A198391 -id:A198391 - cp's OEIS frontend

A199767 Numbers m for which Sum_{i=1..k} (1+1/p_i) + Product_{i=1..k} (1+1/p_i) is an integer, where p_i are the k prime factors of m (with multiplicity).

21, 45, 432, 740, 1728, 3456, 3888, 5616, 12096, 23760, 46656, 52164, 131328, 152064, 186624, 195656, 233280, 311472, 606528, 618192, 746496, 926208, 933120, 979776, 1273536, 1403136, 2985984, 3221456, 3732480, 5178816, 5412096, 5971968, 9704448, 13651200

Offset: 1

Paolo P. Lava, Nov 22 2011

The numbers of the sequence are the solution of the differential equation m’=(a-k)*m-b, which can also be written as A003415(m)=(a-k)*m-A003958(m), where k is the number of prime factors of m, and a is the integer Sum_{i=1..k} (1+1/p_i) + Product_{1=1..k} (1+1/p_i).

The numbers of the sequence satisfy also Sum_{i=1..k} (1-1/p_i) - Product_{i=1..k} (1+1/p_i) = some integer.

			740 has prime factors 2, 2, 5, 37. 1 + 1/2 + 1 + 1/2 + 1 + 1/5 + 1 + 1/37 = 967/185 is the sum over 1+1/p_i. (1+1/2) * (1+1/2) * (1+1/5) * (1+1/37) = 513/185 is the product over 1+1/p_i. 967/185 + 513/185 = 8 is an integer.

J. M. Borwein and E. Wong, A survey of results relating to Giuga’s conjecture on primality, May 8, 1995.
Romeo Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Cf. A003415, A003958, A007850, A198391.

isA199767 := proc(n)
    p := ifactors(n)[2] ;
    add(op(2,d)+op(2,d)/op(1,d),d=p) + mul((1+1/op(1,d))^op(2,d),d=p) ;
    type(%,'integer') ;
end proc:
for n from 20 do
    if isA199767(n) then
           printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Nov 23 2011

A224912 Numbers m for which 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

2, 3, 4, 8, 16, 32, 36, 64, 72, 108, 128, 144, 200, 256, 288, 396, 512, 528, 576, 588, 1024, 1040, 1152, 1296, 2000, 2048, 2304, 2320, 2400, 2592, 3888, 4096, 4160, 4608, 4752, 4800, 5184, 5600, 6552, 7200, 8192, 8448, 9216, 9600, 9936, 10368, 11316, 12000

Offset: 1

Paolo P. Lava, Apr 19 2013

nonn

Apart from 3 all terms are even.

			Prime factors of 11316 are 2^2, 3, 23 and 41.
Sum_{i=1..5} (p(i)/(p(i)-1)) = 2*(2/(2-1)) + 3/(3-1) + 23/(23-1) + 41/(41-1) = 3331/440.
Sroduct_{i=1..5} (p(i)/(p(i)-1)) = 2*(2/(2-1)) * 3/(3-1) * 23/(23-1) * 41/(41-1) = 2829/440.
Their sum is an integer: 3331/440 + 2829/440 = 14.

Paolo P. Lava, Table of n, a(n) for n = 1..200

Cf. A199767, A198391.

with(numtheory);
A224912:=proc(i) local b,c,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:
A224912(10^6);

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

Paolo P. Lava, Oct 09 2013

nonn

Includes 2^(3*a) * 3^(4*b) if 3*a >= 4*b. - Robert Israel, Mar 30 2023

			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.

Robert Israel, Table of n, a(n) for n = 1..155

Cf. A199767, A198391, A227034, A227248, A230111, A230112.

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);

A226365 Composite numbers such that Sum_{i=1..k} (1 + 1/p_i) - Product_{i=1..k} (1 + 1/p_i) is an integer, where p_i are the k prime factors of n (with multiplicity).

Original entry on oeis.org

152, 432, 1620, 1728, 3456, 4752, 22464, 46656, 80892, 139968, 186624, 237168, 326592, 746496, 1651968, 2052864, 2426112, 2985984, 5971968, 10257408, 12177216, 12690432, 14048240, 14183424, 20155392, 20901888, 26127360, 38817792

Offset: 1

Paolo P. Lava, Jun 12 2013

nonn

The numbers of the sequence are the solution of the differential equation n' = (a-k)*n + b, which can also be written as A003415(n) = (a-k)*n + A003958(n), where k is the number of prime factors of n, and a is the integer Sum_{i=1..k} (1 + 1/p_i) - Product_{1=1..k} (1 + 1/p_i).

The numbers of the sequence satisfy also Sum_{i=1..k} (1 - 1/p_i) + Product_{i=1..k} (1 + 1/p_i) = some integer.

			237168 has prime factors 2, 2, 2, 2, 3, 3, 3, 3, 3, 61. 4*(1 + 1/2) + 5*(1 + 1/3) + (1 + 1/61) = 2504/183 is the sum over the 1 + 1/p_i. (1 + 1/2)^4 * (1 + 1/3)^5 * (1 + 1/61) = 3968/183 is the product of the 1 + 1/p_i. The difference over sum and product is 2504/183 - 3968/183 = -8, an integer.

J. M. Borwein and E. Wong, A survey of results relating to Giuga's conjecture on primality, May 8, 1995.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Cf. A003415, A003958, A198391, A199767, A224912.

with(numtheory); ListA226365:=proc(q) local a,d,n,p;
for n from 1 to q do if not isprime(n) then p:=ifactors(n)[2];
a:=add(op(2,d)+op(2,d)/op(1,d),d=p)-mul((1+1/op(1,d))^op(2,d),d=p);
if type(a,integer) then print(n); fi; fi;
od; end: ListA226365(10^9);

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

Paolo P. Lava, Oct 09 2013

nonn

			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.

Cf. A199767, A198391, A227034, A227248, A230110, A230112.

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

Paolo P. Lava, Oct 09 2013

nonn

			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.

Cf. A199767, A198391, A227034, A227248, A230110, A230111.

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);

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A199767 Numbers m for which Sum_{i=1..k} (1+1/p_i) + Product_{i=1..k} (1+1/p_i) is an integer, where p_i are the k prime factors of m (with multiplicity).

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A224912 Numbers m for which 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).

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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).

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A226365 Composite numbers such that Sum_{i=1..k} (1 + 1/p_i) - Product_{i=1..k} (1 + 1/p_i) is an integer, where p_i are the k prime factors of n (with multiplicity).

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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).

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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).

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Maple