A384444
Positive integers k for which the sum of their digits equals the product of their prime digits.
Original entry on oeis.org
1, 2, 3, 5, 7, 10, 20, 22, 30, 50, 70, 100, 123, 132, 200, 202, 213, 220, 231, 300, 312, 321, 500, 700, 1000, 1023, 1032, 1203, 1230, 1247, 1274, 1302, 1320, 1356, 1365, 1427, 1472, 1536, 1563, 1635, 1653, 1724, 1742, 2000, 2002, 2013, 2020, 2031, 2103, 2130, 2147
Offset: 1
1302 is a term, because 1 + 3 + 0 + 2 = 3*2 = 6.
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A384444:=proc(n)
option remember;
local k,c;
if n=1 then
1
else
for k from procname(n-1)+1 do
c:=convert(k,'base',10);
if mul(select(isprime,c))=add(c) then
return k
fi
od
fi;
end proc;
seq(A384444(n),n=1..51);
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Select[Range[2147],Total[IntegerDigits[#]]==Times@@Select[IntegerDigits[#],PrimeQ]&] (* James C. McMahon, Jun 20 2025 *)
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isok(k) = my(d=digits(k)); vecprod(select(isprime, d)) == vecsum(d); \\ Michel Marcus, Jun 04 2025
A384445
a(n) is the number of multisets of n decimal digits where the sum of the digits equals the product of the prime digits.
Original entry on oeis.org
5, 6, 7, 10, 23, 43, 74, 125, 199, 305, 449, 637, 885, 1216, 1649, 2184, 2852, 3664, 4657, 5863, 7298, 9002, 10993, 13312, 16000, 19084, 22613, 26606, 31120, 36192, 41867, 48220, 55317, 63232, 72022, 81746, 92479, 104282, 117229, 131393, 146843, 163652, 181892
Offset: 1
a(3) = 7 because exactly for the 7 multisets with 3 digits {0, 0, 1}, {0, 0, 2}, {0, 0, 3}, {0, 0, 5}, {0, 0, 7}, {0, 2, 2} and {1, 2, 3} their sum equals the product of the prime digits.
a(4) = 10 because exactly for the 10 multisets with 4 digits {0, 0, 0, 1}, {0, 1, 2, 3}, {1, 2, 4, 7}, {1, 3, 5, 6}, {0, 0, 0, 2}, {0, 0, 2, 2}, {0, 0, 0, 3}, {0, 0, 0, 5}, {5, 5, 6, 9} and {0, 0, 0, 7} their sum equals the product of the prime digits.
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f:=proc(p,n)
local c,d,i,l,m,r,s,t,u,w,x,y,z;
m:={0,1,4,6,8,9};
t:=seq(cat(x,i),i in m);
y:={l='Union'(t),w='Set'(l),t=~'Atom'};
d:=(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))};
Order:=p+1;
r:=combstruct:-agfseries(y,d,'unlabeled',z,[[u,s]])[w(z,u)];
r:=collect(convert(r,'polynom'),[z,u],'recursive');
c:=coeff(r,z,p);
coeff(c,u,n)
end proc:
A384445:=proc(n)
local a,k,m,s,p,j,L;
a:=1;
for k from 9*n to 1 by -1 do
L:=ifactors(k)[2];
m:=nops(L);
if m>0 and L[m,1]<=7 then
p:=n-add(L[j,2],j=1..m);
s:=k-add(L[j,1]*L[j,2],j=1..m);
if s=0 and p>=0 then
a:=a+1
elif p>0 and s>0 then
a:=a+f(p,s)
fi
fi
od;
return a
end proc;
seq(A384445(n),n=1..43);
A384505
a(n) is the number of multisets of n positive decimal digits where the sum of the digits equals the product of the prime digits.
Original entry on oeis.org
5, 1, 1, 3, 13, 20, 31, 51, 74, 106, 144, 188, 248, 331, 433, 535, 668, 812, 993, 1206, 1435, 1704, 1991, 2319, 2688, 3084, 3529, 3993, 4514, 5072, 5675, 6353, 7097, 7915, 8790, 9724, 10733, 11803, 12947, 14164, 15450, 16809, 18240, 19757, 21374, 23073, 24876, 26759
Offset: 1
a(1) = 5 because exactly for the 5 multisets with 1 digits {1}, {2}, {3}, {5}, and {7} their sum equals the product of the prime digits.
a(2) = 1 because only for 1 multiset with 2 positive digits {2, 2} their sum equals the product of the prime digits: 2 + 2 = 2*2 = 4.
a(3) = 1 because only for 1 multiset with 3 positive digits {1, 2, 3} their sum equals the product of the prime digits: 1 + 2 + 3 = 2*3 = 6.
a(4) = 3 because exactly for the 3 multisets with 4 digits {1, 2, 4, 7}, {1, 3, 5, 6}, and {5, 5, 6, 9} their sum equals the product of the prime digits: 1 + 2 + 4 + 7 = 2 * 7 = 14, 1 + 3 + 5 + 6 = 3*5 = 15, 5 + 5 + 6 + 9 = 5*5 = 25.
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f:=proc(p,n)
local i,l,m,s,t,u,w,x,z;
m:={1,4,6,8,9};
t:=seq(cat(x,i),i in m);
Order:=p+1;
coeff(coeff(collect(convert(combstruct:-agfseries({l='Union'(t),w='Set'(l),t=~'Atom'},(map2(apply,s,{t})=~m) union {s(w)='Set'(s(l))},'unlabeled',z,[[u,s]])[w(z,u)],'polynom'),[z,u],'recursive'),z,p),u,n)
end proc:
A384505:=proc(n)
local a,k,m,s,p,j,L;
if n=1 then
5
elif n=2 then
1
else
a:=0;
for k from 9*n to 1 by -1 do
L:=ifactors(k)[2];
m:=nops(L);
if m>0 and L[m,1]<=7 then
p:=n-add(L[j,2],j=1..m);
s:=k-add(L[j,1]*L[j,2],j=1..m);
if p>0 and s>0 then
a:=a+f(p,s)
fi
fi
od;
return a
fi;
end proc;
seq(A384505(n),n=1..48);
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