Problems & Puzzles: Puzzles

 Puzzle 774. S(i) and Rad(i) Naohiro Nomoto sent the following nice puzzle:   Definitions: a) s(n) = sigma(n) - n.  ( sigma is sum of divisors of n ) b) rad(n) = Largest squarefree number dividing n.   Puzzle: Find pair of numbers j & k, j≠k such that s( j ) = s( k ) and rad( j ) = rad( k ) One example: j = 160, k= 250.    s(160) = s(250) = 218  rad(160) = rad(250) = 10   More examples here: https://oeis.org/draft/A252997   Q1:  Do j and k exist in an odd number?   Q2: Does a trio exist?, i.e,  Numbers j, k & l such that  s( j ) = s( k ) = s( l ) and rad( j ) = rad( k ) = rad( l ), where j≠k≠l

Contributions came from Fred Schneider

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Fred wrote:

Q1) I found two odd pairs by computing the numbers under a threshold for a particular rad and then using a map to find duplicates:

Solution pair for diff = 517941905: 1473966895 = 5 * 17 * 19 * 97^3 and  1291620475 = 5^2 * 17^2 * 19 * 97^2

Solution pair for diff = 6859688278905: 7169160053895 = 3 * 5 * 7^7 * 31 * 97 * 193 and  5831474212935 = 3^3 * 5 * 7^4 * 31^2 * 97 * 193

He also computed 16 more even pairs, after the first 4 by Naohiro:

Solution pair for diff = 218: 250 = 2 * 5^3 and  160 = 2^5 * 5
Solution pair for diff = 189648: 95832 = 2^3 * 3^2 * 11^3 and  85536 = 2^5 * 3^5 * 11
Solution pair for diff = 720240: 288120 = 2^3 * 3 * 5 * 7^4 and  246960 = 2^4 * 3^2 * 5 * 7^3
Solution pair for diff = 119967120: 38755080 = 2^3 * 3^2 * 5 * 7^2 * 13^3 and  34398000 = 2^4 * 3^3 * 5^3 * 7^2 * 13
Solution pair for diff = 129705984: 60424704 = 2^9 * 3^4 * 31 * 47 and  71614464 = 2^14 * 3 * 31 * 47
Solution pair for diff = 707902440: 280340760 = 2^3 * 3 * 5 * 7^5 * 139 and  240292080 = 2^4 * 3^2 * 5 * 7^4 * 139
Solution pair for diff = 1321744320: 409032000 = 2^6 * 3^2 * 5^3 * 13 * 19 * 23 and  425393280 = 2^7 * 3^2 * 5 * 13^2 * 19 * 23  *** same exponent sum
Solution pair for diff = 99080219520: 36372516480 = 2^7 * 3^3 * 5 * 23^3 * 173 and  34378560000 = 2^9 * 3^3 * 5^4 * 23 * 173
Solution pair for diff = 98890370304: 43374298176 = 2^6 * 3^3 * 11 * 47^2 * 1033 and  49629220608 = 2^8 * 3 * 11^3 * 47 * 1033
Solution pair for diff = 119922568640: 157584203938 = 2 * 7^4 * 193^2 * 881 and  91447828192 = 2^5 * 7^5 * 193 * 881
Solution pair for diff = 139834382688: 49018892832 = 2^5 * 3^2 * 7^5 * 13 * 19 * 41 and  48304331712 = 2^6 * 3^2 * 7^2 * 13^3 * 19 * 41 *** same exponent sum
Solution pair for diff = 347612467648: 234681548864 = 2^6 * 7 * 17 * 31 * 997^2 and  224089101824 = 2^9 * 7^2 * 17^2 * 31 * 997
Solution pair for diff = 580542318720: 153255473280 = 2^7 * 3^5 * 5 * 7^3 * 13^2 * 17 and  150368400000 = 2^7 * 3^5 * 5^5 * 7 * 13 * 17
Solution pair for diff = 952717920000: 302049686400 = 2^7 * 3 * 5^2 * 7 * 11^4 * 307 and  266874854400 = 2^10 * 3^2 * 5^2 * 7^3 * 11 * 307
Solution pair for diff = 1064902900320: 325532420640 = 2^5 * 3^3 * 5 * 7 * 17^2 * 193^2 and  310054500000 = 2^5 * 3^3 * 5^6 * 7 * 17 * 193
Solution pair for diff = 1153644808680: 352660122360 = 2^3 * 3^2 * 5 * 7 * 13 * 17^2 * 193^2 and  335892375000 = 2^3 * 3^2 * 5^6 * 7 * 13 * 17 * 193
Solution pair for diff = 2255573174400: 517621104000 = 2^7 * 3^5 * 5^3 * 7^2 * 11 * 13 * 19 and  538325948160 = 2^8 * 3^5 * 5 * 7^2 * 11 * 13^2 * 19
Solution pair for diff = 3903820736256: 1578089151744 = 2^8 * 3 * 7^3 * 17^2 * 19 * 1091 and  1454948941824 = 2^16 * 3^2 * 7 * 17 * 19 * 1091
Solution pair for diff = 10944640212480: 3386016316800 = 2^7 * 3^2 * 5^2 * 13 * 17 * 31 * 131^2 and  3494575618560 = 2^9 * 3^2 * 5 * 13^3 * 17 * 31 * 131
Solution pair for diff = 14424196864000: 6357213927680 = 2^8 * 5 * 7 * 11^2 * 19^2 * 37 * 439 and  6083458304000 = 2^11 * 5^3 * 7 * 11 * 19 * 37 * 439

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