This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1) = Im(z= 1) with sigma(z) = 1 and m = 1. a(2) = Im(z= 1+3i) with sigma(z) = 5+5i and m = 2-i. a(3) = Im(z= 6+2i) with sigma(z) = -10-10i and m = -1+2i. a(4) = Im(z= 5+5i) with sigma(z) = 20i and m = 2+2i. a(5) = Im(z= 10+10i) with sigma(z) = -40+20i and m = -1+3i. a(6) = Im(z= 12+18i) with sigma(z) = -12+60i and m = 2+2i. a(9) = Im(z= 28+88i) with sigma(z) = -204-32i and m = -1+2i.
a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.
b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
`if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60); # Alois P. Heinz, Jul 03 2014
b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)
a(4) = 4 because in each of the partitions 4, 3+1, 2+2, 1+1+1+1, the frequencies of the summands is nonincreasing as the summands decrease. The partition 2+1+1 is not counted because 2 is used once, but 1 is used twice.
b:= proc(n,i,t) option remember;
if n<0 then 0
elif n=0 then 1
elif i=1 then `if`(n<=t, 1, 0)
elif i=0 then 0
else b(n, i-1, t)
+add(b(n-i*j, i-1, j), j=1..min(t, floor(n/i)))
fi
end:
a:= n-> b(n, n, n):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 21 2011
b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n == 0, 1, i == 1, If[n <= t, 1, 0], i == 0, 0, True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, 1, Min[t, Floor[n/i]]}]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)
a(7) = 4 because in each of the four partitions [7], [3,3,1], [2,2,2,1], [1,1,1,1,1,1,1] the frequency with which a summand is used decreases as the summand decreases.
a100881 = p 0 0 1 where
p m m' k x | x == 0 = if m > m' || m == 0 then 1 else 0
| x < k = 0
| m == 0 = p 1 m' k (x - k) + p 0 m' (k + 1) x
| otherwise = p (m + 1) m' k (x - k) +
if m > m' then p 0 m (k + 1) x else 0
-- Reinhard Zumkeller, Dec 27 2012
b:= proc(n, i, t) option remember;
if n<0 then 0
elif n=0 then 1
elif i=0 then 0
else b(n, i-1, t)
+add(b(n-i*j, i-1, j), j=1..min(t-1, floor(n/i)))
fi
end:
a:= n-> b(n, n, n+1):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 21 2011
b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n==0, 1, i==0, 0, True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, 1, Min[t-1, Floor[n/i]]}]]; a[n_] := b[n, n, n+1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)
2 + i is a norm-perfect number since sigma(2 + i) = 3 + i, and (3^2 + 1^2) = 10 = 2 * 5 = 2 * (2^1 + 1^2). 1 + 3*i is a norm-multiply-perfect number since sigma(1 + 3*i) = 5 + 5*i, and (5^2 + 5^2) = 50 = 5 * 10 = 5 * (1^2 + 3^2).
csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Re[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102531 *)
csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Im[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102532 *)
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