A224346 -id:A224346 - cp's OEIS frontend

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).

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

Paolo P. Lava, Jul 03 2013

nonn

All terms are even numbers.

			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.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A224346, A224912, A226365, A227248.

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);

A227248 Composite numbers 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 n (with multiplicity).

Original entry on oeis.org

4, 72, 102, 216, 260, 264, 270, 432, 462, 504, 612, 720, 1980, 2592, 3672, 5184, 5550, 8352, 10368, 15552, 20736, 22032, 25488, 29016, 30240, 31104, 36288, 38880, 41976, 46656, 49248, 57888, 62208, 93312, 103680, 132192, 186624, 238464, 286848, 373248, 410688

Offset: 1

Paolo P. Lava, Jul 04 2013

nonn

			Prime factors of 270 are 2, 3^3, 5; therefore (2/(2+1)+3*3/(3+1)+5/(5+1))/(2/(2+1)*(3/(3+1))^3*5/(5+1)) = 16.

Paolo P. Lava, Table of n, a(n) for n = 1..65

Cf. A224346, A224912, A226365, A227034.

with(numtheory); ListA227248:=proc(q) local a, d, n, p;
for n from 2 to q do if not isprime(n) then p:=ifactors(n)[2];
a:=add((op(1, d)/(op(1, d)+1))*op(2, d), d=p)/mul((op(1, d)/(op(1, d)+1))^op(2, d), d=p);
if type(a, integer) then print(n); fi; fi;
od; end: ListA227248(10^10);

A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

Original entry on oeis.org

1, 12, 18, 220, 396, 17296, 24016, 287532, 4661056, 64288512, 334144656, 358585488, 555192576, 568719616, 2172649216, 2451538112, 2645953344, 2955423888, 6704333824, 26996772032, 88734733632, 147861504000, 311063879024, 371226582848, 429391876096

Offset: 1

Paolo P. Lava, Mar 07 2014

A212128 and A230164 are subsets of this sequence.

a(26) > 10^12. - Giovanni Resta, Mar 11 2014

			Divisors of 12 are 1, 2, 3, 4, 6, 12 and 1/1 + 1/2 + 1/3 +1/4 + 1/6 + 1/12 = 7/3. Prime factors of 12 are 2^2, 3 and 1/2 + 1/2 + 1/3 = 4/3. Finally 7/3 - 4/3 = 1 that is an integer.

Cf. A003415, A212128, A224346, A230164.

with(numtheory); P:=proc(q) local a,b,c,k,n;
for n from 1 to q do if not isprime(n) then b:=sigma(n)/n;
a:=ifactors(n)[2]; c:=add(a[k][2]/a[k][1],k=1..nops(a));
if type(b-c,integer) then lprint(n,b-c); fi; fi; od; end: P(10^6);

a(9)-a(10), a(13)-a(17), a(19)-a(25) from Giovanni Resta, Mar 11 2014

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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).

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A227248 Composite numbers 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 n (with multiplicity).

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A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

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