A076290 -id:A076290 - cp's OEIS frontend

A227419 Least k such that the sum of the semiprime divisors equals n times the sum of the prime divisors, or 0 if no such k exists.

4, 9, 90, 25, 300, 49, 735, 1770, 7644, 121, 2541, 169, 5187, 6710, 8463, 289, 10982, 361, 11913, 13202, 24339, 529, 18515, 19513, 37851, 20723, 43239, 841, 35322, 961, 43215, 20705, 146595, 270470, 110823, 1369, 62835, 46535, 632316, 1681, 106074, 1849

Offset: 2

Michel Lagneau, Jul 18 2013

nonn

Least k such that A076290(k) = n*A008472(k), or 0 if no such k exists. a(n) = n^2 if n is a prime number => A001248 is a subsequence.

Conjecture: a(n) > 0.

			a(12) = 2541: The divisors of 2541 are {1, 3, 7, 11, 21, 33, 77, 121, 231, 363, 847, 2541}, so the sum of the semiprime divisors is 21 + 33 + 77 + 121 = 252, which is 12 times the sum of prime divisors 3 + 7 + 11 = 21.

Donovan Johnson, Table of n, a(n) for n = 2..1000

Cf. A001248, A076290, A008472.

with(numtheory):for n from 2 to 43 do:ii:=0:for k from 2 to 700000 while(ii=0) do:x:=divisors(k):n1:=nops(x): y:=factorset(k):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if s1=n*s2 then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

A367099 Least positive integer such that the number of divisors having two distinct prime factors is n.

Original entry on oeis.org

1, 6, 12, 24, 36, 60, 72, 120, 144, 216, 288, 360, 432, 960, 720, 864, 1296, 1440, 1728, 2160, 2592, 3456, 7560, 4320, 5184, 7776, 10800, 8640, 10368, 12960, 15552, 17280, 20736, 40320, 25920, 31104, 41472, 60480, 64800, 51840, 62208, 77760, 93312

Offset: 0

Gus Wiseman, Nov 09 2023

nonn

Does this contain every power of six, namely 1, 6, 36, 216, 1296, 7776, ...?

Yes, every power of six is a term, since 6^k = 2^k * 3^k is the least positive integer having n = tau(6^k) - (2k+1) divisors with two distinct prime factors. - Ivan N. Ianakiev, Nov 11 2023

			The divisors of 60 having two distinct prime factors are: 6, 10, 12, 15, 20. Since 60 is the first number having five such divisors, we have a(5) = 60.
The terms together with their prime indices begin:
     1: {}
     6: {1,2}
    12: {1,1,2}
    24: {1,1,1,2}
    36: {1,1,2,2}
    60: {1,1,2,3}
    72: {1,1,1,2,2}
   120: {1,1,1,2,3}
   144: {1,1,1,1,2,2}
   216: {1,1,1,2,2,2}
   288: {1,1,1,1,1,2,2}
   360: {1,1,1,2,2,3}
   432: {1,1,1,1,2,2,2}
   960: {1,1,1,1,1,1,2,3}
   720: {1,1,1,1,2,2,3}
   864: {1,1,1,1,1,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 0..4469

The version for all divisors is A005179 (firsts of A000005).
For all prime factors (A001222) we have A220264, firsts of A086971.
Positions of first appearances in A367098 (counts divisors in A007774).
A000961 lists prime powers, complement A024619.
A001221 counts distinct prime factors.
A001358 lists semiprimes, squarefree A006881, complement A100959.
A367096 lists semiprime divisors, sum A076290.
Cf. A001248, A054753, A056170, A079275, A146289, A366740, A367093.

nn=1000;
w=Table[Length[Select[Divisors[n],PrimeNu[#]==2&]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

a(n) = my(k=1); while (sumdiv(k, d, omega(d)==2) != n, k++); k; \\ Michel Marcus, Nov 11 2023

A224830 Numbers n such that both the sum of the semiprime divisors of n and the sum of the prime divisors of n are prime numbers.

Original entry on oeis.org

36, 72, 108, 144, 165, 210, 216, 273, 288, 324, 345, 385, 399, 432, 462, 561, 576, 595, 648, 651, 665, 715, 795, 798, 858, 864, 885, 957, 972, 1001, 1015, 1110, 1152, 1218, 1281, 1290, 1296, 1335, 1443, 1463, 1495, 1515, 1533, 1547, 1551, 1615, 1645, 1659

Offset: 1

Michel Lagneau, Jul 21 2013

nonn

Numbers n such that A008472(n) and A076290(n) are both prime numbers.
There exists a subsequence of squares {36, 144, 324, 576, 1296, 2304, 2916, 5184, 9216, 11664, 20736, 26244, 36864, ...} and the numbers of the form n = (p*q)^2 or (p^a*q^v)^2 with p and q primes are in the sequence if we have the two conditions:
(1) p+q = p1 is prime => p=2
(2) p^2 + p*q + q^2 = p2 is prime (subsequence of A007645), because p^2, p*q and q^2 are the three possible semiprime divisors of n, but with p=2, the semiprime divisors are 4, 2q and q^2.
(1) and (2) => p2 - 2*p1 = q^2, hence the property:
Let a number n such that the sum of the semiprime divisors is a prime number p1 and the sum of the prime divisors of n is a prime number p2. If n is a perfect square having two prime divisors, then p1 - 2*p2 = 9. Proof:
If q > 3, q == 1 mod 6 => q^2 + 2q + 4 == 1 mod 6 (if q==5 mod 6, q^2 + 2q + 4 == 3 mod 6 is not prime), but q+2 == 3 mod 6 is not prime. Conclusion: q = 3, and q^2 = 9 if a(n) is a square.
Consequence: if a(n) is a square having two prime divisors, the number k*a(n) with k = 2 or 3 is in the sequence.

			72 is in the sequence because the sum of the prime divisors is 2+3 = 5 and the sum of the semiprime divisors is 4 + 2*3 + 9 = 19.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A114522 and A227680.

Cf. A008472, A076290.

with(numtheory):for n from 2 to 2000 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if type(s1,prime)=true and type(s2,prime)=true then printf(`%d, `,n):else fi:od:

primeSum[n_] := Plus @@ First[Transpose[FactorInteger[n]]]; semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[2000], PrimeQ @ primeSum[#] && PrimeQ @ semipSigma[#] &] (* Amiram Eldar, May 10 2020 *)

A345315 a(n) = Sum_{d|n} d^[Omega(d) = 2], where [ ] is the Iverson bracket.

Original entry on oeis.org

1, 2, 2, 6, 2, 9, 2, 7, 11, 13, 2, 14, 2, 17, 18, 8, 2, 19, 2, 18, 24, 25, 2, 16, 27, 29, 12, 22, 2, 36, 2, 9, 36, 37, 38, 25, 2, 41, 42, 20, 2, 46, 2, 30, 28, 49, 2, 18, 51, 39, 54, 34, 2, 21, 58, 24, 60, 61, 2, 43, 2, 65, 34, 10, 68, 66, 2, 42, 72, 64, 2, 28, 2, 77, 44, 46, 80

Offset: 1

Wesley Ivan Hurt, Jun 13 2021

nonn

For each divisor d of n, add d if d is semiprime, otherwise add 1. For example, the divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24, and the only semiprime divisors of 24 are 4 and 6, so a(24) = 1 + 1 + 1 + 4 + 6 + 1 + 1 + 1 = 16.

Inverse Möbius transform of 1 + (n - 1)*c(n), where c = A064911. - Wesley Ivan Hurt, Jul 22 2025

			a(12) = Sum_{d|12} d^[Omega(d) = 2] = 1^0 + 2^0 + 3^0 + 4^1 + 6^1 + 12^0 = 14.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005 (tau), A001222 (Omega), A001358 (semiprimes), A064911, A076290, A086971.

Table[Sum[k^KroneckerDelta[PrimeOmega[k], 2] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]

a(n) = sumdiv(n, d, if (bigomega(d)==2, d, 1)); \\ Michel Marcus, Jun 13 2021

From Wesley Ivan Hurt, Jul 22 2025: (Start)
a(p^k) = k + p^2 for p prime and k >= 2, else k + 1 if k = 0 or 1.
a(n) = tau(n) + Sum_{d|n} (d - 1) * c(d), where c = A064911.
a(n) = A076290(n) + A000005(n) - A086971(n). (End)

A350337 Sum of the divisors of n that are prime or the product of 2 (not necessarily distinct) primes.

Original entry on oeis.org

0, 2, 3, 6, 5, 11, 7, 6, 12, 17, 11, 15, 13, 23, 23, 6, 17, 20, 19, 21, 31, 35, 23, 15, 30, 41, 12, 27, 29, 41, 31, 6, 47, 53, 47, 24, 37, 59, 55, 21, 41, 53, 43, 39, 32, 71, 47, 15, 56, 42, 71, 45, 53, 20, 71, 27, 79, 89, 59, 45, 61, 95, 40, 6, 83, 77, 67, 57, 95, 73, 71, 24, 73

Offset: 1

Wesley Ivan Hurt, Dec 25 2021

nonn

Sum of the divisors of n of the form p, p^2, or p*q, where p and q are prime.

Cf. A008472, A010051, A037143, A064911, A076290, A350331.

a[n_] := DivisorSum[n, # &, 0 < PrimeOmega[#] <= 2 &]; Array[a, 100] (* Amiram Eldar, Dec 26 2021 *)

a(n) = sumdiv(n, d, if (isprime(d) || (bigomega(d)==2), d)); \\ Michel Marcus, Dec 26 2021

a(n) = Sum_{d|n} d * (u(d) + v(d)), where u is the prime characteristic and v is the semiprime characteristic.

a(n) = A008472(n) + A076290(n). - Wesley Ivan Hurt, Jan 02 2024

cp's OEIS Frontend

A227419 Least k such that the sum of the semiprime divisors equals n times the sum of the prime divisors, or 0 if no such k exists.

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A367099 Least positive integer such that the number of divisors having two distinct prime factors is n.

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A224830 Numbers n such that both the sum of the semiprime divisors of n and the sum of the prime divisors of n are prime numbers.

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A345315 a(n) = Sum_{d|n} d^[Omega(d) = 2], where [ ] is the Iverson bracket.

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A350337 Sum of the divisors of n that are prime or the product of 2 (not necessarily distinct) primes.

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