A308683 -id:A308683 - cp's OEIS frontend

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

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

A141621 Numbers that begin a run of 5 consecutive integers of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

10093613546512321, 14414905793929921, 266667848769941521, 562672865058083521, 1579571757660876721, 1841337567664174321, 2737837351207392721, 4456162869973433521, 4683238426747860721, 4993613853242910721, 5037980611623036721, 5174116847290255921

Offset: 1

Matthijs Coster, Aug 23 2008

Old name was "The first number of a series of 5 consecutive numbers with the same signature, i.e., all numbers have the format p^2*q, where p and q are primes. Therefore the number of divisors is the same (6)." [That name could have been confusing in that not every sequence of 5 consecutive integers having the same prime signature has the prime signature p^2*q; e.g., 204323 is the first of 5 consecutive numbers of the form p^2*q*r. - Jon E. Schoenfield, Jun 05 2018]
Each of the five numbers in each such sequence has 6 divisors.
It is easy to prove that any number in this sequence must be congruent to 1 modulo 240. The program below calculates only an element of the sequence. Since the reference A119479 it is the smallest one. If we assume that the first element has the format 7^2*n49, the second number has the format 2*p^2, the third element has the format 3^2*n9 and the fifth element has the format 5^2*n25, then p must be modulo 22050 one out of 1181, 3719, 4219, 9119, 12931, 17831, 18331 or 20869.
It is unclear if these numbers are the smallest ones. - Matthijs Coster, Aug 28 2008 [The terms listed in the Data section are, in fact, the smallest numbers matching the definition. - Jon E. Schoenfield, Jun 05 2018]
The first quintuple not of the aforementioned form starts with 5344962129269790721 = 23^2*prime. - Ivan Neretin, Feb 08 2016
Among the first 200 terms, the frequency with which the squared prime factor p is {7, 17, 23, 31, 41, 47, 73, 127, 193, 1039, 1399} is {171, 10, 6, 4, 3, 1, 1, 1, 1, 1, 1}, respectively. - Jon E. Schoenfield, Jun 09 2018

			a(1) = 10093613546512321, because
10093613546512321 = 7^2 * 205992113194129,
10093613546512322 =   2 * 71040881^2,
10093613546512323 = 3^2 * 1121512616279147,
10093613546512324 = 2^2 * 2523403386628081, and
10093613546512325 = 5^2 * 403744541860493,
so each of the five consecutive integers is of the form p^2*q, and no smaller run of five consecutive integers has this property. [corrected by _Jon E. Schoenfield_, Jun 05 2018]

Ray Chandler, Table of n, a(n) for n = 1..2000 (first 25 terms from Ivan Neretin, through 200 terms from Jon E. Schoenfield)
Richard I. Hess, Problem 1231, Crux Mathematicorum, Vol. 13, No. 4, p. 118, 1987. (takes a long time to download)
Richard I. Hess, Puzzles from Around the World, p. 63, H17.
Carlos Rivera, Problem 20.- Divisors (II) K consecutive numbers with the same number of divisors, The Prime Puzzles & Problems Connection.
StackExchange, Sequence of numbers with prime factorization pq^2
wu :: forums, Same Number of Divisors, Oct 05 2007.
Index to sequences related to prime signature

Cf. A119479, A006558, A005237, A005238, A006601.

Cf. A054753, A074172, A178032, A308683.

## Warning: this program appears to be incorrect [Joerg Arndt, Feb 29 2016]
for m in range(5000):
    p = 22050*m+17831
    if is_prime(p):
        n = 2*p^2-2
        n4 = n/4+1
        if is_prime(n4):
            n49 = floor((n+1)/49)
            if (49*n49 == n+1) and is_prime(n49):
                n9 = floor((n+3)/9)
                if (9*n9 == n+3) and is_prime(n9):
                    n25 = floor((n+5)/25)
                    if (25*n25 == n+5) and is_prime(n25):
                        print(n+1, n49, p, n9, n4, n25)

Two more terms Matthijs Coster, Aug 28 2008
Missing terms added and extended by Ivan Neretin, Feb 08 2016
New name from Jon E. Schoenfield, Jun 05 2018

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

cp's OEIS Frontend

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

A141621 Numbers that begin a run of 5 consecutive integers of the form p^2*q where p and q are distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

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