A349745 -id:A349745 - cp's OEIS frontend

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

Antti Karttunen, Feb 16 2022

nonn

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?

			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.

Cf. A000203, A003961, A286561, A351555.
Subsequences: A351541, A351542, and also conjecturally A351538.
Cf. A351553 (complement among even numbers).
No common terms with A349745.

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);

A353365 Numbers k such that the odd part of sigma(sigma(k)) is equal to the odd part of sigma(k).

Original entry on oeis.org

1, 5, 12, 427, 9120, 9180, 9504, 9720, 9960, 10296, 10620, 10740, 10824, 11070, 11310, 11480, 11484, 11556, 11628, 11748, 11934, 11960, 12024, 12036, 12072, 12084, 12376, 12460, 12510, 12570, 12640, 12924, 12980, 13000, 13216, 13340, 13554, 13804, 13806, 13962, 13984, 14022, 14056, 14094, 14178, 14212, 14336, 14380

Offset: 1

Antti Karttunen, Apr 17 2022

nonn

Numbers k such that sigma(sigma(k)) = 2^e * sigma(k), for some e >= 0.
Numbers k such that sigma(k) is in A336702.
Numbers k for which A000265(A051027(k)) = A161942(k).
If there existed any hypothetical 3-perfect number (A005820) of the form x = 4u+2 and not divisible by 3, then x would be also included in this sequence, as then sigma(sigma(x)) = 12*x = 4*sigma(x). Such x would be also a term of A349745 and of A351458, and x/2 would be a rare odd term of A000396, and also in A336702. See also the diagram in A347392.

Index entries for sequences related to sigma(n)

Cf. A000203, A000265, A000396, A005820, A051027, A161942, A336702.
Subsequence of A066961.
Cf. also A336700, A336701, A347383, A347392, A349745, A351458 and also A353363.

A000265(n) = (n>>valuation(n,2));
isA353365(n) = (A000265(sigma(sigma(n)))==A000265(sigma(n)));

A349746 Numbers k for which k * gcd(sigma(k), u) is equal to sigma(k) * gcd(k, u), where u is obtained by shifting the prime factorization of k two steps toward larger primes [with u = A003961(A003961(k))].

Original entry on oeis.org

1, 11466, 114660, 411264, 804384, 871416, 4999680, 46332000, 176417280, 378069120, 396168192, 485188704, 709430400, 2004912000, 3921372000, 5600534400, 6128179200, 6956471808, 7556976000, 7746979968, 9904204800, 14092001280, 14182439040, 23423662080, 31998395520

Offset: 1

Antti Karttunen, Nov 30 2021

nonn

Sigma preserves both the 2-adic and 3-adic valuation of the terms of this sequence.

All 65 known 5-multiperfect numbers (A046060) are included in this sequence, as well as the smallest 7-multiperfect number, 141310897947438348259849402738485523264343544818565120000 = A007539(7), and probably the majority of other p-multiperfect numbers as well, where p is a prime > 3. However, any term that is in A349747 is not included in this sequence.

Cf. A000203, A003961, A007539, A007814, A007949, A046060, A349747.

Cf. also A349169, A349745.

f[p_, e_] := NextPrime[p, 2]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := n * GCD[(sigma = DivisorSigma[1, n]), (u = s[n])] == sigma * GCD[n, u]; Select[Range[10^6], q] (* Amiram Eldar, Dec 01 2021 *)

A003961twice(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f); };
isA349746(n) = { my(s=sigma(n),u=A003961twice(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)) and A007949(A000203(a(n))) = A007949(a(n)). [See comment]

a(15)-a(25) from Martin Ehrenstein, Dec 17 2021

A351550 Numbers of the form 4u+2 in A351551.

Original entry on oeis.org

2, 10, 34, 106, 1666, 8746, 26242, 134946, 2125762, 3997714, 8298346, 156057642, 323814834, 672166026, 3901441050

Offset: 1

Antti Karttunen, Feb 17 2022

This is the intersection of A016825 and A351551. Possibly after the initial 2 also the intersection of A191217 and A351551, or more precisely, of 2*A191218 and A351551. However, so far no common terms with A351538 encountered.

Cf. A016825, A191217, A191218, A351538, A351551.

Cf. also A349745.

cp's OEIS Frontend

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.

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A353365 Numbers k such that the odd part of sigma(sigma(k)) is equal to the odd part of sigma(k).

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A349746 Numbers k for which k * gcd(sigma(k), u) is equal to sigma(k) * gcd(k, u), where u is obtained by shifting the prime factorization of k two steps toward larger primes [with u = A003961(A003961(k))].

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A351550 Numbers of the form 4u+2 in A351551.

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