A227034
Composite numbers such that product_{i=1..k} (p_i/(p_i-1)) / sum_{i=1..k} (p_i/(p_i-1)) is an integer, where p_i are the k prime factors of n (with multiplicity).
Original entry on oeis.org
4, 16, 72, 132, 256, 800, 1232, 2208, 2960, 5184, 5376, 11904, 19200, 23760, 39040, 41472, 65536, 72000, 76032, 76800, 84816, 203280, 259200, 288768, 332928, 345600, 373248, 383040, 416000, 614400, 628992, 640000, 663552, 691200, 1228800, 1996800, 2013312
Offset: 1
Prime factors of 1232 are 2^4, 7, 11 and ((2/(2-1))^4*7/(7-1)*11/(11-1)) / (4*2/(2-1)+7/(7-1)+11/(11-1)) = 2.
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with(numtheory); ListA226365:=proc(q) local a, d, n, p;
for n from 2 to q do if not isprime(n) then p:=ifactors(n)[2];
a:=mul((op(1,d)/(op(1,d)-1))^op(2,d),d=p)/add((op(1,d)/(op(1,d)-1))*op(2,d),d=p);
if type(a,integer) then print(n); fi; fi;
od; end: ListA226365(10^10);
A230110
Composite numbers m such that Sum_{i=1..k} (p_i/(p_i+1)) + Product_{i=1..k} (p_i/(p_i-1)) is an integer, where p_i are the k prime factors of m (with multiplicity).
Original entry on oeis.org
8, 10, 30, 63, 64, 512, 588, 720, 800, 1320, 3960, 4096, 5184, 5760, 6400, 7200, 21600, 27720, 27900, 32768, 35280, 41472, 46080, 51200, 70840, 84672, 92400, 95040, 105600, 151200, 188160, 262144, 331776, 368640, 376320, 409600, 504000, 518400, 576000, 640000
Offset: 1
Prime factors of 3960 are 2^3, 3^2, 5 and 11.
Sum_{i=1..7} (p(i)/(p(i)+1)) = 3*(2/(2+1)) + 2*(3/(3+1)) + 5/(5+1) + 11/(11+1) = 21/4.
Product_{i=1..7} (p(i)/(p(i)-1)) = (2/(2+1))^3 * (3/(3-1))^2 * 5/(5-1) * 11/(11-1) = 99/4.
Their sum is an integer: 21/4 + 99/4 = 30.
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with(numtheory); P:=proc(i) local b,d,n,p;
for n from 2 to i do p:=ifactors(n)[2];
b:=add(op(2,d)*op(1,d)/(op(1,d)+1),d=p)+mul((op(1,d)/(op(1,d)-1))^op(2,d),d=p);
if trunc(b)=b then print(n); fi; od; end: P(10^7);
A230111
Composite numbers m such that Sum_{i=1..k} (p_i/(p_i+1)) - Product_{i=1..k} (p_i/(p_i-1)) is an integer, where p_i are the k prime factors of m (with multiplicity).
Original entry on oeis.org
8, 10, 64, 512, 720, 800, 1320, 1944, 4096, 5184, 5760, 6400, 7200, 8370, 23520, 32768, 41472, 44000, 46080, 47040, 51200, 69580, 74088, 76096, 84672, 93000, 95040, 105600, 129360, 235200, 240000, 262144, 331776, 368640, 409600, 518400, 546480, 576000, 640000
Offset: 1
Prime factors of 7200 are 2^5, 3^2 and 5^2.
Sum_{i=1..9} (p(i)/(p(i)+1)) = 5*(2/(2+1)) + 2*(3/(3+1)) + 2*(5/(5+1)) = 13/2.
Product_{i=1..9} (p(i)/(p(i)-1)) = (2/(2+1))^5 * (3/(3-1))^2 * (5/(5-1))^2 = 225/2.
Their sum is an integer: 13/2 - 225/2 = -106.
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with(numtheory); P:=proc(i) local b,d,n,p;
for n from 2 to i do p:=ifactors(n)[2];
b:=add(op(2,d)*op(1,d)/(op(1,d)+1),d=p)-mul((op(1,d)/(op(1,d)-1))^op(2,d),d=p);
if trunc(b)=b then print(n); fi; od; end: P(10^7);
A230112
Composite numbers m such that Product_{i=1..k} (p_i/(p_i-1)) / Sum_{i=1..k} (p_i/(p_i+1)) is an integer, where p_i are the k prime factors of m (with multiplicity).
Original entry on oeis.org
4, 8, 16, 64, 256, 720, 800, 2200, 4096, 25600, 33600, 36288, 41472, 46080, 65536, 92400, 104960, 235200, 282240, 338688, 376320, 403200, 419840, 535680, 556640, 576000, 580800, 640000, 844800, 979776, 1088640, 1244160, 1354752, 1382400, 1505280, 1689600, 1995840
Offset: 1
Prime factors of 2200 are 2^3, 5^2 and 11.
Sum_{i=1..6} (p(i)/(p(i)+1)) = 3*(2/(2+1)) + 2*(5/(5+1)) + 11/(11+1) = 55/12.
Product_{i=1..6} (p(i)/(p(i)-1)) = (2/(2-1))^3*(5/(5-1))^2*11/(11-1) = 55/4.
The ratio is an integer: (55/4) / (55/12) = 3.
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with(numtheory); P:=proc(q) local a, d, n, p;
for n from 2 to q do if not isprime(n) then p:=ifactors(n)[2];
a:=mul((op(1, d)/(op(1, d)-1))^op(2, d), d=p)/add((op(1, d)/(op(1, d)+1))*op(2, d), d=p); if type(a, integer) then print(n); fi; fi;
od; end: P(10^7);
Showing 1-4 of 4 results.
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