A351558 -id:A351558 - cp's OEIS frontend

A351549 Numbers k for which k * gcd(sigma(k), A019565(k)) is equal to sigma(k) * gcd(k, A019565(k)).

1, 1456, 15480, 114660, 2244600, 3894768, 25108200, 27052704, 65021040, 112402080, 1973921400

Offset: 1

Antti Karttunen, Feb 19 2022

Numbers k such that their abundancy index [sigma(k)/k] is equal to A351557(k)/A351556(k).
Question: If the above ratio is neither 1 nor 2, must it then be > 2? Are all even terms abundant?
a(12) > 2281701376 if it exists.

Cf. A000203, A005101, A007947, A019565, A080398, A351556, A351557, A351558, A351559.
Subsequence of A386424.
Cf. also A351458, A351548.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
isA351549(n) = { my(s=sigma(n), z=A019565(n)); (n*gcd(s,z))==(s*gcd(n,z)); };

A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

Original entry on oeis.org

0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 3, 8, 9, 2, 3, 0, 1, 2, 1, 0, 1, 2, 3, 0, 0, 10, 1, 8, 5, 2, 1, 0, 1, 2, 3, 32, 1, 6, 1, 0, 9, 2, 1, 8, 33, 2, 3, 0, 0, 2, 3, 0, 1, 6, 3, 0, 1, 2, 3, 8, 1, 2, 33, 0, 1, 2, 65, 0, 1, 2, 3, 0, 1, 2, 1, 12, 1, 10, 5, 0, 16, 2, 3, 0, 1, 18, 7, 0, 1, 2, 9, 8, 1, 2, 7, 0, 1, 2, 35, 0, 65, 2, 33

Offset: 1

Antti Karttunen, Feb 19 2022

Cf. A000203, A004198, A007947, A019565, A080398, A351557, A351560.

Cf. also A324644, A351558.

Table[If[# == 1, 0, Total[#2*2^PrimePi[#1] & @@@ FactorInteger[#]]/2] &@ GCD[DivisorSigma[1, n], Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 103}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A351559(n) = A048675(gcd(sigma(n), A019565(n)));

a(n) = A048675(A351557(n)) = A048675(gcd(sigma(n), A019565(n))).

a(n) = n AND A351560(n), where AND is bitwise-and, A004198.

A351556 a(n) = gcd(n, A019565(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 7, 15, 1, 1, 3, 1, 5, 1, 11, 1, 1, 1, 1, 3, 7, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 39, 1, 1, 21, 1, 1, 5, 1, 1, 1, 1, 1, 3, 13, 1, 3, 55, 7, 1, 1, 1, 5, 1, 1, 21, 1, 1, 3, 1, 17, 1, 5, 1, 1, 1, 1, 3, 1, 7, 3, 1, 1, 1, 1, 1, 1, 85, 1, 3, 11, 1, 3, 7, 1, 1, 1, 5, 1, 1, 1, 3, 5, 1, 51, 1, 13, 7

Offset: 0

Antti Karttunen, Feb 19 2022

Cf. A007947, A019565.

Cf. also A324198, A351549, A351557, A351558.

Table[GCD[n, Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[n, 2], 1]], {n, 0, 105}] (* Michael De Vlieger, Feb 20 2022 *)

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A351556(n) = gcd(n, A019565(n));

a(n) = gcd(n, A019565(n)) = gcd(A007947(n), A019565(n)).
a(n) = A007947(a(n)).
a(n) = A019565(A351558(n)).

cp's OEIS Frontend

A351549 Numbers k for which k * gcd(sigma(k), A019565(k)) is equal to sigma(k) * gcd(k, A019565(k)).

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A351559 a(n) = A048675(gcd(sigma(n), A019565(n))).

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A351556 a(n) = gcd(n, A019565(n)).

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