A141621 -id:A141621 - cp's OEIS frontend

A030515 Numbers with exactly 6 divisors.

12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428

Offset: 1

Jeff Burch

Numbers which are either the 5th power of a prime or the product of a prime and the square of a different prime, i.e., numbers which are in A050997 (5th powers of primes) or A054753. - Henry Bottomley, Apr 25 2000
Also numbers which are the square root of the product of their proper divisors. - Amarnath Murthy, Apr 21 2001
Such numbers are multiplicatively 3-perfect (i.e., the product of divisors of a(n) equals a(n)^3). - Lekraj Beedassy, Jul 13 2005
Since A119479(6)=5, there are never more than 5 consecutive terms. Quintuples of consecutive terms start at 10093613546512321, 14414905793929921, 266667848769941521, ... (A141621). No such quintuple contains a term of the form p^5. - Ivan Neretin, Feb 08 2016

Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

R. J. Mathar, Table of n, a(n) for n = 1..1000
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.12.
Eric Weisstein's World of Mathematics, Divisor Product

Cf. A061117.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, {2,seq(i,i=3..floor(N/4))}):
S:= select(`<=`,{seq(p^5, p = Primes),seq(seq(p*q^2, p=Primes minus {q}),q=Primes)},N):
sort(convert(S,list)); # Robert Israel, Feb 10 2016

f[n_]:=Length[Divisors[n]]==6; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
Select[Range[500],DivisorSigma[0,#]==6&] (* Harvey P. Dale, Oct 02 2014 *)

is(n)=numdiv(n)==6 \\ Charles R Greathouse IV, Jan 23 2014

from sympy import divisor_count
def ok(n): return divisor_count(n) == 6
print([k for k in range(429) if ok(k)]) # Michael S. Branicky, Dec 18 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A030515(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**2) for p in primerange(isqrt(x)+1))+primepi(integer_nthroot(x,3)[0])-primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Union of A050997 and A054753. - Lekraj Beedassy, Jul 13 2005

A000005(a(n))=6. - Juri-Stepan Gerasimov, Oct 10 2009

Definition clarified by Jonathan Sondow, Jan 23 2014

A074172 Smaller of two consecutive numbers of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 244, 332, 387, 507, 548, 603, 604, 724, 844, 908, 931, 963, 1075, 1083, 1251, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 2523, 2524, 2636, 2644, 2763, 3283, 3356, 3411, 3508, 3788, 3987, 4075, 4203, 4204, 4418, 4491, 4804, 4868

Offset: 1

Amarnath Murthy, Aug 30 2002

nonn

From Robert Israel, Dec 06 2018: (Start)
There are four forms of terms, for odd primes p,q,r:
4*p where 4*p+1 = q^2*r, r == 1 (mod 4)
2*p^2 where 2*p^2+1 = q^2*r, r == 3 (mod 4)
p^2*q where p^2*q+1 = 2*r^2, q == 1 (mod 4)
p^2*q where p^2*q+1 = 4*r, q == 3 (mod 4).
Are there infinitely many terms of each type?
(End)

			44 is a member as 44 = 2^2*11 and 45 = 3^2*5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A054753, A074173, A074174, A178032, A308683, A141621.

filter:= proc(n) local F;
F:= map(t -> t[2],ifactors(n)[2]);
F = [2,1] or F = [1,2]
end proc:
A054753:= select(filter, {$1..10000}):
sort(convert(A054753 intersect map(`-`,A054753,1),list)); # Robert Israel, Dec 06 2018

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst

isok1(n) = vecsort(factor(n)[,2]) == [1,2]~;
isok(n) = isok1(n) && isok1(n+1); \\ Michel Marcus, Sep 20 2017

More terms from T. D. Noe, Oct 04 2004

Name clarified by Sean A. Irvine, Jan 13 2025

A178032 Numbers n such that n, n+1, n+2 are all of the form p*q^2 for distinct primes p,q.

Original entry on oeis.org

603, 2523, 4203, 4923, 7442, 10467, 18027, 20402, 54475, 58923, 79011, 97675, 104211, 118323, 120787, 122571, 124891, 132723, 134307, 148075, 200491, 229707, 243602, 246571, 249307, 258507, 303651, 324331, 331387, 370827, 385675

Offset: 1

John L. Drost, Dec 16 2010

nonn

			603=3*3*67, 604=2*2*151, 605=5*11*11
2523=3*29*29, 2524=2*2*631, 2525=5*5*101

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 500 terms from Harvey P. Dale)
Index to sequences related to prime signature

Cf. A054753, A074172, A308683, A141621.

SequencePosition[Table[If[Sort[FactorInteger[n][[All,2]]]=={1,2},1,0],{n,400000}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 30 2019 *)

is_A178032 = lambda n: all(sorted(m for p,m in factor(k)) == [1, 2] for k in (n, n+1, n+2)) # D. S. McNeil, Dec 17 2010

Corrected and extended by D. S. McNeil, Dec 16 2010

A308683 Numbers k such that k, k+1, k+2, k+3 are all of the form p*q^2 for distinct primes p,q.

Original entry on oeis.org

17042641441, 180383003522, 231242161921, 313187053922, 349881182161, 717767333521, 740230547281, 907336510801, 1080505860722, 1646310367681, 1682581387922, 1896109379522, 2638936079522, 2919248012401, 3121595927522

Offset: 1

Ray Chandler, Jun 21 2019

nonn

See StackExchange link for pseudo-code to generate four or five consecutive values of the form.

			17042641441 = 7 * 7 * 347809009,
17042641442 = 2 * 92311 * 92311,
17042641443 = 3 * 3 * 1893626827,
17042641444 = 2 * 2 * 4260660361.

Ray Chandler, Table of n, a(n) for n = 1..10000
StackExchange, Sequence of numbers with prime factorization pq^2
Index to sequences related to prime signature

Cf. A054753, A074172, A178032, A141621.

A178810 Largest possible number of consecutive integers with the same prime signature as A025487(n).

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 1, 3, 3, 1, 1, 3, 7, 1

Offset: 1

Will Nicholes, Jun 16 2010

The corresponding smallest integers that begin these largest runs of consecutive integers are in A178811. - Bernard Schott, Feb 16 2021

a(16) = 3 (see A178811). - Jon E. Schoenfield, Dec 02 2023

			A025487(2) = 2, prime signature {1}. There are a maximum of 2 consecutive integers with that prime signature: 2 and 3.
A025487(4) = 6, prime signature {1,1}. There are a maximum of 3 consecutive integers with that prime signature (e.g., 33, 34 and 35).
A025487(6) = 12, prime signature {1,2}. There are a maximum of 5 consecutive integers with that prime signature (e.g., 10093613546512321, 10093613546512322, 10093613546512323, 10093613546512324, and 10093613546512325). Compare A141621.
A025487(13) = 60, prime signature {1,1,2}. There are a maximum of 7 possible consecutive integers, between two multiples of 8, with that prime signature; the smallest such run starts at 932537185321. - _Bernard Schott_, Feb 16 2021

Diophante, A1845, Les squelettes (in French).

Cf. A025487, A141621, A178811.

Minor edits by Ray Chandler, Jul 29 2010
a(6) corrected by Bobby Jacobs, Sep 25 2016
a(12)-a(14) from Bernard Schott, Feb 16 2021

A178811 The smallest integer that begins the longest run of consecutive integers with the prime signature of A025487(n).

Original entry on oeis.org

1, 2, 4, 33, 8, 10093613546512321, 16, 28375, 1309, 32, 36, 7939375, 932537185321, 64

Offset: 1

Will Nicholes, Jun 16 2010

From Bernard Schott, Feb 17 2021: (Start)
The corresponding lengths of these longest runs of consecutive integers are in A178810.
If a(n) = 2^k for some k <> 1, then a(n) = A025487(n) and A178810(n) = 1; for k = 1, a(2) = A025487(2) = A178810(2) = 2 because there exists a run of two consecutive primes (2,3).
a(18) = 128, a(22) = 203433. [corrected by Jon E. Schoenfield, Nov 30 2023] (End)
From Jon E. Schoenfield, Dec 02 2023: (Start)
a(16) = 3302209375 = 5^5 * 1056707. (3302209376 = 2^5 * 103194043, 3302209377 = 3^5 * 13589339.) No run of four consecutive integers of the form p^5 * q with p,q distinct primes can exist: one of the two even numbers would be 32*q and the other would be 2*p^5, and they would differ by 2, yielding either (1) 2*p^5 + 2 = 32*q -> p^5 + 1 = 16*q -> (p^4 - p^3 + p^2 - p + 1)*(p+1) = 16*q, so p+1 = 16, but then p = 15 would not be a prime, or (2) 2*p^5 - 2 = 32*q -> p^5 - 1 = 16*q -> (p^4 + p^3 + p^2 + p + 1)*(p-1) = 16*q, so p-1 = 16, so p = 17, but then the odd number between 2*p^5 and 32*q would be 2*17^5 - 1 = 2839713 = 3 * 37 * 25583 (which would not have the required prime signature).
a(17) <= 921198089181020748838245 (which starts a run of seven consecutive integers of the form p^3*q*r; no run of eight or more can exist, as any set of eight consecutive integers includes an odd multiple of 4).
a(n) = A025487(n) if A025487(n) is a proper power (i.e., a number of the form b^e where b,e > 1). (Thus a(3) = 4, a(5) = 8, a(7) = 16, a(10) = 32, a(11) = 36, a(14) = 64, a(18) = 128, a(19) = 144, a(23) = 216, a(25) = 256, a(32) = 512, a(33) = 576, a(38) = 900, a(40) = 1024, a(44) = 1296, a(48) = 1728, a(51) = 2048, a(53) = 2304.)
Conjecture: a(n) = A025487(n) if A025487(n) is a powerful number (A001694); i.e., if A025487(n) is a powerful number, then there exists no run of two or more consecutive integers with the same prime signature as that of A025487(n). (E.g., if this conjecture holds, a(15) = 72 (cf. A367781), a(26) = 288, a(30) = 432, a(37) = 864, a(42) = 1152, a(49) = 1800.) (End)

			For n = 3, A025487(3) = 4, corresponding to a prime signature of {2}. Since the maximum number of consecutive integers with that prime signature is 1, a(3) is 4, the smallest integer that starts a "run" of 1.
A025487(4) = 6 whose prime signature is {1,1}; a(4) = 33 because 33 is the smallest integer where starts a run of A178810(4) = 3 consecutive integers with prime signature {1,1}: (33=3*11, 34=2*17, 35=5*7). - _Bernard Schott_, Feb 16 2021

Diophante, A1845, Les squelettes (in French).

Cf. A001694, A025487, A060355, A178810 (maximum size of such runs), A141621.

Minor edits by Ray Chandler, Jul 29 2010
a(6) corrected by Bobby Jacobs, Sep 25 2016
a(12) from Hugo van der Sanden, May 20 2019
a(13)-a(14) from Bernard Schott, Feb 16 2021

A323743 Table read by rows: row n lists the numbers k for which there exist only finitely many runs of n consecutive integers whose number-of-divisors function sums to k.

Original entry on oeis.org

1, 3, 4, 5, 5, 7, 8, 9, 8, 9, 11, 12, 13, 14, 15, 10, 13, 15, 17, 18, 19, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 16, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 20, 22, 24, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39

Offset: 1

Jon E. Schoenfield, Apr 02 2019

Row n lists the numbers k such that
  0 < |{m : Sum_j={m..m+n-1} tau(j) = k}| < infinity
where tau(j) = A000005(j) is the number of divisors of j.

			There is only one number with exactly 1 divisor (namely, k=1), but there are infinitely many numbers with j divisors for every j >= 2, so row 1 consists only of the single term 1.
The sequence of values tau(k) for k >= 1 is A000005, which begins 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ..., from which the sums of two consecutive terms are 1+2=3, 2+2=4, 2+3=5, 3+2=5, 2+4=6, 4+2=6, 2+4=6, 4+3=7, 3+4=7, ...; no number j < 3 appears as such a sum, every j >= 6 appears infinitely many times as such a sum, and each j in {3,4,5} appears as such a sum only finitely many times, so row 2 is {3, 4, 5}.
Row 3 does not contain 6 as a term because there exists no run of 3 consecutive numbers whose sum of tau values is exactly 6.
The first six rows of the table are as follows:
  row 1: {1};
  row 2: {3, 4, 5};
  row 3: {5, 7, 8, 9};
  row 4: {8, 9, 11, 12, 13, 14, 15};
  row 5: {10, 13, 15, 17, 18, 19};
  row 6: {14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27}.

Cf. A000005, A005237, A006558, A048892, A072507, A100366, A119479, A141621, A284596, A284597, A292580, A319037, A319045, A319046.

A373560 a(n) is the smallest multiple of prime(n)^2 that starts a run of 5 consecutive integers with 6 divisors, or -1 if no such multiple exists.

Original entry on oeis.org

-1, -1, -1, 10093613546512321, -1, -1, 7700031346933907521, -1, 5344962129269790721, -1, 20453982425165652721, -1, 8163195338222675521, -1, 2467958104789157112721, -1, -1, -1, -1, 14666767069023896053921, 212170739123852995921, 287954235303137500060321, -1, 84769922583214545304321

Offset: 1

Jon E. Schoenfield, Jun 09 2024

sign

Terms were obtained using the b-file at A141621.
a(n) = -1 if prime(n) is not in A001132.
Conjecture: the converse is also true.

			a(1) = a(2) = a(3) = -1 because the first of five consecutive integers having six divisors is never a multiple of 2^2, 3^2, or 5^2.
a(4) = 10093613546512321 because it is the smallest term in A141621 that is a multiple of prime(4)^2 = 49.
a(9) = 5344962129269790721 because it is the smallest term in A141621 that is a multiple of prime(9)^2 = 23^2.

Cf. A001132, A030515, A042999, A054753, A141621.

cp's OEIS Frontend

A030515 Numbers with exactly 6 divisors.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A074172 Smaller of two consecutive numbers of the form p^2*q where p and q are distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A178032 Numbers n such that n, n+1, n+2 are all of the form p*q^2 for distinct primes p,q.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Extensions

A308683 Numbers k such that k, k+1, k+2, k+3 are all of the form p*q^2 for distinct primes p,q.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A178810 Largest possible number of consecutive integers with the same prime signature as A025487(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A178811 The smallest integer that begins the longest run of consecutive integers with the prime signature of A025487(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A323743 Table read by rows: row n lists the numbers k for which there exist only finitely many runs of n consecutive integers whose number-of-divisors function sums to k.

Views

Author

Keywords

Comments

Examples

Crossrefs

A373560 a(n) is the smallest multiple of prime(n)^2 that starts a run of 5 consecutive integers with 6 divisors, or -1 if no such multiple exists.

Views

Author

Keywords

Comments

Examples