A332218 -id:A332218 - cp's OEIS frontend

A332221 a(n) = A156552(sigma(n)).

0, 2, 3, 8, 5, 11, 7, 10, 32, 13, 11, 35, 17, 23, 23, 1024, 13, 66, 19, 37, 31, 27, 23, 43, 1024, 37, 39, 71, 21, 55, 31, 38, 47, 29, 47, 72, 257, 43, 71, 45, 37, 95, 67, 75, 133, 55, 47, 4099, 258, 2050, 55, 49, 29, 87, 55, 87, 79, 45, 43, 151, 2049, 95, 263, 1073741824, 75, 111, 259, 77, 95, 111, 55, 138, 4097, 517, 4099, 83

Offset: 1

Antti Karttunen, Feb 12 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to sigma(n)

Cf. A000203, A156552, A058063, A332216 (fixed points), A332218, A332222.

Array[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ DivisorSigma[1, #]]] &, 76] (* Michael De Vlieger, Feb 12 2020 *)

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332221(n) = A156552(sigma(n));

a(n) = A156552(A000203(n)).

A000120(a(n)) = A058063(n).

A332216 Fixed points of A332221: Numbers k such that A156552(sigma(k)) is equal to k.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 31, 47, 55, 79, 87, 127, 191, 383, 1279, 5119, 6143, 8191, 20479, 81919, 131071, 524287, 786431, 1310719, 2147483647

Offset: 1

Antti Karttunen, Feb 10 2020

Equally, numbers k such that sigma(k) is equal to A005940(1+k).
The primes in this sequence are obtained by subtracting 1 from those terms of A029747 that are one more than a prime.
Questions: Are there other composite terms than 55 and 87? Are there other even terms than 2? (All such even terms should also occur in A332218).

Cf. A000203, A005940, A029747, A156552, A332217, A332218, A332221.

Subsequences: A000668, A007505, A050522.

Select[Range[10^5], DivisorSigma[1, #] == Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2], q}, Times @@ Flatten[Table[q = Take[p, -i]; Prime[Count[Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}]]] &] (* Michael De Vlieger, Feb 12 2020, after Robert G. Wilson v at A005940 *)

A332217 Numbers k for which the 2-adic valuation of sigma(k) is zero and its 3-adic valuation is 1 (so that sigma(k) is an odd multiple of 3, but not of 9).

Original entry on oeis.org

2, 8, 18, 49, 50, 72, 128, 162, 169, 196, 200, 242, 361, 441, 450, 512, 578, 648, 676, 784, 961, 968, 1058, 1152, 1225, 1250, 1369, 1444, 1458, 1521, 1682, 1764, 1800, 1849, 2178, 2312, 2704, 3136, 3200, 3249, 3362, 3721, 3844, 3969, 4050, 4225, 4232, 4418, 4489, 4608, 4802, 4900, 5000

Offset: 1

Antti Karttunen, Feb 11 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Subsequence of A067051, which is a subsequence of A028982.
Cf. A332218 (a subsequence).
Cf. A000203, A007949, A332216.

Select[Range[5*10^3], IntegerExponent[DivisorSigma[1, #], {2, 3}] === {0, 1} &] (* Michael De Vlieger, Feb 12 2020 *)

isA332217(n) = ((sigma(n)%2)&&(valuation(sigma(n),3)==1));

{n: A000035(A000203(n))*A007949(A000203(n))==1}.

cp's OEIS Frontend

A332221 a(n) = A156552(sigma(n)).

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A332216 Fixed points of A332221: Numbers k such that A156552(sigma(k)) is equal to k.

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A332217 Numbers k for which the 2-adic valuation of sigma(k) is zero and its 3-adic valuation is 1 (so that sigma(k) is an odd multiple of 3, but not of 9).

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