A332216 -id:A332216 - 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).

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}.

A332218 Numbers k such that A332221(k) = A156552(sigma(k)) is 2*{an odd square}.

Original entry on oeis.org

2, 162, 441, 2704, 4225, 275194921

Offset: 1

Antti Karttunen, Feb 11 2020

Any even term of A332216 must occur also in this sequence.

			  a(n)              -> sigma(a(n))              -> A156552(sigma(a(n)))
     2 = 2^1 * 1^2  ->    3 = 3^1               ->      2 = 2^1 * 1^1,
   162 = 2^1 * 3^4  ->  363 = 3^1 * 11^2        ->     98 = 2^1 * 7^2,
   441 = 3^2 * 7^2  ->  741 = 3^1 * 13^1 * 19^1 ->    578 = 2^1 * 17^2,
  2704 = 2^4 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,
  4225 = 5^2 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,
and
275194921 = 53^2 * 313^2 -> 281384229 = 3^1 * 7^1 * 181^2 * 409^1 -> 9671406556943421676716050 = 2^1 * 5^2 * 7^2 * 62829235873^2.

Index entries for sequences related to sigma(n)

Cf. A000203, A156552, A332216, A332217, A332221.

Subsequence of A332217 ⊂ A067051 ⊂ A028982.

Select[Range@ 5000, And[IntegerQ[#], OddQ[#]] &@ Sqrt[#/2] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ DivisorSigma[1, #]]] &] (* Michael De Vlieger, Feb 12 2020 *)

\\ Needs also code from A156552:
istosq(n) = ((1==valuation(n,2))&&issquare(n/2));
for(n=1,2^25,if(istosq(A156552(sigma(n*n))),print1(n*n,", ")); if(istosq(A156552(sigma(2*n*n))),print1(2*n*n,", ")));

cp's OEIS Frontend

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

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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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A332218 Numbers k such that A332221(k) = A156552(sigma(k)) is 2*{an odd square}.

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