A056550 -id:A056550 - cp's OEIS frontend

A326488 Numbers m such that A327566(m) = Sum_{k=1..m} isigma(k) is divisible by m, where isigma(k) is the sum of infinitary divisors of k (A049417).

1, 2, 160, 285, 2340, 2614, 8903, 81231, 171710, 182712, 434887, 2651907, 56517068, 143714354, 922484770, 5162883263, 39421525873

Offset: 1

Amiram Eldar, Sep 20 2019

The infinitary version of A056550.

The corresponding quotients, A327566(a(n))/a(n), are 1, 2, 118, 209, 1711, 1910, 6506, 59357, 125473, 133513, 317781, 1937798, 41298052, 105014703, 674076450, 3772612983, 28806028088, ...

			2 is in the sequence since isigma(1) + isigma(2) = 1 + 3 = 4 is divisible by 2.

Cf. A049417 (isigma), A327566 (sums of isigma).

Cf. A056550 (corresponding with sigma), A064611 (unitary), A307043 (exponential), A307161 (bi-unitary).

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); seq = {}; s = 0; Do[s = s + isigma [n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

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

A229883 Numbers k such that Sum_{j=1..k} sigma_(j) == 0 (mod k), where sigma_(j) is the sum of the anti-divisors of j (A066417).

Original entry on oeis.org

1, 2, 5, 8, 11, 30, 34, 172, 311, 498, 562, 602, 630, 1742, 4608, 4842, 13664, 16386, 24659, 29150, 56357, 58185, 86267, 88114, 242156, 245325, 839756, 947942, 2524087, 2963552, 4218803, 18281326, 28292036, 30023108, 46376824, 52058844, 85990503, 139548984

Offset: 1

Paolo P. Lava, Oct 02 2013

nonn

Tested up to k = 10^6.

			The sum of the anti-divisors of the numbers from 1 to 8 is 0 + 0 + 2 + 3 + 5 + 4 + 10 + 8 = 32 and 32/8 = 4.

Cf. A056550, A066417.

with(numtheory); P:=proc(q) local a,b,j,k,n; b:=0;
for n from 1 to q do a:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
b:=b+a; if b mod n=0 then print(n); fi; od; end: P(10^6);

a(29)-a(38) from Donovan Johnson, Oct 12 2013

A303900 Numbers k such that the average of all strong divisors of all positive integers <= k is an integer.

Original entry on oeis.org

2, 8, 12, 16, 67, 924122, 1067239

Offset: 1

Ilya Gutkovskiy, May 02 2018

We say d is a strong divisor of k iff d is a divisor of k and d > 1.
Numbers k such that A002541(k) | A024917(k).
a(8) > 10^12. - Giovanni Resta, May 05 2018

Index entries for sequences related to divisors of numbers

Cf. A024917, A027750, A032741, A039653, A056550, A226647, A284288, A284755, A303480, A303482, A303659.

a(3)-a(7) corrected by Daniel Suteu, May 03 2018

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

A332270 Numbers k such that Sum_{j=1..k} j*floor(k/j) is divisible by k+1.

Original entry on oeis.org

3, 4, 14, 29, 82, 67117, 86249, 140064, 185699, 392081, 2915083, 6315155, 9723681, 17754993, 820165642, 9388829301, 143904506919, 192738887697

Offset: 1

Seiichi Manyama, May 04 2020

a(19) > 5*10^11. - Giovanni Resta, May 05 2020

Cf. A000203, A024916, A056550.

q[n_] := Divisible[Sum[j * Floor[n/j], {j, 1, n}], n + 1]; Select[Range[100], q] (* Amiram Eldar, May 03 2021 *)

for(k=1, 1e4, if(sum(j=1, k, k\j*j)%(k+1)==0, print1(k", ")))

s=0; for(k=1, 1e7, s+=sigma(k); if(s%(k+1)==0, print1(k", ")))

a(16)-a(18) from Giovanni Resta, May 05 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

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

A326488 Numbers m such that A327566(m) = Sum_{k=1..m} isigma(k) is divisible by m, where isigma(k) is the sum of infinitary divisors of k (A049417).

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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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A229883 Numbers k such that Sum_{j=1..k} sigma_*(j) == 0 (mod k), where sigma_*(j) is the sum of the anti-divisors of j (A066417).

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A303900 Numbers k such that the average of all strong divisors of all positive integers <= k is an integer.

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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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A332270 Numbers k such that Sum_{j=1..k} j*floor(k/j) is divisible by k+1.

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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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A229883 Numbers k such that Sum_{j=1..k} sigma_(j) == 0 (mod k), where sigma_(j) is the sum of the anti-divisors of j (A066417).