25:
321:
belongs to both lists with the same sum of prime factors. The intersection also contains pairs that are not square-free, for example (7129199, 7129200) = (7×11×19×443, 2×3×5×13×457). Here 7+11+19+443 = 2+3+5+13+457 = 480, and also 7+11+11+19+443 = 2+2+2+2+3+5+5+13+457 = 491.
54:
306:
285:
232:
159:) broke Ruth's record, and the student of one of Pomerance's colleagues noticed that the sums of the prime factors of 714 and 715 were equal.
128:
There are different variations in the definition, depending on how many times to count primes that appear multiple times in a factorization.
76:
416:
330:
Ruth-Aaron pairs are sparse (that is, they have density 0). This was conjectured by Nelson et al. in 1974 and proven in 1978 by
147:
total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Pomerance was a
37:
47:
41:
33:
243:
Counting repeated prime factors (e.g., 8 = 2×2×2 and 9 = 3×3 with 2+2+2 = 3+3), the first few Ruth–Aaron pairs are:
345:(overlapping Ruth–Aaron pairs) also exist. The first and possibly the second when counting distinct prime factors:
58:
463:
519:
152:
318:
372:
and 3 + 11 + 83 + 2081 + 2411 = 2 + 5 + 7 + 293 + 1193 + 3089 = 23 + 29 + 157 + 359 + 4021 = 4589.
524:
403:
and 2 + 2 + 37 + 89 + 101 + 5197 = 5 + 283 + 1259 + 3881 = 2 + 3 + 167 + 2549 + 2707 = 5428.
445:
Nelson, C.; Penney, D. E.; and
Pomerance, C. "714 and 715." J. Recr. Math. 7, 87–89, 1974.
156:
132:
513:
331:
148:
481:
499:
495:
276:
272:
223:
219:
215:
211:
168:
102:
16:
Two consecutive integers for which the sums of the prime factors of each are equal
268:
264:
207:
203:
199:
195:
191:
187:
98:
90:
434:
260:
256:
252:
248:
183:
179:
140:
358:
and 2 + 3 + 7 + 11 + 23 + 8419 = 5 + 4201 + 4259 = 2 + 31 + 43 + 8389 = 8465.
486:
136:
144:
376:
The first two Ruth–Aaron triplets when counting repeated prime factors:
171:
106:
389:
and 2 + 3 + 251 + 277 = 17 + 53 + 463 = 2 + 2 + 11 + 19 + 499 = 533.
226:), (369, 370), (492, 493), (714, 715), (1682, 1683), (2107, 2108)
18:
454:
Erdős, P. and
Pomerance, C. "On the Largest Prime Factors of
310:
289:
236:
101:
integers (e.g., 714 and 715) for which the sums of the
366:151165960540 = 2 × 2 × 5 × 7 × 293 × 1193 × 3089,
174:are counted, the first few Ruth–Aaron pairs are:
407:As of 2006 only the 4 above triplets are known.
46:but its sources remain unclear because it lacks
363:151165960539 = 3 × 11 × 11 × 83 × 2081 × 2411,
8:
504:The prime puzzles & problems connection
300:(5, 6), (77, 78), (714, 715), (5405, 5406).
394:6913943284 = 2 × 2 × 37 × 89 × 101 × 5197,
369:151165960541 = 23 × 29 × 157 × 359 × 4021,
296:The intersection of the two lists begins:
155:at the time Aaron (a member of the nearby
77:Learn how and when to remove this message
427:
400:6913943286 = 2 × 3 × 167 × 2549 × 2707,
279:), (714, 715), (948, 949), (1330, 1331)
355:89460296 = 2 × 2 × 2 × 31 × 43 × 8389,
349:89460294 = 2 × 3 × 7 × 11 × 23 × 8419,
304:(The lesser of each pair is listed in
283:(The lesser of each pair is listed in
230:(The lesser of each pair is listed in
7:
397:6913943285 = 5 × 283 × 1259 × 3881,
143:, as Ruth's career regular-season
124:2 + 3 + 7 + 17 = 5 + 11 + 13 = 29.
14:
23:
386:417164 = 2 × 2 × 11 × 19 × 499,
1:
506:. Retrieved November 9, 2006.
500:"Ruth–Aaron pairs revisited"
435:Aaron Numbers -- Numberphile
380:417162 = 2 × 3 × 251 × 277,
352:89460295 = 5 × 4201 × 4259,
541:
464:Aequationes Mathematicae
32:This article includes a
417:Maris–McGwire–Sosa pair
383:417163 = 17 × 53 × 463,
317:Any Ruth–Aaron pair of
61:more precise citations.
131:The name was given by
496:"Ruth–Aaron Triplets"
153:University of Georgia
113:714 = 2 × 3 × 7 × 17,
319:square-free integers
480:Weisstein, Eric W.
343:Ruth–Aaron triplets
338:Ruth–Aaron triplets
466:17, 311–321, 1978.
116:715 = 5 × 11 × 13,
34:list of references
482:"Ruth-Aaron pair"
462: + 1."
167:If only distinct
87:
86:
79:
532:
492:
491:
467:
452:
446:
443:
437:
432:
313:
292:
239:
97:consists of two
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
540:
539:
535:
534:
533:
531:
530:
529:
510:
509:
479:
478:
475:
470:
453:
449:
444:
440:
433:
429:
425:
413:
340:
334:and Pomerance.
328:
305:
284:
231:
165:
95:Ruth–Aaron pair
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
538:
536:
528:
527:
522:
512:
511:
508:
507:
493:
474:
473:External links
471:
469:
468:
447:
438:
426:
424:
421:
420:
419:
412:
409:
405:
404:
401:
398:
395:
391:
390:
387:
384:
381:
374:
373:
370:
367:
364:
360:
359:
356:
353:
350:
339:
336:
327:
324:
302:
301:
281:
280:
271:), (77, 78), (
228:
227:
164:
161:
157:Atlanta Braves
133:Carl Pomerance
126:
125:
118:
117:
114:
85:
84:
42:external links
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
537:
526:
523:
521:
520:Prime numbers
518:
517:
515:
505:
501:
497:
494:
489:
488:
483:
477:
476:
472:
465:
461:
457:
451:
448:
442:
439:
436:
431:
428:
422:
418:
415:
414:
410:
408:
402:
399:
396:
393:
392:
388:
385:
382:
379:
378:
377:
371:
368:
365:
362:
361:
357:
354:
351:
348:
347:
346:
344:
337:
335:
333:
325:
323:
320:
315:
312:
308:
299:
298:
297:
294:
291:
287:
278:
274:
270:
266:
262:
258:
254:
250:
246:
245:
244:
241:
238:
234:
225:
221:
217:
213:
209:
205:
201:
197:
193:
189:
185:
181:
177:
176:
175:
173:
170:
162:
160:
158:
154:
150:
149:mathematician
146:
142:
138:
134:
129:
123:
122:
121:
115:
112:
111:
110:
108:
104:
103:prime factors
100:
96:
92:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
503:
485:
459:
455:
450:
441:
430:
406:
375:
342:
341:
329:
316:
303:
295:
282:
242:
229:
166:
130:
127:
119:
94:
88:
73:
64:
53:Please help
45:
109:are equal:
99:consecutive
91:mathematics
59:introducing
514:Categories
423:References
332:Paul Erdős
141:Hank Aaron
525:Babe Ruth
487:MathWorld
137:Babe Ruth
411:See also
163:Examples
145:home run
105:of each
67:May 2015
326:Density
311:A039753
309::
290:A039752
288::
237:A006145
235::
172:factors
151:at the
107:integer
55:improve
169:prime
40:, or
498:and
458:and
307:OEIS
286:OEIS
263:), (
255:), (
233:OEIS
218:), (
210:), (
202:), (
194:), (
186:), (
139:and
135:for
120:and
93:, a
314:).
293:).
277:126
273:125
240:).
224:154
220:153
216:105
212:104
89:In
516::
502:.
484:.
275:,
269:16
267:,
265:15
259:,
251:,
222:,
214:,
208:78
206:,
204:77
200:50
198:,
196:49
192:25
190:,
188:24
182:,
44:,
36:,
490:.
460:n
456:n
261:9
257:8
253:6
249:5
247:(
184:6
180:5
178:(
80:)
74:(
69:)
65:(
51:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.