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.
For n = 8 the partitions of 8 that contain more nonprime parts than prime parts are [8], [4, 4], [4, 3, 1], [6, 1, 1], [4, 2, 1, 1], [5, 1, 1, 1], [3, 2, 1, 1, 1], [4, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]. There are 12 of these partitions so a(8) = 12.
a(n) = my(nb=0); forpart(p=n, if (#select(x->!isprime(x), Vec(p)) > #p/2, nb++)); nb; \\ Michel Marcus, Jun 25 2022
from sympy import isprime from sympy.utilities.iterables import partitions def c(p): return 2*sum(p[i] for i in p if not isprime(i)) > sum(p.values()) def a(n): return sum(1 for p in partitions(n) if c(p)) print([a(n) for n in range(51)]) # Michael S. Branicky, Jun 28 2022
For n = 8 the partitions of 8 that contain more prime parts than nonprime parts are [5, 3], [3, 3, 2], [4, 2, 2], [2, 2, 2, 2], [5, 2, 1], [3, 2, 2, 1], [2, 2, 2, 1, 1]. There are seven of these partitions so a(8) = 7.
a(n) = my(nb=0); forpart(p=n, if (#select(isprime, Vec(p)) > #p/2, nb++)); nb; \\ Michel Marcus, Jun 25 2022
from sympy import isprime from sympy.utilities.iterables import partitions def c(p): return 2*sum(p[i] for i in p if isprime(i)) > sum(p.values()) def a(n): return sum(1 for p in partitions(n) if c(p)) print([a(n) for n in range(51)]) # Michael S. Branicky, Jun 28 2022
The terms together with their prime indices begin:
2: {1}
3: {2}
4: {1,1}
5: {3}
6: {1,2}
8: {1,1,1}
9: {2,2}
10: {1,3}
11: {5}
12: {1,1,2}
14: {1,4}
15: {2,3}
16: {1,1,1,1}
17: {7}
18: {1,2,2}
20: {1,1,3}
21: {2,4}
22: {1,5}
24: {1,1,1,2}
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]
a(48) = 3 because we have [39, 9], [33, 15] and [27, 21].
nmax = 105; CoefficientList[Series[(1 + x^2)/(1 + x) Product[(1 + x^k)/((1 + x^(2 k)) (1 + x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x]
b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
b(n, i-1)+`if`(isprime(i), 0, b(n-i, min(n-i, i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..60); # Alois P. Heinz, Apr 28 2018
nmax = 55; CoefficientList[Series[1/(1 - x) Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a(10) = 5 because we have [10], [9, 1], [6, 4], [4, 6] and [1, 9].
a(9) = 3 because we have [8, 1], [6, 1, 1, 1] and [4, 1, 1, 1, 1, 1].
nmax = 60; CoefficientList[Series[(1/2) (Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a(9) = 3 because we have [9], [4, 4, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].
nmax = 60; CoefficientList[Series[(1/2) (Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a(10)=10, as 10 can be expressed in the following ways: 1+1+1+1+1+1+1+1+1+1 (1+1)*(1+1)+1+1+1+1+1+1 (1+1)*(1+1)+(1+1)*(1+1)+1+1 (1+1)*(1+1)*(1+1)+1+1 (1+1)*(1+1+1)+1+1+1+1 (1+1)*(1+1+1)+(1+1)*(1+1) (1+1)*(1+1+1+1)+1+1 (1+1+1)*(1+1+1)+1 (1+1)*(1+1+1+1+1) (1+1)*((1+1)*(1+1)+1).
from itertools import count,islice
from collections import Counter
from math import comb
from sympy import divisors
def euler_transform(x):
xlist = []
z = []
y = []
for n,x in enumerate(x,1):
xlist.append(x)
z.append(sum(d*xlist[d-1] for d in divisors(n)))
yy = (z[-1]+sum(zz*yy for zz,yy in zip(z,reversed(y))))//n
yield yy
y.append(yy)
def factorizations(n,fmin=2):
if n == 1:
yield []
return
for d in divisors(n,generator=True):
if d < fmin: continue
for f in factorizations(n//d,d):
yield [d]+f
def A373446_generator():
alist = []
def bgen():
blist = []
for n in count(1):
b = 0
for p in factorizations(n):
if len(p) == 1: continue
m = 1
for k,c in Counter(p).items():
m *= comb(alist[k-1]-blist[k-1]+c-1,c)
b += m
yield b
blist.append(b)
for a in euler_transform(bgen()):
yield a
alist.append(a)
print(list(islice(A373446_generator(),60))) # Pontus von Brömssen, Jun 13 2024
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