cp's OEIS Frontend

A382215 MM-numbers of multiset partitions into constant blocks with a common sum.

%I A382215 #17 Apr 03 2025 03:36:21
%S A382215 1,2,3,4,5,7,8,9,11,16,17,19,23,25,27,31,32,35,41,49,53,59,64,67,81,
%T A382215 83,97,103,109,121,125,127,128,131,157,175,179,191,209,211,227,241,
%U A382215 243,245,256,277,283,289,311,331,343,353,361,367,391,401,419,431,461
%N A382215 MM-numbers of multiset partitions into constant blocks with a common sum.
%C A382215 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%F A382215 Equals A326534 /\ A302492.
%e A382215 The terms together with their prime indices of prime indices begin:
%e A382215    1: {}
%e A382215    2: {{}}
%e A382215    3: {{1}}
%e A382215    4: {{},{}}
%e A382215    5: {{2}}
%e A382215    7: {{1,1}}
%e A382215    8: {{},{},{}}
%e A382215    9: {{1},{1}}
%e A382215   11: {{3}}
%e A382215   16: {{},{},{},{}}
%e A382215   17: {{4}}
%e A382215   19: {{1,1,1}}
%e A382215   23: {{2,2}}
%e A382215   25: {{2},{2}}
%e A382215   27: {{1},{1},{1}}
%e A382215   31: {{5}}
%e A382215   32: {{},{},{},{},{}}
%e A382215   35: {{2},{1,1}}
%e A382215   41: {{6}}
%e A382215   49: {{1,1},{1,1}}
%e A382215   53: {{1,1,1,1}}
%e A382215   59: {{7}}
%t A382215 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A382215 Select[Range[100],SameQ@@Total/@prix/@prix[#] && And@@SameQ@@@prix/@prix[#]&]
%o A382215 (PARI) is(k) = my(f=factor(k)[, 1]~, k, p, v=vector(#f, i, primepi(f[i]))); for(i=1, #v, k=isprimepower(v[i], &p); if(k||v[i]==1, v[i]=k*primepi(p), return(0))); #Set(v)<2; \\ _Jinyuan Wang_, Apr 02 2025
%Y A382215 Twice-partitions of this type are counted by A279789.
%Y A382215 For just constant blocks we have A302492, counted by A000688.
%Y A382215 For sets of constant multisets we have A302496, counted by A050361.
%Y A382215 For just common sums we have A326534, counted by A321455.
%Y A382215 Factorizations of this type are counted by A381995.
%Y A382215 For strict blocks and distinct sums we have A382201, counted by A381633.
%Y A382215 Normal multiset partitions of this type are counted by A382204.
%Y A382215 For strict instead of constant blocks we have A382304, counted by A382080.
%Y A382215 For sets of constant multisets with distinct sums A382426, counted by A381635.
%Y A382215 A055396 gives least prime index, greatest A061395.
%Y A382215 A056239 adds up prime indices, row sums of A112798.
%Y A382215 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A382215 Cf. A000720, A000961, A001055, A302242, A302601, A368100, A381715, A381716, A381636, A381871.
%K A382215 nonn
%O A382215 1,2
%A A382215 _Gus Wiseman_, Mar 21 2025