 115140 114966-62 
534 
176 
86766 
86677-03 
575 
440 
100704 
100497-95 
616 
240 115380 115340-80 
535 
424 
87190 
87001-97 
576 
192 
100896 
100847-81 
617 
616 1115996 
11571559 
536 
264 
87454 
87327-51 
577 
576 
101472 
101198-28 
618 
204 1116200 
116090-99 
537 
356 
87810 
87653-66 
578 
272 
101744 
101549-36 
619 
618 116818 
116466-99 
538 
268 
88078 
87980-42 
579 
384 
102128 
10190105 
620 
240 117058 
116843-60 
539 
420 
88498 
88307-79 
580 
224 
102352 
10225334 
621 
396 117454 
117220-82 
540 
144 
88642 
83635-77 
581 
492 
102844 
102606-24 
622 
310 117764 
11759865 
541 
540 
89182 
88964-35 
582 
192 
103036 
102959-75 
623 
528 118292 
11797708 
* Continued from Phil. Mag. for April, p. 257, where the Table extends 
from 1 to 500 inclusive, and in which the following misprints occur : — 
In the column of ^n 2 , opposite 33, for 333-01 read 331 01 
„ „ "* „ 303, 'for 28906-59 read 27906-59 
„ „ „ 401, for 48879-64 read 48877-64 
In the extended as well as in the original Table it will he seen that the 
sum-totient is always intermediate between 3/*r 2 .n 2 and 3/7r 2 . (t?4-1) 2 . 
The formula of verification applied at every tenth step to the T column 
precludes the possibility of the existence of other than typographical errors 
or errors of transcription. Accumulative errors are rendered impossible. 
