cp's OEIS Frontend

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

%I A141621 #68 Apr 07 2025 12:26:59
%S A141621 10093613546512321,14414905793929921,266667848769941521,
%T A141621 562672865058083521,1579571757660876721,1841337567664174321,
%U A141621 2737837351207392721,4456162869973433521,4683238426747860721,4993613853242910721,5037980611623036721,5174116847290255921
%N A141621 Numbers that begin a run of 5 consecutive integers of the form p^2*q where p and q are distinct primes.
%C A141621 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]
%C A141621 Each of the five numbers in each such sequence has 6 divisors.
%C A141621 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.
%C A141621 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]
%C A141621 The first quintuple not of the aforementioned form starts with 5344962129269790721 = 23^2*prime. - _Ivan Neretin_, Feb 08 2016
%C A141621 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
%H A141621 Ray Chandler, <a href="/A141621/b141621.txt">Table of n, a(n) for n = 1..2000</a> (first 25 terms from Ivan Neretin, through 200 terms from Jon E. Schoenfield)
%H A141621 Richard I. Hess, <a href="https://cms.math.ca/crux/backfile/Crux_v13n04_Apr.pdf">Problem 1231</a>, Crux Mathematicorum, Vol. 13, No. 4, p. 118, 1987. (takes a long time to download)
%H A141621 Richard I. Hess, <a href="https://web.archive.org/web/20111121204944/http://totalgadha.com/file.php/1/moddata/forum/17/20234/mm-hess.pdf">Puzzles from Around the World</a>, p. 63, H17.
%H A141621 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_020.htm">Problem 20.- Divisors (II) K consecutive numbers with the same number of divisors</a>, The Prime Puzzles & Problems Connection.
%H A141621 StackExchange, <a href="http://math.stackexchange.com/questions/64892/sequence-of-numbers-with-prime-factorization-pq2">Sequence of numbers with prime factorization pq^2</a>
%H A141621 wu :: forums, <a href="http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1191596291">Same Number of Divisors</a>, Oct 05 2007.
%H A141621 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%e A141621 a(1) = 10093613546512321, because
%e A141621 10093613546512321 = 7^2 * 205992113194129,
%e A141621 10093613546512322 =   2 * 71040881^2,
%e A141621 10093613546512323 = 3^2 * 1121512616279147,
%e A141621 10093613546512324 = 2^2 * 2523403386628081, and
%e A141621 10093613546512325 = 5^2 * 403744541860493,
%e A141621 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]
%o A141621 (Sage)
%o A141621 ## Warning: this program appears to be incorrect [_Joerg Arndt_, Feb 29 2016]
%o A141621 for m in range(5000):
%o A141621     p = 22050*m+17831
%o A141621     if is_prime(p):
%o A141621         n = 2*p^2-2
%o A141621         n4 = n/4+1
%o A141621         if is_prime(n4):
%o A141621             n49 = floor((n+1)/49)
%o A141621             if (49*n49 == n+1) and is_prime(n49):
%o A141621                 n9 = floor((n+3)/9)
%o A141621                 if (9*n9 == n+3) and is_prime(n9):
%o A141621                     n25 = floor((n+5)/25)
%o A141621                     if (25*n25 == n+5) and is_prime(n25):
%o A141621                         print(n+1, n49, p, n9, n4, n25)
%Y A141621 Cf. A119479, A006558, A005237, A005238, A006601.
%Y A141621 Cf. A054753, A074172, A178032, A308683.
%K A141621 hard,nonn
%O A141621 1,1
%A A141621 _Matthijs Coster_, Aug 23 2008
%E A141621 Two more terms _Matthijs Coster_, Aug 28 2008
%E A141621 Missing terms added and extended by _Ivan Neretin_, Feb 08 2016
%E A141621 New name from _Jon E. Schoenfield_, Jun 05 2018