A379923 -id:A379923 - cp's OEIS frontend

A379922 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * sigma_2(k).

1, 2, 3, 42, 329, 633, 1039, 5689, 26621, 39245, 1101875, 1216075, 40088584, 67244920, 104332211, 549673265, 777631064, 19879301756

Offset: 1

Amiram Eldar, Jan 06 2025

Numbers m such that m | A379921(m).
The corresponding quotients, A379921(m)/m, are -1, 2, -2, 120, 5228, ... (see the link for more values).
a(19) > 5*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A379921(a(n))/a(n) for n = 1..18.

Cf. A001157 (sigma_2), A379921.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379923, A379924.

With[{m = 40000}, Position[Accumulate[Table[(-1)^n * DivisorSigma[2, n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * sigma(k, 2); if(!(s % k), print1(k, ", ")));

A379924 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * usigma(k).

Original entry on oeis.org

1, 2, 9, 54, 101, 178, 189, 2071, 3070, 9171, 11450, 12794, 21405, 27553, 35285, 251974, 2069863, 2395894, 155931488, 387586437, 758519827, 1202435693, 9859113494, 42703260442

Offset: 1

Amiram Eldar, Jan 06 2025

Numbers m such that m | A370898(m).
The corresponding quotients, A370898(m)/m, are -1, 1, 0, 6, 9, ... (see the link for more values).
a(25) > 5*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A370898(a(n))/a(n) for n = 1..24.

Cf. A034448 (usigma), A370898.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379922, A379923.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; With[{m = 260000}, Position[Accumulate[Table[(-1)^n * usigma[n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]); }
list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * usigma(k); if(!(s % k), print1(k, ", ")));

cp's OEIS Frontend

A379922 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^(k+1) * sigma_2(k).

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