A085156 -id:A085156 - cp's OEIS frontend

A322551 Primes indexed by squarefree semiprimes.

13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 199, 233, 257, 269, 271, 293, 313, 347, 373, 389, 421, 439, 443, 449, 467, 487, 491, 499, 577, 607, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947, 983, 1051, 1061, 1093

Offset: 1

Gus Wiseman, Dec 15 2018

nonn

A squarefree semiprime is a product of two distinct prime numbers.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of non-loop edges.

			The sequence of edges whose MM-numbers belong to the sequence begins: {{1,2}}, {{1,3}}, {{1,4}}, {{2,3}}, {{2,4}}, {{1,5}}, {{1,6}}, {{2,5}}, {{1,7}}, {{3,4}}, {{1,8}}, {{2,6}}, {{1,9}}, {{2,7}}, {{3,5}}, {{2,8}}.

Cf. A001358, A003963, A006881, A056239, A085156, A106349, A112798, A302242, A302491, A320458, A320459, A320461.

Select[Range[100],PrimeOmega[#]==1&&PrimeOmega[PrimePi[#]]==2&&SquareFreeQ[PrimePi[#]]&]

isok(p) = isprime(p) && (ip=primepi(p)) && (omega(ip)==2) && (bigomega(ip) == 2); \\ Michel Marcus, Dec 16 2018

A085155 Powers of semiprimes.

Original entry on oeis.org

1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 25, 26, 33, 34, 35, 36, 38, 39, 46, 49, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166

Offset: 1

Reinhard Zumkeller, Jun 21 2003

nonn

Numbers of form A001358(i)^j;

m>1 is a term iff A067029(m)=A071178(m) and (A001221(m)=2 or A067029(m) is even).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Semiprime analog of A000961 = prime powers.

Cf. A001358, A074985, A085156.

Select[Range@ 166, Function[n, Or[n == 1, And[Length@ # == 1, EvenQ@ First@ #], And[Length@ # == 2, SameQ @@ #]] &[FactorInteger[n][[All, -1]]]]] (* Michael De Vlieger, Mar 04 2017 *)

is(n)=my(f=factor(n)[,2]); #f==0 || (#f==2 && f[1]==f[2]) || (#f==1 && f[1]%2==0) \\ Charles R Greathouse IV, Oct 19 2015

{1} UNION {A001358 semiprimes} UNION {A074985 squares of semiprimes} UNION {cubes of semiprimes} UNION {4th powers of semiprimes} UNION ... - Jonathan Vos Post, Sep 06 2006

A102466 Numbers such that the number of divisors is the sum of numbers of prime factors with and without repetitions.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109

Offset: 1

Reinhard Zumkeller, Jan 09 2005

nonn

A000005(a(n)) = A001221(a(n)) + A001222(a(n)); prime powers are a subsequence (A000961); complement of A102467; not the same as A085156.
Equals { n | omega(n)=1 or Omega(n)=2 }, that is, these are exactly the prime powers (>1) and semiprimes. - M. F. Hasler, Jan 14 2008
For n > 1: A086971(a(n)) <= 1. - Reinhard Zumkeller, Dec 14 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005, A001221, A001222, A000961, A102467, A085156, A086971, A135767.

a102466 n = a102466_list !! (n-1)
a102466_list = [x | x <- [1..], a000005 x == a001221 x + a001222 x]
-- Reinhard Zumkeller, Dec 14 2012

with(numtheory):
q:= n-> is(tau(n)=bigomega(n)+nops(factorset(n))):
select(q, [$1..200])[];  # Alois P. Heinz, Jul 14 2023

Select[Range[110],DivisorSigma[0,#]==PrimeOmega[#]+PrimeNu[#]&] (* Harvey P. Dale, Mar 09 2016 *)

is(n)=my(f=factor(n)[,2]); #f==1 || f==[1,1]~ \\ Charles R Greathouse IV, Oct 19 2015

def is_A102466(n) :
    return bool(sloane.A001221(n) == 1 or sloane.A001222(n) == 2)
def A102466_list(n) :
    return [k for k in (1..n) if is_A102466(k)]
A102466_list(109)  # Peter Luschny, Feb 08 2012

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A322551 Primes indexed by squarefree semiprimes.

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A085155 Powers of semiprimes.

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A102466 Numbers such that the number of divisors is the sum of numbers of prime factors with and without repetitions.

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