A331371
Numbers k such that k and k+1 are both half-Zumkeller numbers (A246198).
Original entry on oeis.org
224, 440, 1224, 2024, 3968, 5624, 11024, 18224, 35720, 38024, 50624, 53360, 65024, 74528, 81224, 140624, 148224, 159200, 164024, 184040, 189224, 194480, 207024, 216224, 233288, 245024, 314720, 354024, 370880, 378224, 416024, 423800, 442224, 455624, 497024, 511224
Offset: 1
224 is a term since both 224 and 225 are half-Zumkeller numbers: the proper divisors of 224 are {1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112} and 1 + 2 + 4 + 7 + 8 + 14 + 16 + 32 + 56 = 28 + 112, and the proper divisors of 225 are {1, 3, 5, 9, 15, 25, 45, 75} and 1 + 3 + 15 + 25 + 45 = 5 + 9 + 75.
-
hzQ[n_] := Module[{d = Most @ Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; hzq1 = False; s = {}; Do[hzq2 = hzQ[n]; If[hzq1 && hzq2, AppendTo[s, n - 1]]; hzq1 = hzq2, {n, 2, 6000}]; s
A345704
Zumkeller numbers k (A083207) such that the next Zumkeller number is k + 12.
Original entry on oeis.org
282, 840, 1596, 1794, 1920, 2496, 2928, 3108, 3522, 3540, 3594, 4008, 4188, 4602, 4620, 4998, 5268, 5862, 6060, 6708, 6888, 7086, 7788, 7968, 8382, 8400, 9048, 9840, 10362, 10542, 10920, 11100, 11568, 12126, 12162, 12180, 13422, 14106, 14322, 14394, 14880, 15348
Offset: 1
282 is a term since it is a Zumkeller number, and the next Zumkeller number is 282 + 12 = 294.
- Robert Israel, Table of n, a(n) for n = 1..648
- Robert Gerbicz, A083207 On an observation of Frank Buss, posts to the SeqFan list, July 2010.
- Pankaj Jyoti Mahanta, Manjil P. Saikia and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, Journal of Number Theory, Vol. 217 (2020), pp. 218-236.
-
iszum:= proc(n) local D,s,P,d;
D:= numtheory:-divisors(n);
s:= convert(D,`+`);
if s::odd then return false fi;
P:= mul(1+x^d,d=D);
coeff(P,x,s/2) > 0
end proc:
last:= 6: R:= NULL: count:= 0:
for i from 7 while count < 60 do
if iszum(i) then
if i-last = 12 then R:= R, last; count:= count+1 fi;
last:= i;
fi
od:
R; # Robert Israel, Feb 13 2023
-
zumQ[n_] := Module[{d = Divisors[n], sum, x}, sum = Plus @@ d; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; z = Select[Range[5000], zumQ]; z[[Position[Differences[z], 12] // Flatten]]
-
from itertools import count, islice
from sympy import divisors
def A345704_gen(startvalue=1): # generator of terms >= startvalue
m = -20
for n in count(max(startvalue,1)):
d = divisors(n)
s = sum(d)
if s&1^1 and n<<1<=s:
d = d[:-1]
s2, ld = (s>>1)-n, len(d)
z = [[0 for in range(s2+1)] for in range(ld+1)]
for i in range(1, ld+1):
y = min(d[i-1], s2+1)
z[i][:y] = z[i-1][:y]
for j in range(y,s2+1):
z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
if z[i][s2] == s2:
if m == n-12:
yield m
m = n
break
A345704_list = list(islice(A345704_gen(),10)) # Chai Wah Wu, Feb 13 2023
A361934
Numbers k such that k and k+1 are both primitive Zumkeller numbers (A180332).
Original entry on oeis.org
82004, 84524, 158235, 516704, 2921535, 5801984, 10846016, 12374144, 12603824, 18738224, 24252074, 24887655, 25691984, 32409530, 33696975, 35356544, 36149295, 41078114, 42541190, 43485584
Offset: 1
82004 is a term since 82004 and 82005 are both primitive Zumkeller numbers.
-
q[n_, d_, s1_, m1_] := Module[{s = s1, m = m1}, If[m == 0, False, While[d[[m]] > n, s -= d[[m]]; m--]; d[[m]] == n || If[s > n, q[n - d[[m]], d, s - d[[m]], m - 1] || q[n, d, s - d[[m]], m - 1], n == s]]];
(* after M. F. Hasler's pari code at A006037 *)
zumQ[n_] := Module[{d = Most[Divisors[n]], m, s}, m = Length[d]; s = Total[d]; If[OddQ[s + n], False, q[(s + n)/2, d, s, m]]];
primZumQ[n_] := zumQ[n] && AllTrue[Most[Divisors[n]], ! zumQ[#] &];
seq[kmax_] := Module[{s = {}, zq1 = False, zq2}, Do[zq2 = primZumQ[k]; If[zq1 && zq2, AppendTo[s, k - 1]]; zq1 = zq2, {k, 2, kmax}]; s]; seq[3*10^6]
-
is1(n,d,s,m) = {m||return; while(d[m]>n, s-=d[m]; m--||return); d[m]==n || if(nM. F. Hasler at A006037
isZum(n) = {my(d = divisors(n)[^-1], s = vecsum(d), m = #d); if((s+n)%2, return(0), is1((s+n)/2, d, s, m)); }
isPrimZum(n) = {if(!isZum(n), return(0)); fordiv(n, d, if(d < n && isZum(d), return(0))); 1;}
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isPrimZum(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2);}
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