A064611 -id:A064611 - cp's OEIS frontend

A309272 Numbers m such that m divides A173290(m) = Sum_{k=1..m} psi(k), where psi is the Dedekind psi function (A001615).

1, 2, 5, 15, 31, 40, 66, 81, 315, 966, 1398, 1768, 30166, 32335, 98734, 388033, 591597, 1375056, 14966304, 15160528, 50793208, 51302236, 99253376, 110994356, 230465053, 402340268, 497982399, 2027319577, 2879855394, 18450762682, 29922126368, 31711273834, 40583934786

Offset: 1

Amiram Eldar, Oct 23 2019

nonn

The corresponding quotients are 1, 2, 4, 12, 24, 31, 51, 62, 240, 735, 1063, 1344, 22924, 24572, 75029, 294870, 449560, 1044918, 11373028, 11520620, 38598210, 38985025, 75423522, 84345597, 175132440, 305741942, 378421246, 1540578144, 2188427680, 14020898356, 22738089456, 24097678498, 30840092321, ...

			2 is in the sequence since psi(1) + psi(2) = 1 + 3 = 4 is divisible by 2.
5 is in the sequence since psi(1) + psi(2) + ... + psi(5) = 1 + 3 + 4 + 6 + 6 = 20 is divisible by 5.

Cf. A001615, A173290.

Cf. A045345, A048290, A050226, A056550, A063971, A064605, A064606, A064607, A064610, A064611, A064612, A306950, A307043, A307161, A326488.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); seq = {}; s = 0; Do[s += psi[n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^4}]; seq

a(31)-a(33) from Giovanni Resta, Oct 24 2019

A355541 Numbers k such that A061201(k) is divisible by k.

Original entry on oeis.org

1, 2, 7, 31, 1393, 5012, 7649, 50235, 147296, 426606, 611769, 3491681, 9324642, 11815109, 53962364, 82680301, 96789197, 230882246, 378444764, 1489280093, 1489280606, 3651325650, 5891877914, 5891877947, 5891877966, 58604540872

Offset: 1

Amiram Eldar, Jul 06 2022

Numbers k such that the mean value of A007425 over the range 1..k is an integer.
The corresponding quotients are 1, 2, 4, 9, 32, 43, 47, 67, 80, 94, 99, 125, 141, 145, 172, 180, 183, 200, 210, 239, 239, 259, 270, 270, 270, 326, ... .
a(27) > 7.5*10^10, if it exists.

			7 is a term since A061201(7) = 28 = 4 * 7 is divisible by 7.

Cf. A007425, A061201.

Similar sequences: A045345, A050226, A056550, A064605, A064606, A064607, A064610, A064611, A064612, A048290.

f[p_, e_] := (e+1)*(e+2)/2;  d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; sum = 0; seq = {}; Do[sum += d3[n]; If[Divisible[sum, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

A355542 Numbers k such that A272718(k) is divisible by k.

Original entry on oeis.org

1, 2, 3, 11, 13, 50, 81, 96, 395, 640, 59136, 65719, 632621, 1342813, 2137073, 2755370, 3446370, 10860093, 321939569, 1872591111, 8858043355

Offset: 1

Amiram Eldar, Jul 06 2022

Numbers k such that the mean value of A018804 over the range 1..k is an integer.
The corresponding quotients are 1, 2, 3, 13, 16, 80, 141, 172, 865, 1500, 219530, 246058, 2804048, 6259092, 10263121, 13445321, 17051542, 57521176, 2036840289, 12849666590, 64967828053, ... .
a(22) > 6.5*10^10, if it exists.

			11 is a term since A061201(11) = 143 = 11 * 13 is divisible by 11.

Cf. A018804, A272718.

Similar sequences: A045345, A050226, A056550, A064605, A064606, A064607, A064610, A064611, A064612, A048290.

f[p_,e_] := (e*(p-1)/p+1)*p^e; pillai[1] = 1; pillai[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; sum = 0; Do[sum += pillai[n]; If[Divisible[sum, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

cp's OEIS Frontend

A309272 Numbers m such that m divides A173290(m) = Sum_{k=1..m} psi(k), where psi is the Dedekind psi function (A001615).

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A355541 Numbers k such that A061201(k) is divisible by k.

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A355542 Numbers k such that A272718(k) is divisible by k.

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