A056077 Indices n of terms of sequence A001142, Product_{k=0..n} binomial(n,k), that are divisible by all primes <= n.
1, 2, 4, 6, 10, 11, 12, 16, 18, 22, 23, 28, 29, 30, 35, 36, 39, 40, 42, 44, 46, 47, 52, 55, 58, 59, 60, 62, 66, 69, 70, 71, 72, 78, 79, 82, 83, 88, 89, 95, 96, 100, 102, 104, 106, 107, 108, 111, 112, 119, 125, 126, 130, 131, 134, 136, 138, 139, 143, 148, 149, 150, 153
Offset: 1
Examples
11 is included because Product_{k=0..11} binomial(11, k) is divisible by 2, 3, 5, 7 and 11.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 69.
Programs
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Maple
isA056077 := proc(n) local radh; radh := proc(n) option remember; mul(k, k = numtheory:-factorset(mul(k^k/factorial(k), k=0..n))) end; type(radh(n)/radh(n-1), integer) end: # isA056077(0) = true. select(isA056077, [$1..153]); # Peter Luschny, Dec 21 2019
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Mathematica
With[{s = Select[Range@ 154, Function[n, (n/Apply[Power, Last@ #]) > #[[-1, 1]] &@ FactorInteger[n]]]}, -1 + Union[s, Prime@ Range@ PrimePi@ Max@ s]] (* Michael De Vlieger, Sep 23 2017 *)
Formula
Let h(m) = Product(PrimeDivisors(Product_{k=0..m} k^k/k!)). If h(m-1) divides h(m) then m is in this sequence. # Peter Luschny, Dec 21 2019
Extensions
Extended by Ray Chandler, Nov 17 2008
Comments