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A057949 Numbers with more than one factorization into S-primes. See A054520 and A057948 for definition.

%I A057949 #23 May 27 2021 17:16:19
%S A057949 441,693,1089,1197,1449,1617,1881,1953,2205,2277,2541,2709,2793,2961,
%T A057949 3069,3249,3381,3465,3717,3933,3969,4221,4257,4389,4473,4557,4653,
%U A057949 4761,4977,5229,5301,5313,5445,5733,5841,5929,5985,6237,6321,6417,6489,6633
%N A057949 Numbers with more than one factorization into S-primes. See A054520 and A057948 for definition.
%C A057949 Numbers with k >= 4 prime factors (with multiplicity) that are congruent to 3 mod 4, no k-1 of which are equal. - _Charlie Neder_, Nov 03 2018
%H A057949 Eric M. Schmidt, <a href="/A057949/b057949.txt">Table of n, a(n) for n = 1..10000</a>
%e A057949 2205 is in S = {1,5,9, ... 4i+1, ...}, 2205 = 5*9*49 = 5*21^2; 5, 9, 21 and 49 are S-primes (A057948).
%o A057949 (Sage)
%o A057949 def A057949_list(bound) :
%o A057949     numterms = (bound-1)//4 + 1
%o A057949     M = [1] * numterms
%o A057949     for k in range(1, numterms) :
%o A057949         if M[k] == 1 :
%o A057949             kpower = k
%o A057949             while kpower < numterms :
%o A057949                 step = 4*kpower+1
%o A057949                 for j in range(kpower, numterms, step) :
%o A057949                     M[j] *= 4*k+1
%o A057949                 kpower = 4*kpower*k + kpower + k
%o A057949     # Now M[k] contains the product of the terms p^e where p is an S-prime
%o A057949     # and e is maximal such that p^e divides 4*k+1
%o A057949     return [4*k+1 for k in range(numterms) if M[k] > 4*k+1]
%o A057949 # _Eric M. Schmidt_, Dec 11 2016
%o A057949 (PARI) ok(n)={if(n%4==1, my(f=factor(n)); my(s=[f[i,2] | i<-[1..#f~], f[i,1]%4==3]); vecsum(s)>=4 && vecmax(s)<vecsum(s)-1, 0)} \\ _Andrew Howroyd_, Nov 25 2018
%Y A057949 Cf. A054520, A057948, A057950.
%Y A057949 Cf. A343826 (only 1 way), A343827 (exactly 2 ways), A343828 (exactly 3 ways).
%K A057949 nonn
%O A057949 1,1
%A A057949 _Jud McCranie_, Oct 14 2000
%E A057949 Offset corrected by _Eric M. Schmidt_, Dec 11 2016