A324256 -id:A324256 - cp's OEIS frontend

A326257 MM-numbers of weakly nesting multiset partitions.

49, 91, 98, 133, 147, 169, 182, 196, 203, 245, 247, 259, 266, 273, 294, 299, 301, 338, 343, 361, 364, 371, 377, 392, 399, 406, 427, 441, 455, 481, 490, 494, 497, 507, 518, 529, 532, 539, 546, 551, 553, 559, 588, 598, 602, 609, 623, 637, 665, 667, 676, 686, 689

Offset: 1

Gus Wiseman, Jun 21 2019

nonn

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. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

A multiset partition is weakly nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x <= z and t <= y or z <= x and y <= t.

			The sequence of terms together with their multiset multisystems begins:
   49: {{1,1},{1,1}}
   91: {{1,1},{1,2}}
   98: {{},{1,1},{1,1}}
  133: {{1,1},{1,1,1}}
  147: {{1},{1,1},{1,1}}
  169: {{1,2},{1,2}}
  182: {{},{1,1},{1,2}}
  196: {{},{},{1,1},{1,1}}
  203: {{1,1},{1,3}}
  245: {{2},{1,1},{1,1}}
  247: {{1,2},{1,1,1}}
  259: {{1,1},{1,1,2}}
  266: {{},{1,1},{1,1,1}}
  273: {{1},{1,1},{1,2}}
  294: {{},{1},{1,1},{1,1}}
  299: {{1,2},{2,2}}
  301: {{1,1},{1,4}}
  338: {{},{1,2},{1,2}}
  343: {{1,1},{1,1},{1,1}}
  361: {{1,1,1},{1,1,1}}

MM-numbers of crossing multiset partitions are A324170.
MM-numbers of nesting multiset partitions are A324256.
MM-numbers of capturing multiset partitions are A326255.
Nesting set partitions are A016098.
Cf. A001055, A034827, A056239, A112798, A122880, A302242, A324171.
Cf. A326211, A326243, A326258, A326260.

wknXQ[stn_]:=MatchQ[stn,{_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;(x<=z&&y>=t)||(x>=z&&y<=t)]
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],wknXQ[primeMS/@primeMS[#]]&]

A324255 Lesser of super amicable pair m < n defined by sigma(sigma(m)) = sigma(sigma(n)) = m + n.

Original entry on oeis.org

23, 14999, 34673, 55373, 2056961, 2458187, 4586987, 5174363, 6204767, 15788453, 20699927, 22239647, 25748273, 43428947, 53636627, 74005637, 92626307, 132253139, 134131379, 158486747, 167677187, 183614987, 307412933, 385042487, 413859143, 418168343, 683772017

Offset: 1

Amiram Eldar, Feb 19 2019

nonn

The larger numbers in each pair are in A324256.

Analogous to A002025 as A019279 is analogous to A000396.

			(23, 37) are the first pair since sigma(sigma(23)) = sigma(sigma(37)) = 60 = 23 + 37.

Cf. A000203, A000396, A002025, A019279, A045613 (unitary analog), A051027, A324256.

seq={}; s[n_]:=DivisorSigma[1,DivisorSigma[1,n]]-n; Do[m=s[n];If[m>n && s[m]==n, AppendTo[seq, n]], {n,1,60000}]; seq

f(n) = sigma(sigma(n)) - n;
lista(nn) = {for (n=1, nn, my(fn = f(n)); if ((fn > n) && (f(fn) == n), print1(n, ", ")););} \\ Michel Marcus, Feb 20 2019

cp's OEIS Frontend

A326257 MM-numbers of weakly nesting multiset partitions.

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