A048290 -id:A048290 - cp's OEIS frontend

A379923 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^k * A000005(k).

1, 5, 18, 22, 25, 29, 197, 1350, 1360, 1362, 1368, 1381, 1391, 1395, 10200, 75486, 75490, 557768, 557843, 557853, 557898, 4121846, 4122064, 4122112, 4122222, 30457732, 30457773, 30457835, 30458040, 30458133, 30458138, 30458140, 30458335, 225056911, 225056919, 225056925, 225056989

Offset: 1

Amiram Eldar, Jan 06 2025

nonn

Numbers m such that m | A307704(m).
The corresponding quotients, A307704(m)/m, are -1, 0, 1, 1, 1, 1, 2, 3, 3, 3, ... (see the link for more values).
a(38) > 2*10^10, if it exists.

Amiram Eldar, Table of n, a(n), A307704(a(n))/a(n) for n = 1..37.

Cf. A000005, A307704.

Similar sequences: A050226, A048290, A056550, A064605, A067929, A067931, A379922, A379924.

With[{m = 10000}, Position[Accumulate[Table[(-1)^n * DivisorSigma[0, n], {n, 1, m}]]/Range[m], _?IntegerQ] // Flatten]

list(lim) = my(s = 0); for(k = 1, lim, s += (-1)^k * numdiv(k); 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, ", ")));

A144857 Numbers k that divide Sum_{i=1..k} phi(i)^2, where phi(i) = totient function A000010.

Original entry on oeis.org

1, 2, 3, 6, 26, 190, 610, 2078, 2670, 7038, 16466, 89973, 150374, 157298, 163367, 419090, 640627, 879702, 3479689, 5618437, 11304721, 74106171, 471591726, 475915439, 1198344149, 2270643086, 3051266010

Offset: 1

Ctibor O. Zizka, Sep 23 2008

Does a number k exist such that RootMeanSquare(phi(1), ..., phi(k)) is an integer?

Cf. A000010, A057434, A048290, A140480.

lst = {}; s = 0; Do[ s = s + EulerPhi[n]^2; If[ Mod[s, n] == 0, AppendTo[lst, n]], {n, 10^9}]; lst (* Robert G. Wilson v, Oct 02 2008 *)

s=0;for(n=1,1e6,s+=eulerphi(n)^2;if(s%n==0,print1(n", "))) \\ Charles R Greathouse IV, Mar 05 2013

{k: k | A057434(k)}. - R. J. Mathar, Sep 29 2008

a(8)-a(11) from R. J. Mathar, Sep 29 2008
a(12)-a(24) from Robert G. Wilson v, Oct 02 2008
a(25)-a(27) from Donovan Johnson, Aug 21 2011

A145190 Numbers k such that phi(1)phi(2)...*phi(k) / k is an integer, where phi(k) is the totient function (A000010).

Original entry on oeis.org

1, 4, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 65, 66, 69, 70, 72, 75, 77, 78, 80, 81, 84, 87, 88, 90, 91, 92, 96, 98, 99, 100, 104, 105, 108, 110, 112, 115, 116, 117, 119, 120, 121, 123

Offset: 1

Ctibor O. Zizka, Oct 03 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A048290, A144857.

Select[Range[200], IntegerQ[Product[EulerPhi[i], {i, 1, #}]/#] &] (* Carl Najafi, Aug 19 2011 *)
nn=150;With[{ep=Rest[FoldList[Times,1,EulerPhi[Range[nn]]]]}, Flatten[ Position[Table[ep[[i]]/i,{i,nn}],?IntegerQ]]] (* _Harvey P. Dale, May 12 2012 *) (* this program is more than 200 times faster than the first Mathematica program above *)

isok(n)=prod(k=1,n,eulerphi(k))%n==0 \\ Anders Hellström, Aug 22 2015

More terms from Carl Najafi, Aug 19 2011

A355544 Numbers k such that the arithmetic mean of the first k squarefree numbers is an integer.

Original entry on oeis.org

1, 3, 6, 37, 75, 668, 1075, 37732, 742767, 1811865, 3140083, 8937770, 108268896, 282951249, 633932500, 1275584757, 60455590365

Offset: 1

Amiram Eldar, Jul 06 2022

Numbers k such that A173143(k) is divisible by k.

The corresponding quotients A173143(k)/k are 1, 2, 4, 29, ..., and the corresponding values of A005117(k) are 1, 3, 7, 59, ... (see the link for more values).

			3 is a term since the arithmetic mean of the first 3 squarefree numbers, (1+2+3)/3 = 2, is an integer.

Amiram Eldar, Table of n, a(n), A173143(a(n))/a(n), A005117(a(n)) for n = 1..16.

Cf. A005117, A173143.

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

s = Select[Range[10^6], SquareFreeQ]; r = Accumulate[s]/Range[Length[s]]; ind = Position[r, _?IntegerQ] // Flatten

upto(n) = my(s=0,k=0); forsquarefree(m=1, n, s+=m[1]; k+=1; if(s%k == 0, print1(k, ", "))); \\ Daniel Suteu, Jul 06 2022

a(17) from Daniel Suteu, Jul 06 2022

A356747 Numbers m that divide A306070(m) = Sum_{k=1..m} bphi(k), where bphi is the bi-unitary totient function (A116550).

Original entry on oeis.org

1, 2, 141, 1035, 2388, 3973, 5157, 14160, 37023, 68861, 99889, 116106, 117939, 627400, 1561944, 1626983, 5901444, 10054091, 12260525, 32619981, 49775099

Offset: 1

Amiram Eldar, Aug 25 2022

The corresponding quotients A306070(m)/m are 1, 1, 57, 418, ... (see the link for more values).

a(22) > 6.5*10^8, if it exists.

Amiram Eldar, Table of n, a(n), A306070(a(n))/a(n) for n = 1..21.

Cf. A116550, A306070.

Similar sequences: A048290, A306950.

phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; seq = {}; s = 0; Do[s = s + bphi[n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

A227975 Numbers m such that m divides Sum_{k=1..m} lambda(k).

Original entry on oeis.org

1, 2, 5, 6, 10, 18, 30, 82, 4866, 8784, 10170, 23364, 76296, 247166, 585570, 735480, 848754, 1559520, 2884840, 11272940, 35642420, 56652788, 174935486, 196398413, 679063441, 1398826844, 1542228164, 1665703953, 2699813692, 5734751503

Offset: 1

Michel Lagneau, Jun 17 2016

nonn

lambda(n) is the Carmichael lambda function (A002322). The corresponding ratios (Sum_{k=1..m} lambda(k))/m are given by the sequence {1, 1, 2, 2, 3, 5, 8, 19, 711, 1221, 1399, 3011, 9034, 27187, 61246, 75971, 86971, 154710, 277344, 1015576,...}.

a(31) > 10^10. - Dana Jacobsen, Jul 07 2016

			5 is in the sequence because 5 divides Sum_{k=1..5} lambda(k) = 1 + 1 + 2 + 2 + 4 = 2*5.

Cf. A002322, A048290, A162578.

s = 0; Do[s = s + CarmichaelLambda[n]; If[IntegerQ[s/n], Print[n]], {n, 1, 10^9}]

use ntheory ":all"; my $v=0; for my $m (1..1e6) { $v=vecsum($v,carmichael_lambda($m)); say $m unless $v % $m; } # Dana Jacobsen, Jul 07 2016

More terms from Dana Jacobsen, Jul 07 2016

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

Original entry on oeis.org

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

A339009 Numbers k such that the average number of odd divisors of {1..k} is an integer.

Original entry on oeis.org

1, 2, 165, 170, 1274, 9437, 69720, 69732, 69734, 69736, 515230, 515236, 515246, 28132043, 28132063, 28132079

Offset: 1

Ilya Gutkovskiy, Nov 18 2020

Numbers k that divide A060831(k) where A060831(k) = Sum_{j=1..k} A001227(j).

The sequence also includes: 83860580242, 4578632504347, 4578632504465, 4578632504515. - Daniel Suteu, Nov 24 2020

			165 is in the sequence because the average number of odd divisors of {1..165} is an integer: A060831(165) / 165 = 495 / 165 = 3.

Eric Weisstein's World of Mathematics, Odd Divisor Function.

Cf. A001227, A048290, A050226, A060831, A063971, A064610, A064612.

s[n_] := Module[{c = 0, k = 1, sum = 0, seq = {}}, While[c < n, sum += DivisorSigma[0, k/2^IntegerExponent[k, 2]]; If[Divisible[sum, k], c++; AppendTo[seq, k]]; k++]; seq]; s[13] (* Amiram Eldar, Nov 18 2020 *)

f(n) = my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2; \\ A060831
isok(k) = (f(k) % k) == 0; \\ Michel Marcus, Nov 25 2020

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

cp's OEIS Frontend

A379923 Numbers m that divide the alternating sum Sum_{k=1..m} (-1)^k * A000005(k).

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

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A144857 Numbers k that divide Sum_{i=1..k} phi(i)^2, where phi(i) = totient function A000010.

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A145190 Numbers k such that phi(1)*phi(2)*...*phi(k) / k is an integer, where phi(k) is the totient function (A000010).

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A355544 Numbers k such that the arithmetic mean of the first k squarefree numbers is an integer.

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A356747 Numbers m that divide A306070(m) = Sum_{k=1..m} bphi(k), where bphi is the bi-unitary totient function (A116550).

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A227975 Numbers m such that m divides Sum_{k=1..m} lambda(k).

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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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A145190 Numbers k such that phi(1)phi(2)...*phi(k) / k is an integer, where phi(k) is the totient function (A000010).