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.

4, 8, 12, 16, 18, 26, 32, 36, 38, 44, 48, 50, 52, 56, 58, 64, 68, 72, 74, 76, 78, 80, 82, 86, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 118, 122, 124, 126, 128, 132, 134, 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

Offset: 1

Author

Antti Karttunen, Feb 16 2022

Keywords

• nonn

Comments

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.

Even numbers k for which A351555(k) > 0.

Question: Is A351538 subsequence of this sequence?

Examples

			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.
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.
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..29826
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A003961, A286561, A351555.

Subsequences: A351541, A351542, and also conjecturally A351538.

Cf. A351553 (complement among even numbers).

No common terms with A349745.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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)); };
isA351543(n) = (!(n%2) && A351555(n)>0);