cp's OEIS Frontend

A331029 Least integer of each composite prime signature where primes ending in 1 or 9 are treated distinctly from those ending in 3 or 7.

%I A331029 #20 Mar 23 2025 20:53:43
%S A331029 1,3,9,11,21,27,33,63,81,99,121,189,209,231,243,273,297,363,441,567,
%T A331029 627,693,729,819,891,1089,1323,1331,1701,1881,2079,2187,2299,2457,
%U A331029 2541,2673,3003,3267,3969,3993,4389,4641,4851,5103,5643,5733,6061,6237,6561,6897
%N A331029 Least integer of each composite prime signature where primes ending in 1 or 9 are treated distinctly from those ending in 3 or 7.
%C A331029 This sequence is analogous to A025487. The two primes 2 and 5 are ignored and the remainder are divided into two distinct classes depending on the final digit of the prime. A combined prime signature is then created from the prime signatures of the two classes of prime.
%C A331029 Consider the problem of finding the smallest number having n divisors ending with 1 or 9 (sequence A085645). Solutions must lie in this sequence since numbers with the same composite prime signature as defined here will have the same number of divisors ending with 1 or 9.
%C A331029 Primes ending in either 1 or 9 are 11, 19, 29, 31, 41, 59, ... (A045468).
%C A331029 Primes ending in either 3 or 7 are 3, 7, 13, 17, 23, 37, ... (A097957).
%C A331029 The partial products of these two sequences form two sequences analogous to the primorial numbers (11, 11*19, 11*19*29, ... and 3, 3*7, 3*7*13, ...). In the same manner that A025487 can be defined as products of primorial numbers, an alternative description of this sequence is that it is the set of all products of the two primorial analogs.
%H A331029 Andrew Howroyd, <a href="/A331029/b331029.txt">Table of n, a(n) for n = 1..1000</a>
%e A331029 Primes in this sequence are 3 and 11 because these are the smallest primes in the two classes.
%e A331029 Semiprimes in this sequence are 9 = 3^2, 21 = 3*7, 33 = 3*11, 121 = 11^2,  209 = 11*19 because 3, 7 are the smallest primes ending with either 3 or 7 and 11, 19 are the smallest primes ending with either 1 or 9.
%o A331029 (PARI)
%o A331029 GenS(lim, pred)={my(L=List(), S=[1]); forprime(p=2, oo, if(pred(p), listput(L,S); my(pp=vector(logint(lim, p), i, p^i)); S=concat([k*pp[1..min(if(k>1, my(f=factor(k)[,2]); f[#f], oo), logint(lim\k, p))] | k<-S]); if(!#S, return(Set(concat(L)))) ))}
%o A331029 Merge(s1, s2, lim)={Set(concat(vector(#s1, i, [t | t<-s1[i]*s2, t<=lim])))}
%o A331029 lista331029(lim)={Merge(GenS(lim, k->abs(k%10-5)==2), GenS(lim, k->abs(k%10-5)==4), lim)}
%o A331029 { lista331029(10^4) }
%Y A331029 Cf. A025487, A045468 (primes ending in 1 or 9), A085645, A097957 (primes ending in 3 or 7), A331082.
%K A331029 nonn,base
%O A331029 1,2
%A A331029 _Andrew Howroyd_, Jan 07 2020