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A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

%I A351543 #19 Feb 18 2022 11:10:29
%S A351543 4,8,12,16,18,26,32,36,38,44,48,50,52,56,58,64,68,72,74,76,78,80,82,
%T A351543 86,88,90,92,96,98,100,104,108,112,116,118,122,124,126,128,132,134,
%U A351543 136,144,146,148,150,152,156,158,162,164,166,172,176,178,180,184,188,192,194,196,200,202,204,206,208,212,218,222,226
%N A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.
%C A351543 Even numbers k such that sigma(k) has an odd prime factor prime(i), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351542 in that prime(i) is not here required to be a factor of k itself. The condition implies also that if there is any such odd prime factor prime(i) of sigma(k), it must be >= 5.
%C A351543 Even numbers k for which A351555(k) > 0.
%C A351543 Question: Is A351538 subsequence of this sequence?
%H A351543 Antti Karttunen, <a href="/A351543/b351543.txt">Table of n, a(n) for n = 1..29826</a>
%H A351543 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A351543 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A351543 12 = 2^2 * 3 is present as sigma(12) = 28 = 2^2 * 7, whose prime factorization contains an odd prime 7 such that neither it nor the immediately previous prime, which is 5, divide 12 itself.
%e A351543 196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.
%e A351543 364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.
%o A351543 (PARI)
%o A351543 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A351543 A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
%o A351543 isA351543(n) = (!(n%2) && A351555(n)>0);
%Y A351543 Cf. A000203, A003961, A286561, A351555.
%Y A351543 Subsequences: A351541, A351542, and also conjecturally A351538.
%Y A351543 Cf. A351553 (complement among even numbers).
%Y A351543 No common terms with A349745.
%K A351543 nonn
%O A351543 1,1
%A A351543 _Antti Karttunen_, Feb 16 2022