A189975 -id:A189975 - cp's OEIS frontend

A101296 n has the a(n)-th distinct prime signature.

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 5, 6, 2, 9, 2, 10, 4, 4, 4, 11, 2, 4, 4, 8, 2, 9, 2, 6, 6, 4, 2, 12, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 13, 2, 4, 6, 14, 4, 9, 2, 6, 4, 9, 2, 15, 2, 4, 6, 6, 4, 9, 2, 12, 7, 4, 2, 13, 4, 4, 4, 8, 2, 13, 4, 6, 4, 4, 4, 16, 2, 6, 6, 11, 2, 9, 2, 8, 9, 4, 2, 15, 2, 9, 4, 12, 2, 9, 4, 6, 6, 4, 4, 17

Offset: 1

David Wasserman, Dec 21 2004

From Antti Karttunen, May 12 2017: (Start)
Restricted growth sequence transform of A046523, the least representative of each prime signature. Thus this partitions the natural numbers to the same equivalence classes as A046523, i.e., for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j), and for that reason satisfies in that respect all the same conditions as A046523. For example, we have, for all i, j: if a(i) = a(j), then:
    A000005(i) = A000005(j), A008683(i) = A008683(j), A286605(i) = A286605(j).
So, this sequence (instead of A046523) can be used for finding sequences where a(n)'s value is dependent only on the prime signature of n, that is, only on the multiset of prime exponents in the factorization of n. (End)
This is also the restricted growth sequence transform of many other sequences, for example, that of A181819. See further comments there. - Antti Karttunen, Apr 30 2022

			From _David A. Corneth_, May 12 2017: (Start)
1 has prime signature (), the first distinct prime signature. Therefore, a(1) = 1.
2 has prime signature (1), the second distinct prime signature after (1). Therefore, a(2) = 2.
3 has prime signature (1), as does 2. Therefore, a(3) = a(2) = 2.
4 has prime signature (2), the third distinct prime signature after () and (1). Therefore, a(4) = 3. (End)
From _Antti Karttunen_, May 12 2017: (Start)
Construction of restricted growth sequences: In this case we start with a(1) = 1 for A046523(1) = 1, and thereafter, for all n > 1, we use the least so far unused natural number k for a(n) if A046523(n) has not been encountered before, otherwise [whenever A046523(n) = A046523(m), for some m < n], we set a(n) = a(m).
For n = 2, A046523(2) = 2, which has not been encountered before (first prime), thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n = 3, A046523(2) = 2, which was already encountered as A046523(1), thus we set a(3) = a(2) = 2.
For n = 4, A046523(4) = 4, not encountered before (first square of prime), thus we allot for a(4) the least so far unused number, which is 3, thus a(4) = 3.
For n = 5, A046523(5) = 2, as for the first time encountered at n = 2, thus we set a(5) = a(2) = 2.
For n = 6, A046523(6) = 6, not encountered before (first semiprime pq with distinct p and q), thus we allot for a(6) the least so far unused number, which is 4, thus a(6) = 4.
For n = 8, A046523(8) = 8, not encountered before (first cube of a prime), thus we allot for a(8) the least so far unused number, which is 5, thus a(8) = 5.
For n = 9, A046523(9) = 4, as for the first time encountered at n = 4, thus a(9) = 3.
(End)
From _David A. Corneth_, May 12 2017: (Start)
(Rough) description of an algorithm of computing the sequence:
Suppose we want to compute a(n) for n in [1..20].
We set up a vector of 20 elements, values 0, and a number m = 1, the minimum number we haven't checked and c = 0, the number of distinct prime signatures we've found so far.
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
We check the prime signature of m and see that it's (). We increase c with 1 and set all elements up to 20 with prime signature () to 1. In the process, we adjust m. This gives:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]. The least number we haven't checked is m = 2. 2 has prime signature (1). We increase c with 1 and set all elements up to 20 with prime signature (1) to 2. In the process, we adjust m. This gives:
[1, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0]
We check the prime signature of m = 4 and see that its prime signature is (2). We increase c with 1 and set all numbers up to 20 with prime signature (2) to 3. This gives:
[1, 2, 2, 3, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0]
Similarily, after m = 6, we get
[1, 2, 2, 3, 2, 4, 2, 0, 3, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0], after m = 8 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0], after m = 12 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 0, 2, 6, 2, 0], after m = 16 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 0], after m = 20 we get:
[1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 8]. Now, m > 20 so we stop. (End)
The above method is inefficient, because the step "set all elements a(n) up to n = Nmax with prime signature s(n) = S[c] to c" requires factoring all integers up to Nmax (or at least comparing their signature, once computed, with S[c]) again and again. It is much more efficient to run only once over each m = 1..Nmax, compute its prime signature s(m), add it to an ordered list in case it did not occur earlier, together with its "rank" (= new size of the list), and assign that rank to a(m). The list of prime signatures is much shorter than [1..Nmax]. One can also use m'(m) := the smallest n with the prime signature of m (which is faster to compute than to search for the signature) as representative for s(m), and set a(m) := a(m'(m)). Then it is sufficient to have just one counter (number of prime signatures seen so far) as auxiliary variable, in addition to the sequence to be computed. - _M. F. Hasler_, Jul 18 2019

Michel Marcus (terms 1..10000) & Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A025487, A046523, A064839 (ordinal transform of this sequence), A181819, and arrays A095904, A179216.
Cf. A000005, A008683.
Sequences that are unions of finite number (>= 2) of equivalence classes determined by the values that this sequence obtains (i.e., sequences mentioned in David A. Corneth's May 12 2017 formula): A001358 (A001248 U A006881, values 3 & 4), A007422 (values 1, 4, 5), A007964 (2, 3, 4, 5), A014612 (5, 6, 9), A030513 (4, 5), A037143 (1, 2, 3, 4), A037144 (1, 2, 3, 4, 5, 6, 9), A080258 (6, 7), A084116 (2, 4, 5), A167171 (2, 4), A217856 (6, 9).
Cf. also A077462, A305897 (stricter variants, with finer partitioning) and A254524, A286603, A286605, A286610, A286619, A286621, A286622, A286626, A286378 for other similarly constructed sequences.

A101296 := proc(n)
    local a046523, a;
    a046523 := A046523(n) ;
    for a from 1 do
        if A025487(a) = a046523 then
            return a;
        elif A025487(a) > a046523 then
            return -1 ;
        end if;
    end do:
end proc: # R. J. Mathar, May 26 2017

With[{nn = 120}, Function[s, Table[Position[Keys@s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], {n, nn}] ] (* Michael De Vlieger, May 12 2017, Version 10 *)

find(ps, vps) = {for (k=1, #vps, if (vps[k] == ps, return(k)););}
lisps(nn) = {vps = []; for (n=1, nn, ps = vecsort(factor(n)[,2]); ips = find(ps, vps); if (! ips, vps = concat(vps, ps); ips = #vps); print1(ips, ", "););} \\ Michel Marcus, Nov 15 2015; edited by M. F. Hasler, Jul 16 2019

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(100000,n,A046523(n))),"b101296.txt");
\\ Antti Karttunen, May 12 2017

A025487(a(n)) = A046523(n).
Indices of records give A025487. - Michel Marcus, Nov 16 2015
From David A. Corneth, May 12 2017: (Start) [Corresponding characteristic function in brackets]
a(A000012(n)) = 1 (sig.: ()).      [A063524]
a(A000040(n)) = 2 (sig.: (1)).     [A010051]
a(A001248(n)) = 3 (sig.: (2)).     [A302048]
a(A006881(n)) = 4 (sig.: (1,1)).   [A280710]
a(A030078(n)) = 5 (sig.: (3)).
a(A054753(n)) = 6 (sig.: (1,2)).   [A353472]
a(A030514(n)) = 7 (sig.: (4)).
a(A065036(n)) = 8 (sig.: (1,3)).
a(A007304(n)) = 9 (sig.: (1,1,1)). [A354926]
a(A050997(n)) = 10 (sig.: (5)).
a(A085986(n)) = 11 (sig.: (2,2)).
a(A178739(n)) = 12 (sig.: (1,4)).
a(A085987(n)) = 13 (sig.: (1,1,2)).
a(A030516(n)) = 14 (sig.: (6)).
a(A143610(n)) = 15 (sig.: (2,3)).
a(A178740(n)) = 16 (sig.: (1,5)).
a(A189975(n)) = 17 (sig.: (1,1,3)).
a(A092759(n)) = 18 (sig.: (7)).
a(A189988(n)) = 19 (sig.: (2,4)).
a(A179643(n)) = 20 (sig.: (1,2,2)).
a(A189987(n)) = 21 (sig.: (1,6)).
a(A046386(n)) = 22 (sig.: (1,1,1,1)).
a(A162142(n)) = 23 (sig.: (2,2,2)).
a(A179644(n)) = 24 (sig.: (1,1,4)).
a(A179645(n)) = 25 (sig.: (8)).
a(A179646(n)) = 26 (sig.: (2,5)).
a(A163569(n)) = 27 (sig.: (1,2,3)).
a(A179664(n)) = 28 (sig.: (1,7)).
a(A189982(n)) = 29 (sig.: (1,1,1,2)).
a(A179666(n)) = 30 (sig.: (3,4)).
a(A179667(n)) = 31 (sig.: (1,1,5)).
a(A179665(n)) = 32 (sig.: (9)).
a(A189990(n)) = 33 (sig.: (2,6)).
a(A179669(n)) = 34 (sig.: (1,2,4)).
a(A179668(n)) = 35 (sig.: (1,8)).
a(A179670(n)) = 36 (sig.: (1,1,1,3)).
a(A179671(n)) = 37 (sig.: (3,5)).
a(A162143(n)) = 38 (sig.: (2,2,2)).
a(A179672(n)) = 39 (sig.: (1,1,6)).
a(A030629(n)) = 40 (sig.: (10)).
a(A179688(n)) = 41 (sig.: (1,3,3)).
a(A179689(n)) = 42 (sig.: (2,7)).
a(A179690(n)) = 43 (sig.: (1,1,2,2)).
a(A189991(n)) = 44 (sig.: (4,4)).
a(A179691(n)) = 45 (sig.: (1,2,5)).
a(A179692(n)) = 46 (sig.: (1,9)).
a(A179693(n)) = 47 (sig.: (1,1,1,4)).
a(A179694(n)) = 48 (sig.: (3,6)).
a(A179695(n)) = 49 (sig.: (2,2,3)).
a(A179696(n)) = 50 (sig.: (1,1,7)).
(End)

Data section extended to 120 terms by Antti Karttunen, May 12 2017

Minor edits/corrections by M. F. Hasler, Jul 18 2019

A179642 Product of exactly 5 primes, 3 of which are distinct.

Original entry on oeis.org

120, 168, 180, 252, 264, 270, 280, 300, 312, 378, 396, 408, 440, 450, 456, 468, 520, 552, 588, 594, 612, 616, 680, 684, 696, 700, 702, 728, 744, 750, 760, 828, 882, 888, 918, 920, 945, 952, 980, 984, 1026, 1032, 1044, 1064, 1100, 1116, 1128, 1144, 1160

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

nonn

			120=2^3*3*5, 168=2^3*3*7, 180=2^2*3^2*5, 252=2^2*3^2*7, 264=2^3*3*11, 270=2*3^3*5

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Union of A189975 and A179643.

Cf. A006881, A007304, A085986, A085987, A178739.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3} || Sort[Last/@FactorInteger[n]]=={1,2,2}; Select[Range[2000], f]

list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/3),forprime(q=2,sqrt(lim\p^3),if(p==q,next);t=p^3*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));forprime(p=2,sqrt(lim\12),forprime(q=p+1,sqrt(lim\p^2\2),t=(p*q)^2;forprime(r=2,lim\t,if(p==r||q==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 19 2011

A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.

Original entry on oeis.org

420, 630, 660, 780, 924, 990, 1020, 1050, 1092, 1140, 1170, 1380, 1386, 1428, 1470, 1530, 1540, 1596, 1638, 1650, 1710, 1716, 1740, 1820, 1860, 1932, 1950, 2070, 2142, 2220, 2244, 2380, 2394, 2436, 2460, 2508, 2550, 2574, 2580, 2604, 2610, 2652, 2660, 2790

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

Theorem 4 in Goldston-Graham-Pintz-Yildirim proves that a(n+1) = a(n) + 1 for infinitely many n. - Charles R Greathouse IV, Jul 17 2015, corrected by M. F. Hasler, Jul 17 2019

T. D. Noe, Table of n, a(n) for n = 1..1000
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers (2008)
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,2}; Select[Range[4000],f]

is(n)=vecsort(factor(n)[,2])==[1, 1, 1, 2]~ \\ Charles R Greathouse IV, Jul 17 2015

Definition reworded by M. F. Hasler, Jul 17 2019

A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

Original entry on oeis.org

1260, 1980, 2100, 2340, 2772, 2940, 3060, 3150, 3276, 3300, 3420, 3900, 4140, 4284, 4410, 4788, 4950, 5100, 5148, 5220, 5580, 5700, 5796, 5850, 6468, 6660, 6732, 6900, 7260, 7308, 7350, 7380, 7524, 7644, 7650, 7700, 7740, 7812, 7956, 8460, 8550, 8700

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296. - M. F. Hasler, Jul 17 2019

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,2}; Select[Range[10000], f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\60), t1=p^2; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 6, 0, 8, 0, 8, 2, 2, 0, 18, 1, 2, 3, 8, 0, 15, 0, 10, 2, 2, 2, 24, 0, 2, 2, 18, 0, 15, 0, 8, 8, 2, 0, 32, 1, 8, 2, 8, 0, 18, 2, 18, 2, 2, 0, 44, 0, 2, 8, 15, 2, 15, 0, 8, 2, 15, 0, 49, 0, 2, 8, 8, 2, 15, 0, 32, 6, 2

Offset: 1

Michel Lagneau, Aug 03 2016

nonn

Number of elements in the set {(x, y): x|n, y|n, x < y, gcd(x, y) > 1}.
Every element of the sequence is repeated indefinitely, for instance:
a(n)=0 if n prime;
a(n)=1 if n = p^2 for p prime (A001248);
a(n)=2 if n is a squarefree semiprime (A006881);
a(n)=3 if n = p^3 for p prime (A030078);
a(n)=6 if n = p^4 for p prime (A030514);
a(n)=8 if n is a number which is the product of a prime and the square of a different prime (A054753);
a(n)=10 if n = p^5 for p prime (A050997);
a(n)=15 if n is in the set {A007304} union {64} = {30, 42, 64, 66, 70,...} = {Sphenic numbers} union {64};
a(n)=18 if n is the product of the cube of a prime (A030078) and a different prime (see A065036);
a(n)=21 if n = p^7 for p prime (A092759);
a(n)=24 if n is square of a squarefree semiprime (A085986);
a(n)=32 if n is the product of the 4th power of a prime (A030514) and a different prime (see A178739);
a(n)=36 if n = p^9 for p prime (A179665);
a(n)=44 if n is the product of exactly four primes, three of which are distinct (A085987);
a(n)=45 if n is a number with 11 divisors (A030629);
a(n)=49 if n is of the form p^2*q^3, where p,q are distinct primes (A143610);
a(n)=50 if n is the product of the 5th power of a prime (A050997) and a different prime (see A178740);
a(n)=55 if n if n = p^11 for p prime(A079395);
a(n)=72 if n is a number with 14 divisors (A030632);
a(n)=80 if n is the product of four distinct primes (A046386);
a(n)=83 if n is a number with 15 divisors (A030633);
a(n)=89 if n is a number with prime factorization pqr^3 (A189975);
a(n)=96 if n is a number that are the cube of a product of two distinct primes (A162142);
a(n)=98 if n is the product of the 7th power of a prime and a distinct prime (p^7*q) (A179664);
a(n)=116 if n is the product of exactly 2 distinct squares of primes and a different prime (p^2*q^2*r) (A179643);
a(n)=126 if n is the product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2) (A179646);
a(n)=128 if n is the product of the 8th power of a prime and a distinct prime (p^8*q) (A179668);
a(n)=150 if n is the product of the 4th power of a prime and 2 different distinct primes (p^4*q*r) (A179644);
a(n)=159 if n is the product of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3) (A179666).
It is possible to continue with a(n) = 162, 178, 209, 224, 227, 238, 239, 260, 289, 309, 320, 333,...

			a(12) = 8 because the divisors of 12 are {1, 2, 3, 4, 6, 12} and GCD(d_i, d_j)>1 for the 8 following pairs of divisors: (2,4), (2,6), (2,12), (3,6), (3,12), (4,6), (4,12) and (6,12).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001248, A006881, A007304, A030078, A030514, A030632, A046386, A050997, A054753, A063647, A065036, A066446, A079395, A085986, A085987, A092759, A143610, A162142, A178739, A178740, A179644, A179646, A179664, A189975.

Cf. A333976 (same with d <= e).

with(numtheory):nn:=100:
for n from 1 to nn do:
x:=divisors(n):n0:=nops(x):it:=0:
for i from 1 to n0 do:
  for j from i+1 to n0 do:
   if gcd(x[i],x[j])>1
    then
    it:=it+1:
    else
   fi:
  od:
od:
  printf(`%d, `,it):
od:

Table[Sum[Sum[(1 - KroneckerDelta[GCD[i, k], 1]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k - 1}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2021 *)

a(n)=my(d=divisors(n)); sum(i=2,#d, sum(j=1,i-1, gcd(d[i],d[j])>1)) \\ Charles R Greathouse IV, Aug 03 2016

a(n)=my(f=factor(n)[,2],t=prod(i=1,#f,f[i]+1)); t*(t-1)/2 - (prod(i=1,#f,2*f[i]+1)+1)/2 \\ Charles R Greathouse IV, Aug 03 2016

a(n) = A066446(n) - A063647(n).

a(n) = Sum_{d1|n, d2|n, d1Wesley Ivan Hurt, Jan 01 2021

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

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

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] % 2 && e[1] > 1 && (#e == 1 || e[2] == 1));

A381315 Numbers whose prime factorization exponents include exactly one 3 and no exponent greater than 3.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A176297 and A375072, and first differs from them at n = 20: A176297(20) = A375072(20) = 216 = 2^3 * 3^3 is not a term of this sequence.

The asymptotic density of this sequence is (1/zeta(3)) * Sum_{p prime} 1/(p+p^2+p^3) = 0.089602607198058453295... .

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

Subsequence of A046100, A176297, A375072 and A375145.
Subsequences: A030078, A048109, A065036, A143610, A163569, A179670, A179695, A179700, A189975, A189984, A190109, A190378, A190382.
Cf. A002117.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, MemberQ[e, 3] && Count[e, _?(# < 3 &)] == Length[e] - 1]; Select[Range[600], q]

isok(k) = {my(e = factor(k)[, 2]~); select(x -> x > 2, e) == [3];}

A382292 Numbers k such that A382290(k) = 1.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 64, 72, 88, 96, 104, 108, 120, 125, 135, 136, 152, 160, 168, 184, 189, 192, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 320, 328, 343, 344, 351, 352, 360, 375, 376, 378, 392, 408, 416, 424, 432, 440, 448, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Mar 21 2025

First differs from A374590 and A375432 at n = 25: A374590(25) = A375432(25) = 216 is not a term of this sequence.
Numbers k such that A382291(k) = 2, i.e., numbers whose number of infinitary divisors is twice the number of their unitary divisors.
Numbers whose prime factorization has a single exponent that is a sum of two distinct powers of 2 (A018900) and all the other exponents, if they exist, are powers of 2. Equivalently, numbers of the form p^e * m, where p is a prime, e is a term in A018900, and m is a term in A138302 that is coprime to p.
If k is a term then k^2 is also a term. If m is a term in A138302 that is coprime to k then k * m is also a term. The primitive terms, i.e., the terms that cannot be generated from smaller terms using these rules, are the numbers of the form p^(2^i+1), where p is prime and i >= 1.
Analogous to A060687, which is the sequence of numbers k with prime excess A046660(k) = 2.
The asymptotic density of this sequence is A271727 * Sum_{p prime} (((1 - 1/p)/f(1/p)) * Sum_{k>=1} 1/p^A018900(k)) = 0.11919967112489084407..., where f(x) = 1 - x^3 + Sum_{k>=2} (x^(2^k)-x^(2^k+1)).

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

Cf. A018900, A046660, A060687, A271727, A374590, A375432, A382290, A382291.

Subsequences (numbers of the form): A030078 (p^3), A050997 (p^5), A030516 (p^6), A179665 (p^9), A030629 (p^10), A030631 (p^12), A065036 (p^3*q), A178740 (p^5*q), A189987 (p^6*q), A179692 (p^9*q), A143610 (p^2*q^3), A179646 (p^5*q^2), A189990 (p^2*q^6), A179702 (p^4*q^5), A179666 (p^4*q^3), A190464 (p^4*q^6), A163569 (p^3*q^2*r), A189975 (p*q*r^3), A190115 (p^2*q^3*r^4), A381315, A048109.

f[p_, e_] := DigitCount[e, 2, 1] - 1; q[1] = False; q[n_] := Plus @@ f @@@ FactorInteger[n] == 1; Select[Range[500], q]

isok(k) = vecsum(apply(x -> hammingweight(x) - 1, factor(k)[, 2])) == 1;

A275345 Characteristic polynomials of a square matrix based on A051731 where A051731(1,N)=1 and A051731(N,N)=0 and where N=size of matrix, analogous to the Redheffer matrix.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, 0, 2, -1, 0, 0, 2, -3, 1, -1, 2, 1, -5, 4, -1, 1, -3, 5, -8, 9, -5, 1, -1, 4, -4, -5, 15, -14, 6, -1, 0, -1, 6, -17, 29, -31, 20, -7, 1, 0, 0, 2, -13, 36, -55, 50, -27, 8, -1, 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1

Offset: 0

Mats Granvik, Jul 24 2016

From Mats Granvik, Sep 30 2017: (Start)
Conjecture: The largest absolute value of the eigenvalues of these characteristic polynomials appear to have the same prime signature in the factorization of the matrix sizes N.
In other words: Let b(N) equal the sequence of the largest absolute values of the eigenvalues of the characteristic polynomials of the matrices of size N. b(N) is then a sequence of truncated eigenvalues starting:
b(N=1..infinity)
= 1.00000, 1.61803, 1.61803, 2.00000, 1.61803, 2.20557, 1.61803, 2.32472, 2.00000, 2.20557, 1.61803, 2.67170, 1.61803, 2.20557, 2.20557, 2.61803, 1.61803, 2.67170, 1.61803, 2.67170, 2.20557, 2.20557, 1.61803, 3.08032, 2.00000, 2.20557, 2.32472, 2.67170, 1.61803, 2.93796, 1.61803, 2.89055, 2.20557, 2.20557, 2.20557, 3.21878, 1.61803, 2.20557, 2.20557, 3.08032, 1.61803, 2.93796, 1.61803, 2.67170, 2.67170, 2.20557, 1.61803, 3.45341, 2.00000, 2.67170, 2.20557, 2.67170, 1.61803, 3.08032, 2.20557, 3.08032, 2.20557, 2.20557, 1.61803, 3.53392, 1.61803, 2.20557, 2.67170, ...
It then appears that for n = 1,2,3,4,5,...,infinity we have the table:
Prime signature: b(Axxxxxx(n)) = Largest abs(eigenvalue):
p^0      : b(1)          = 1.0000000000000000000000000000...
p        : b(A000040(n)) = 1.6180339887498949025257388711...
p^2      : b(A001248(n)) = 2.0000000000000000000000000000...
p*q      : b(A006881(n)) = 2.2055694304005917238953315973...
p^3      : b(A030078(n)) = 2.3247179572447480566665944934...
p^2*q    : b(A054753(n)) = 2.6716998816571604358216518448...
p^4      : b(A030514(n)) = 2.6180339887498917939012699207...
p^3*q    : b(A065036(n)) = 3.0803227214906021558249449299...
p*q*r    : b(A007304(n)) = 2.9379558827528557962693867011...
p^5      : b(A050997(n)) = 2.8905508875432590620846440288...
p^2*q^2  : b(A085986(n)) = 3.2187765853016649941764626419...
p^4*q    : b(A178739(n)) = 3.4534111136673804054453285061...
p^2*q*r  : b(A085987(n)) = 3.5339198574905377192578725953...
p^6      : b(A030516(n)) = 3.1478990357047909043330946587...
p^3*q^2  : b(A143610(n)) = 3.7022736187975437971431347250...
p^5*q    : b(A178740(n)) = 3.8016448153137023524550386355...
p^3*q*r  : b(A189975(n)) = 4.0600260453688532535920785448...
p^7      : b(A092759(n)) = 3.3935083220984414431597997463...
p^4*q^2  : b(A189988(n)) = 4.1453038440113498808159420150...
p^2*q^2*r: b(A179643(n)) = 4.2413382309993874486053755390...
p^6*q    : b(A189987(n)) = 4.1311805192254587026923218218...
p*q*r*s  : b(A046386(n)) = 3.8825338629275134572083061357...
...
b(Axxxxxx(1)) in the sequences above, is given by A025487.
(End)
First column in the coefficients of the characteristic polynomials is the Möbius function A008683.
Row sums of coefficients start: 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
Third diagonal is a signed version of A000096.
Most of the eigenvalues are equal to 1. The number of eigenvalues equal to 1 are given by A075795 for n>1.
The first three of the eigenvalues above can be calculated as nested radicals. The fourth eigenvalue 2.205569430400590... minus 1 = 1.205569430400590... is also a nested radical.

			{
{ 1},
{ 1, -1},
{-1, -1,  1},
{-1,  0,  2,  -1},
{ 0,  0,  2,  -3,  1},
{-1,  2,  1,  -5,  4,   -1},
{ 1, -3,  5,  -8,  9,   -5,   1},
{-1,  4, -4,  -5, 15,  -14,   6,  -1},
{ 0, -1,  6, -17, 29,  -31,  20,  -7,  1},
{ 0,  0,  2, -13, 36,  -55,  50, -27,  8, -1},
{ 1, -7, 23, -50, 84, -112, 112, -78, 35, -9, 1}
}

Mats Granvik, Mathematica program to verify that eigenvalues determine prime signature
OEIS Wiki, Prime signatures
Eric Weisstein, Prime signature
Index to sequences related to prime signature

Cf. A051731, A008683, A000040, A001248, A006881, A030078, A030514, A054753, A000096, A001622, A272874, A075795.

Cf. A025487. - Mats Granvik, Sep 30 2017

Clear[x, AA, nn, s]; Monitor[AA = Flatten[Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; MatrixForm[A]; a = A[[1, nn]]; A[[1, nn]] = A[[nn, nn]]; A[[nn, nn]] = a; CoefficientList[CharacteristicPolynomial[A, x], x], {nn, 1, 10}]], nn]

A257042 a(n) = (3n+7)n^2.

Original entry on oeis.org

0, 10, 52, 144, 304, 550, 900, 1372, 1984, 2754, 3700, 4840, 6192, 7774, 9604, 11700, 14080, 16762, 19764, 23104, 26800, 30870, 35332, 40204, 45504, 51250, 57460, 64152, 71344, 79054, 87300, 96100, 105472, 115434, 126004, 137200, 149040, 161542, 174724

Offset: 0

Garrett Frandson, Apr 14 2015

Consider a natural number r such that r has 15 proper divisors and 5 prime factors (note that these prime factors do not have to be distinct). The difference between these two values, say d(r), is in this case 10. Where n is a positive integer, d(r^n)=(3*n+7)*n^2.

			The smallest integer that satisfies this is 120: it has 15 proper divisors (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60) and 5 prime factors (2, 2, 2, 3, 5), so d(120)=10. The square of 120, 14400, we would expect to have a difference of 52 between the number of its proper divisors and prime factors, and with respectively 62 and 10, d(120)=52 indeed. Checking this with further integer powers of 120 will continue to generate terms in this sequence.
The integers which satisfy the proper-divisor-prime-factor requirement are those of A189975.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A189975.

[(3*n+7)*n^2: n in [0..65]]; // Vincenzo Librandi, Apr 15 2015

A257042:=n->(3*n+7)*n^2: seq(A257042(n), n=0..50); # Wesley Ivan Hurt, Apr 16 2015

Table[(3 n + 7) n^2, {n, 40}] (* or *) CoefficientList[Series[(10 + 12 x - 4 x^2) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 15 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,10,52,144},40] (* Harvey P. Dale, Mar 27 2025 *)

lista(nn) = {v = 1; while(!((numdiv(v)-1 == 15) && (bigomega(v) == 5)), v++); for (n=0, nn, vn = v^n; nb = numdiv(vn)-1-bigomega(vn); print1(nb, ", "););} \\ Michel Marcus, Apr 16 2015

From Vincenzo Librandi, Apr 15 2015: (Start)
G.f.: x*(10+12*x-4*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
From Amiram Eldar, Jul 30 2024: (Start)
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/98 + Pi^2/42 + 9*log(3)/98 - 351/1372.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/84 - sqrt(3)*Pi/49 - 6*log(2)/49 + 225/1372. (End)

More terms from Vincenzo Librandi, Apr 15 2015

cp's OEIS Frontend

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A179642 Product of exactly 5 primes, 3 of which are distinct.

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A189982 Numbers with prime signature (2,1,1,1), i.e., factorization p*q*r*s^2 with distinct primes p, q, r, s.

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A179690 Numbers of the form p^2*q^2*r*s where p, q, r, and s are distinct primes.

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A275387 Numbers of ordered pairs of divisors d < e of n such that gcd(d, e) > 1.

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A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

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A381315 Numbers whose prime factorization exponents include exactly one 3 and no exponent greater than 3.

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A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.

A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

A257042 a(n) = (3n+7)n^2.