2020.12.25 17:58
the sum of the reciprocals of two consecutive factorials often has 1 in the numerator
I noticed today that the sum of the reciprocals of two consecutive factorials often has 1 in the numerator. This means that if I pick a natural number "a" at random and then calculate "(1/a!) + (1/(a+1)!)", for many values of "a", I'll end up with a fraction whose numerator is 1.
For example, for the first thousand natural numbers (i.e., for a from 1 to 1000) the denominators are, one by one:
{3, 2, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, \
1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, \
1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, \
67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, \
1, 89, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1, 101, 1, 103, 1, 1, 1, 107, \
1, 109, 1, 1, 1, 113, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 1, {3, 2, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, \
1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, \
1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, \
67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, \
1, 89, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1, 101, 1, 103, 1, 1, 1, 107, \
1, 109, 1, 1, 1, 113, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 1, \
1, 1, 131, 1, 1, 1, 1, 1, 137, 1, 139, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
149, 1, 151, 1, 1, 1, 1, 1, 157, 1, 1, 1, 1, 1, 163, 1, 1, 1, 167, 1, \
1, 1, 1, 1, 173, 1, 1, 1, 1, 1, 179, 1, 181, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 191, 1, 193, 1, 1, 1, 197, 1, 199, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 211, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 223, 1, 1, 1, 227, 1, 229, \
1, 1, 1, 233, 1, 1, 1, 1, 1, 239, 1, 241, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
251, 1, 1, 1, 1, 1, 257, 1, 1, 1, 1, 1, 263, 1, 1, 1, 1, 1, 269, 1, \
271, 1, 1, 1, 1, 1, 277, 1, 1, 1, 281, 1, 283, 1, 1, 1, 1, 1, 1, 1, 1,
1, 293, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 307, 1, 1, 1, 311, \
1, 313, 1, 1, 1, 317, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 331, 1, \
1, 1, 1, 1, 337, 1, 1, 1, 1, 1, 1, 1, 1, 1, 347, 1, 349, 1, 1, 1, \
353, 1, 1, 1, 1, 1, 359, 1, 1, 1, 1, 1, 1, 1, 367, 1, 1, 1, 1, 1, \
373, 1, 1, 1, 1, 1, 379, 1, 1, 1, 383, 1, 1, 1, 1, 1, 389, 1, 1, 1, \
1, 1, 1, 1, 397, 1, 1, 1, 401, 1, 1, 1, 1, 1, 1, 1, 409, 1, 1, 1, 1, \
1, 1, 1, 1, 1, 419, 1, 421, 1, 1, 1, 1, 1, 1, 1, 1, 1, 431, 1, 433, \
1, 1, 1, 1, 1, 439, 1, 1, 1, 443, 1, 1, 1, 1, 1, 449, 1, 1, 1, 1, 1, \
1, 1, 457, 1, 1, 1, 461, 1, 463, 1, 1, 1, 467, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 479, 1, 1, 1, 1, 1, 1, 1, 487, 1, 1, 1, 491, 1, 1, 1, 1, \
1, 1, 1, 499, 1, 1, 1, 503, 1, 1, 1, 1, 1, 509, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 521, 1, 523, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 541, 1, 1, 1, 1, 1, 547, 1, 1, 1, 1, 1, 1, 1, 1, 1, 557, 1, 1, \
1, 1, 1, 563, 1, 1, 1, 1, 1, 569, 1, 571, 1, 1, 1, 1, 1, 577, 1, 1, \
1, 1, 1, 1, 1, 1, 1, 587, 1, 1, 1, 1, 1, 593, 1, 1, 1, 1, 1, 599, 1, \
601, 1, 1, 1, 1, 1, 607, 1, 1, 1, 1, 1, 613, 1, 1, 1, 617, 1, 619, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 631, 1, 1, 1, 1, 1, 1, 1, 1, 1, 641, 1, \
643, 1, 1, 1, 647, 1, 1, 1, 1, 1, 653, 1, 1, 1, 1, 1, 659, 1, 661, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 673, 1, 1, 1, 677, 1, 1, 1, 1, 1, 683, \
1, 1, 1, 1, 1, 1, 1, 691, 1, 1, 1, 1, 1, 1, 1, 1, 1, 701, 1, 1, 1, 1, \
1, 1, 1, 709, 1, 1, 1, 1, 1, 1, 1, 1, 1, 719, 1, 1, 1, 1, 1, 1, 1, \
727, 1, 1, 1, 1, 1, 733, 1, 1, 1, 1, 1, 739, 1, 1, 1, 743, 1, 1, 1, \
1, 1, 1, 1, 751, 1, 1, 1, 1, 1, 757, 1, 1, 1, 761, 1, 1, 1, 1, 1, 1, \
1, 769, 1, 1, 1, 773, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 787, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 797, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 809, 1, \
811, 1, 1, 1, 1, 1, 1, 1, 1, 1, 821, 1, 823, 1, 1, 1, 827, 1, 829, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 839, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
853, 1, 1, 1, 857, 1, 859, 1, 1, 1, 863, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 877, 1, 1, 1, 881, 1, 883, 1, 1, 1, 887, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 907, 1, 1, 1, 911, 1, 1, 1, \
1, 1, 1, 1, 919, 1, 1, 1, 1, 1, 1, 1, 1, 1, 929, 1, 1, 1, 1, 1, 1, 1, \
937, 1, 1, 1, 941, 1, 1, 1, 1, 1, 947, 1, 1, 1, 1, 1, 953, 1, 1, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 967, 1, 1, 1, 971, 1, 1, 1, 1, 1, 977, \
1, 1, 1, 1, 1, 983, 1, 1, 1, 1, 1, 1, 1, 991, 1, 1, 1, 1, 1, 997, 1, \
1, 1, 1, 1}\
1, 1, 131, 1, 1, 1, 1, 1, 137, 1, 139, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
149, 1, 151, 1, 1, 1, 1, 1, 157, 1, 1, 1, 1, 1, 163, 1, 1, 1, 167, 1, \
1, 1, 1, 1, 173, 1, 1, 1, 1, 1, 179, 1, 181, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 191, 1, 193, 1, 1, 1, 197, 1, 199, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 211, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 223, 1, 1, 1, 227, 1, 229, \
1, 1, 1, 233, 1, 1, 1, 1, 1, 239, 1, 241, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
251, 1, 1, 1, 1, 1, 257, 1, 1, 1, 1, 1, 263, 1, 1, 1, 1, 1, 269, 1, \
271, 1, 1, 1, 1, 1, 277, 1, 1, 1, 281, 1, 283, 1, 1, 1, 1, 1, 1, 1, 1,
1, 293, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 307, 1, 1, 1, 311, \
1, 313, 1, 1, 1, 317, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 331, 1, \
1, 1, 1, 1, 337, 1, 1, 1, 1, 1, 1, 1, 1, 1, 347, 1, 349, 1, 1, 1, \
353, 1, 1, 1, 1, 1, 359, 1, 1, 1, 1, 1, 1, 1, 367, 1, 1, 1, 1, 1, \
373, 1, 1, 1, 1, 1, 379, 1, 1, 1, 383, 1, 1, 1, 1, 1, 389, 1, 1, 1, \
1, 1, 1, 1, 397, 1, 1, 1, 401, 1, 1, 1, 1, 1, 1, 1, 409, 1, 1, 1, 1, \
1, 1, 1, 1, 1, 419, 1, 421, 1, 1, 1, 1, 1, 1, 1, 1, 1, 431, 1, 433, \
1, 1, 1, 1, 1, 439, 1, 1, 1, 443, 1, 1, 1, 1, 1, 449, 1, 1, 1, 1, 1, \
1, 1, 457, 1, 1, 1, 461, 1, 463, 1, 1, 1, 467, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 479, 1, 1, 1, 1, 1, 1, 1, 487, 1, 1, 1, 491, 1, 1, 1, 1, \
1, 1, 1, 499, 1, 1, 1, 503, 1, 1, 1, 1, 1, 509, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 521, 1, 523, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 541, 1, 1, 1, 1, 1, 547, 1, 1, 1, 1, 1, 1, 1, 1, 1, 557, 1, 1, \
1, 1, 1, 563, 1, 1, 1, 1, 1, 569, 1, 571, 1, 1, 1, 1, 1, 577, 1, 1, \
1, 1, 1, 1, 1, 1, 1, 587, 1, 1, 1, 1, 1, 593, 1, 1, 1, 1, 1, 599, 1, \
601, 1, 1, 1, 1, 1, 607, 1, 1, 1, 1, 1, 613, 1, 1, 1, 617, 1, 619, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 631, 1, 1, 1, 1, 1, 1, 1, 1, 1, 641, 1, \
643, 1, 1, 1, 647, 1, 1, 1, 1, 1, 653, 1, 1, 1, 1, 1, 659, 1, 661, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 673, 1, 1, 1, 677, 1, 1, 1, 1, 1, 683, \
1, 1, 1, 1, 1, 1, 1, 691, 1, 1, 1, 1, 1, 1, 1, 1, 1, 701, 1, 1, 1, 1, \
1, 1, 1, 709, 1, 1, 1, 1, 1, 1, 1, 1, 1, 719, 1, 1, 1, 1, 1, 1, 1, \
727, 1, 1, 1, 1, 1, 733, 1, 1, 1, 1, 1, 739, 1, 1, 1, 743, 1, 1, 1, \
1, 1, 1, 1, 751, 1, 1, 1, 1, 1, 757, 1, 1, 1, 761, 1, 1, 1, 1, 1, 1, \
1, 769, 1, 1, 1, 773, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 787, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 797, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 809, 1, \
811, 1, 1, 1, 1, 1, 1, 1, 1, 1, 821, 1, 823, 1, 1, 1, 827, 1, 829, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 839, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
853, 1, 1, 1, 857, 1, 859, 1, 1, 1, 863, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 877, 1, 1, 1, 881, 1, 883, 1, 1, 1, 887, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 907, 1, 1, 1, 911, 1, 1, 1, \
1, 1, 1, 1, 919, 1, 1, 1, 1, 1, 1, 1, 1, 1, 929, 1, 1, 1, 1, 1, 1, 1, \
937, 1, 1, 1, 941, 1, 1, 1, 1, 1, 947, 1, 1, 1, 1, 1, 953, 1, 1, 1, \
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 967, 1, 1, 1, 971, 1, 1, 1, 1, 1, 977, \
1, 1, 1, 1, 1, 983, 1, 1, 1, 1, 1, 1, 1, 991, 1, 1, 1, 1, 1, 997, 1, \
1, 1, 1, 1}
At least that's what Mathematica tells me when I ask it to compute:
Table[Numerator[(1/a!) + (1/(a + 1)!)], {a, 1, 1000}]
comments:
2020.12.25 18:13 P.
Wszystko jest na OEIS: https://oeis.org/A014973
2021.01.02 09:52 Piotrek
Można by robić taką sztuczkę na imprezach: pomyśl dowolną liczbę naturalną, policz odwrotność jej silni, policz odwrotność silni jej następnika, zsumuj te odwrotności, a ja zgadnę, jaki wyszedł licznik.
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