A074985 -id:A074985 - cp's OEIS frontend

A085155 Powers of semiprimes.

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

A177492 Products of squares of 2 or more distinct primes.

Original entry on oeis.org

36, 100, 196, 225, 441, 484, 676, 900, 1089, 1156, 1225, 1444, 1521, 1764, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 10404, 11025, 11236, 12100, 12321, 12996, 13225, 13924

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2010

nonn

			36=2^2*3^2, 100=2^2*5*2, 196=2^2*7^2,..900=2^2*3^2*5^2,..

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000469, A074985, A085986, A120944, A162142.

q:= n-> not isprime(n) and numtheory[issqrfree](n):
map(x-> x^2, select(q, [$4..120]))[];  # Alois P. Heinz, Aug 02 2024

f1[n_]:=Length[Last/@FactorInteger[n]]; f2[n_]:=Union[Last/@FactorInteger[n]]; lst={};Do[If[f1[n]>1&&f2[n]=={2},AppendTo[lst,n]],{n,0,8!}];lst
Reap[Do[{p, e} = Transpose[FactorInteger[n]]; If[Length[p]>1 && Union[e]=={2}, Sow[n]], {n, 13225}]][[2, 1]]
(* Second program *)
Select[Range[120], And[CompositeQ[#], SquareFreeQ[#]] &]^2 (* Michael De Vlieger, Aug 17 2023 *)

from math import isqrt
from sympy import primepi, mobius
def A177492(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return m**2 # Chai Wah Wu, Aug 02 2024

a(n) = A120944(n)^2. - R. J. Mathar, Dec 06 2010

Definition corrected by R. J. Mathar, Dec 06 2010

A173082 Semiprimes q such that q^2+2 and q^2-2 are also semiprimes.

Original entry on oeis.org

6, 51, 65, 115, 133, 141, 159, 187, 201, 209, 213, 287, 291, 295, 327, 339, 361, 407, 411, 413, 471, 493, 511, 519, 537, 559, 579, 597, 633, 649, 687, 695, 723, 799, 813, 831, 835, 871, 917, 939, 1007, 1041, 1047, 1079, 1135, 1167, 1189, 1195, 1199, 1227

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

nonn

			6^2-2=34 = 2*17 and 6^2+2=38 = 2*19 are semiprimes derived from the semiprime q=6, so q=6 is in the sequence.

Cf. A001358

f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; lst={}; Do[If[f[n], a=n^2-2;b=n^2+2;If[f[a]&&f[b],AppendTo[lst,n]]],{n,8!}]; lst

{A001358(j): A074985(j)-2 in A001358 and A074985(j)+2 in A001358}. [R. J. Mathar, Mar 11 2010]

Definition rephrased - R. J. Mathar, Mar 11 2010

A173083 Semiprimes q such that q^2+3 and q^2-3 are also semiprimes.

Original entry on oeis.org

6, 226, 262, 302, 314, 334, 346, 382, 466, 482, 514, 538, 562, 734, 778, 842, 866, 886, 898, 926, 974, 1006, 1046, 1282, 1306, 1318, 1322, 1438, 1466, 1478, 1574, 1622, 1774, 1822, 1838, 1858, 1894, 1994, 2302, 2342, 2446, 2518, 2578, 2582, 2602, 2638

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 09 2010

nonn

			6^2-3=33 = 3*11 and 6^2+3=39 = 3*13 are semiprimes derived from q=6, so q=6 is in the sequence.

Cf. A001358, A173082

f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2}; lst={}; Do[If[f[n], a=n^2-3;b=n^2+3;If[f[a]&&f[b],AppendTo[lst,n]]],{n,8!}]; lst

{A001358(j): A074985(j)-3 in A001358 and A074985(j)+3 in A001358}. [R. J. Mathar, Mar 11 2010]

Definition rephrased - R. J. Mathar, Mar 11 2010

A128303 Indices of squares (of semiprimes) in the 4-almost primes.

Original entry on oeis.org

1, 3, 8, 12, 24, 29, 59, 66, 90, 97, 162, 172, 187, 224, 234, 335, 385, 412, 489, 531, 551, 630, 692, 791, 921, 997, 1128, 1223, 1256, 1285, 1420, 1484, 1518, 1549, 1937, 2146, 2315, 2441, 2483, 2556, 2606, 2651, 2915, 3124, 3175, 3542, 3587, 3645, 3751, 3800

Offset: 1

Rick L. Shepherd, Mar 04 2007

nonn

			a(5) = 24 as 196 = 14^2 = semiprime(5)^2, the 5th square in the 4-almost primes, is the 24th 4-almost prime.

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

Cf. A014613, A074985, A001358, A128302, A128304.

Position[Select[Range[3*10^4], PrimeOmega[#] == 4 &], ?(IntegerQ[Sqrt[#]] &)] // Flatten (* _Amiram Eldar, Apr 13 2025 *)

list(lim) = {my(f, c); for(k = 1, lim, f = factor(k); if(bigomega(f) == 4, c++; if(vecprod(f[,2]) == 4, print1(c, ", "))));} \\ Amiram Eldar, Apr 13 2025

A014613(a(n)) = A074985(n) = A001358(n)^2.

A108655 Primes that are sums of the squares of two semiprimes.

Original entry on oeis.org

97, 181, 241, 277, 421, 457, 541, 641, 661, 709, 757, 821, 1109, 1117, 1237, 1301, 1381, 1597, 1621, 1669, 1709, 1901, 2069, 2341, 2381, 2417, 2437, 2557, 2617, 2677, 2741, 2797, 3041, 3061, 3221, 3557, 3637, 3701, 3733, 3989, 4241, 4261, 4421, 4517

Offset: 1

Reinhard Zumkeller, Jul 07 2005

nonn

Subsequence of A002144.

a(n) = A074985(i) + A074985(j) for appropriate i and j.

			A000040(733) = 5557 = 81 + 5476 = (3*3)^2 + (2*37)^2 =
A001358(3)^2 + A001358(25)^2 = A074985(3) + A074985(25),
therefore 5557 is a term.

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

Cf. A000404, A001358, A000290.

a108655 n = a108655_list !! (n-1)
a108655_list = filter f a000040_list where
   f p = any (> 0) $ map (a064911 . a037213 . (p -)) $
                         takeWhile (< p) a074985_list
-- Reinhard Zumkeller, Aug 09 2012

A154928 Decimal expansion of Sum_{q in A001358} log(q)/q^2 over the semiprimes q = 4,6,9,...

Original entry on oeis.org

0, 2, 8, 3, 6, 0, 6, 8, 1, 5, 4, 0, 7, 9, 8, 0, 6, 5, 2, 2, 2, 4, 2, 5, 8, 2, 2, 2, 5, 4, 8, 2, 7, 8, 3, 3, 6, 0, 7, 9, 3, 5, 0, 5, 7, 8, 2, 3, 7, 8, 1, 4, 0, 1, 3, 4, 1, 1, 1, 1

Offset: 0

R. J. Mathar, Jan 17 2009

Semiprime analog of A136271. The absolute value of the first derivative of the semiprime zeta function at 2.

			Equals 0.0283606815... = log(4)/16 + log(6)/36 + log(9)/81 + ....

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009; see Table 4 in version 3.

Equals Sum_{j>=1} log(A001358(j))/A074985(j).

Missing zero inserted. Artur Jasinski, Jul 29 2025

A217736 Sum of first n squares of semiprimes.

Original entry on oeis.org

16, 52, 133, 233, 429, 654, 1095, 1579, 2204, 2880, 3969, 5125, 6350, 7794, 9315, 11431, 13832, 16433, 19458, 22707, 26071, 29915, 34140, 38901, 44377, 50306, 57030, 64255, 71651, 79220, 87501, 96150, 104986, 114011, 125247, 137568, 150793, 164717, 178878

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Mar 22 2013

nonn

There are 68 such numbers less than one million; a(68) = 988849.

			a(3) = 4^2 + 6^2 + 9^2 = 133.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A062198. Equals sum(A001358^2), and sum(A074985).

Accumulate[Select[Range[200],PrimeOmega[#]==2&]^2] (* Harvey P. Dale, Mar 13 2018 *)

A349241 Numbers N = pqrs such that |pqr - s| > |ps - qr|, where p <= q <= r <= s are the 4 prime factors of N.

Original entry on oeis.org

16, 24, 36, 54, 60, 81, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 308, 315, 330, 350, 364, 375, 390, 441, 462, 484, 490, 495, 525, 546, 550, 572, 585, 625, 650, 676, 686, 693, 714, 726, 735, 748, 770, 798, 819, 825, 836, 850, 858, 875, 884, 910, 950, 975, 988

Offset: 1

M. F. Hasler, Nov 12 2021

nonn

The set A014613 of numbers n with bigomega(n) = A001222(n) = 4, can be partitioned in these here and their complement A349242. It was suggested (cf. math-fun post in LINKS) to call these here the "trans"- and the others the "cis"-type.
These here include squares of semiprimes (A074985), and in particular 4th powers of primes (A030514), for which |ps - qr| = 0.
Within the 4-almost primes below 10^k, k = 2, 3, ...,8, we have (8, 57, 497, 4960, 49228, 491397, 4869917, ...) of trans type, and more than twice (or even three times) as many of cis type.

			16 = 2^4 = u*v with u = v = 2*2 closer (equal) than u = 2*2*2, v = 2 (difference 8 - 2 = 6).
24 = 2^3*3 = u*v with u = 2*2, v = 2*3 closer (distance 6 - 4 = 2) than u = 2*2*2, v = 3 (distance 8 - 3 = 5).
36 = 2^2*3^2 = u*v with u = v = 2*3 closer (equal) than u = 2^2*3, v = 3 (difference 12 - 3 = 9).
The 4-almost prime 40 = 2^3*5 is not in this sequence because the factorization 40 = u*v with u = 2^3, v = 5 has closer factors (distance 8 - 5 = 3) than u = 2*2, v = 2*5 (distance 10 - 4 = 6).

Marc LeBrun, four factor fun, math-fun mailing list (available for subscribers only), Nov 10 2021

Cf. A014613, A349242, A006530, A020639.

select( {is_A349241(n,a(u)=abs(u-n\u))=bigomega(n)==4 && a((s=factor(n)[,1])[#s])>a(s[1]*s[#s])}, [1..1000])

from itertools import chain
from sympy import factorint
def expand(n):
    return list(chain.from_iterable([[i[0] for j in range(i[1])] for i in factorint(n).items()]))
def is_ok(p,q,r,s):
    return abs(p*q*r-s) > abs(p*s-q*r)
print([i for  i in range(2, 1000) if len(expand(i)) == 4 and is_ok(*expand(i))]) # Gleb Ivanov, Nov 12 2021

{ N in A014613 | |g - N/g| > |sg - N/sg| }, where g = gpf(N) = A006530(N) is the greatest, and s = spf(N) = A020639(N) is the smallest prime factor.

cp's OEIS Frontend

A085155 Powers of semiprimes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A177492 Products of squares of 2 or more distinct primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A173082 Semiprimes q such that q^2+2 and q^2-2 are also semiprimes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A173083 Semiprimes q such that q^2+3 and q^2-3 are also semiprimes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A128303 Indices of squares (of semiprimes) in the 4-almost primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A108655 Primes that are sums of the squares of two semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A154928 Decimal expansion of Sum_{q in A001358} log(q)/q^2 over the semiprimes q = 4,6,9,...

Views

Author

Keywords

Comments

Examples

Links

Formula

Extensions

A217736 Sum of first n squares of semiprimes.

Views

Author

Keywords

Comments