cp's OEIS Frontend

A335402 Numbers m such that the only normal integer partition of m whose run-lengths are a palindrome is (1)^m.

%I A335402 #26 Mar 02 2021 03:18:35
%S A335402 0,1,2,4,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T A335402 89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,
%U A335402 179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269
%N A335402 Numbers m such that the only normal integer partition of m whose run-lengths are a palindrome is (1)^m.
%C A335402 An integer partition is normal if it covers an initial interval of positive integers.
%C A335402 Conjecture: The sequence consists of 0, 1, 4, and all primes except 3.
%C A335402 From _Chai Wah Wu_, Jun 22 2020: (Start)
%C A335402 The above conjecture is true.
%C A335402 Proof: The cases of 0, 1, 4 can be checked by inspection. Next we show that if n is prime and not equal to 3, then n is a term. Let n be prime and consider a palindromic normal partition of n covering the integers 1,...,k with k > 1. Then the multiplicity of 1 and k are the same and the multiplicities of 2 and k-1 are the same, etc.
%C A335402 If k is even, then n is of the form (k+1)r. Since n is prime, this implies that n = k+1. Since n >= k(k+1)/2. this means that k = 2 and n = 3.
%C A335402 If k is odd, then n is of the form (k+1)r + w(k+1)/2. Let m = (k+1)/2, then n = m(2r+w). Since n is prime and r,w > 0, this means that m = 1, k = 1, a contradiction.
%C A335402 Next we show that if n is composite and not equal to 4, then n is not a term.
%C A335402 Suppose n = pq for 1 < p <= q. If p is odd, let k = p-1 > 1.
%C A335402 Consider the partition covering 1,..,k where the multiplicity is 1 except for 1 and k where the multiplicity is q-k/2 + 1 > 0. This is a normal palindromic partition summing up to pq = n.
%C A335402 If p is even, without loss of generality we can choose p = 2. Since n != 4, q >= 3. In this case, choosing k = 3 with 1 and 3 having multiplicity 1 and 2 having multiplicity q-2 > 0 results in a normal palindromic partition of 2q = n. QED
%C A335402 It is clear that if n is not a term, then any multiple of n is also not a term.
%C A335402 (End)
%F A335402 n is a term if and only if n = 0, 1, 2, 4 or a prime > 3. - _Chai Wah Wu_, Jun 22 2020
%e A335402 There are 4 normal integer partitions of 10 whose sequence of multiplicities is a palindrome, namely (4321), (33211), (32221), (1111111111), so 10 does not belong to the sequence. The normal integer partitions of 7 are (3211), (2221), (22111), (211111), (1111111), none of which has palindromic multiplicities except the last, so 7 belongs to the sequence.
%t A335402 Select[Range[0,30],Length[Select[IntegerPartitions[#],And[Or[#=={},Union[#]==Range[First[#]]],Length/@Split[#]==Reverse[Length/@Split[#]]]&]]==1&]
%o A335402 (Python)
%o A335402 # from definition
%o A335402 from sympy.utilities.iterables import partitions
%o A335402 from sympy import integer_nthroot
%o A335402 A335402_list = []
%o A335402 for m in range(0,101):
%o A335402     for d in partitions(m,k=integer_nthroot(2*m,2)[0]):
%o A335402         l = len(d)
%o A335402         if l > 0 and not(l == 1 and 1 in d):
%o A335402             k = max(d)
%o A335402             if l == k:
%o A335402                 for i in range(k//2):
%o A335402                     if d[i+1] != d[k-i]:
%o A335402                         break
%o A335402                 else:
%o A335402                     break
%o A335402     else:
%o A335402         A335402_list.append(m) # _Chai Wah Wu_, Jun 22 2020
%o A335402 (Python)
%o A335402 # from formula
%o A335402 from sympy import prime
%o A335402 A335402_list = [0,1,2,4] + [prime(i) for i in range(3,100)] # _Chai Wah Wu_, Jun 22 2020
%Y A335402 Positions of 1's in A317086.
%Y A335402 Palindromic-multiplicity partitions are counted by A317085.
%Y A335402 Normal integer partitions are counted by A000009.
%Y A335402 Heinz numbers of normal palindromic-multiplicity partitions are A317087.
%Y A335402 Cf. A000041, A000837, A025065, A046022, A124010, A242414.
%K A335402 nonn,easy
%O A335402 1,3
%A A335402 _Gus Wiseman_, Jun 06 2020
%E A335402 a(22)-a(59) from _Chai Wah Wu_, Jun 22 2020