A050226
Numbers m such that m divides Sum_{k = 1..m} A000005(k).
Original entry on oeis.org
1, 4, 5, 15, 42, 44, 47, 121, 336, 340, 347, 930, 2548, 6937, 6947, 51322, 379097, 379131, 379133, 2801205, 20698345, 56264090, 56264197, 152941920, 152942012, 8350344420, 61701166395, 455913379395, 455913379831, 1239301050694, 3368769533660, 3368769533812
Offset: 1
For k = 15 the sum is 1 + 2 + 2 + 3 + 2 + 4 + 2 + 4 + 3 + 4 + 2 + 6 + 2 + 4 + 4 = 45 which is divisible by 15.
- Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.
-
s = 0; Do[ s = s + DivisorSigma[ 0, n ]; If[ Mod[ s, n ] == 0, Print[ n ] ], {n, 1, 2*10^9} ]
k=10^6; a[1]=1;a[n_]:=a[n]=DivisorSigma[0,n]+a[n-1]; nd=a/@Range@k; Select[Range@k,Divisible[nd[[#]],#]&] (* Ivan N. Ianakiev, Apr 30 2016 *)
Module[{nn=400000},Select[Thread[{Range[nn],Accumulate[DivisorSigma[0,Range[nn]]]}],Divisible[#[[2]],#[[1]]]&]][[All,1]] (* The program generates the first 19 terms of the sequence. To generate more, increase the nn constant. *) (* Harvey P. Dale, Jul 03 2022 *)
-
lista(nn) = {my(s = 0); for (n=1, nn, s += numdiv(n); if (!(s % n), print1(n, ", ")););} \\ Michel Marcus, Dec 14 2015
-
def A050226_list(len):
a, L = 0, []
for n in (1..len):
a += sigma(n,0)
if n.divides(a): L.append(n)
return L
A050226_list(10000) # Peter Luschny, Dec 18 2015
A085829
a(n) = least k such that the average number of divisors of {1..k} is >= n.
Original entry on oeis.org
1, 4, 15, 42, 120, 336, 930, 2548, 6930, 18870, 51300, 139440, 379080, 1030484, 2801202, 7614530, 20698132, 56264040, 152941824, 415739030, 1130096128, 3071920000, 8350344420, 22698590508, 61701166395, 167721158286, 455913379324, 1239301050624, 3368769533514
Offset: 1
a(20) = 415739030 because the average number of divisors of {1..415739030} is >= 20.
- Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.
-
s = 0; k = 1; Do[ While[s = s + DivisorSigma[0, k]; s < k*n, k++ ]; Print[k]; k++, {n, 1, 20}]
-
A085829(n) = {local(s,k);s=1;k=1;while(sMichael B. Porter, Oct 23 2009
A085831
a(n) = Sum_{k=1..2^n} d(k) where d(n) = number of divisors of n (A000005).
Original entry on oeis.org
1, 3, 8, 20, 50, 119, 280, 645, 1466, 3280, 7262, 15937, 34720, 75108, 161552, 345785, 736974, 1564762, 3311206, 6985780, 14698342, 30850276, 64607782, 135030018, 281689074, 586636098, 1219788256, 2532608855, 5251282902, 10874696106, 22493653324, 46475828418
Offset: 0
-
k = s = 0; Do[ While[ k < 2^n, k++; s = s + DivisorSigma[ 0, k ]]; Print[s], {n, 0, 29} ]
-
a(n) = sum(k=1, 2^n, numdiv(k)); \\ Michel Marcus, Oct 10 2021
-
from math import isqrt
def A085831(n): return (lambda m, r: 2*sum(r//k for k in range(1, m+1))-m*m)(isqrt(2**n),2**n) # Chai Wah Wu, Oct 08 2021
A085567
Least m such that the average number of divisors of all integers from 1 to m equals n, or 0 if no such number exists.
Original entry on oeis.org
1, 4, 15, 42, 121, 336, 930, 2548, 6937, 0, 51322, 0, 379097, 0, 2801205, 0, 20698345, 56264090, 152941920, 0, 0, 0, 8350344420, 0, 61701166395, 0, 455913379395, 1239301050694, 3368769533660, 0, 24892027072619, 0, 183928584450999, 0, 0, 0
Offset: 1
a(2) = 4 because (1/4)*(1+2+2+3) = 2.
- Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.
Missing terms a(16)-a(17) and a(20)-a(29) added by
Donovan Johnson, Dec 21 2008
A095256
Number of numbers not divisible by 10 that stay multiples of themselves when freed of their last n digits.
Original entry on oeis.org
23, 473, 7053, 93643, 1166714, 13969985, 162725300, 1857511487, 20877697534, 231802823099, 2548286736153, 27785452448917, 300880375389561, 3239062263180829, 34693207724723990, 369957928177109127, 3929837791070240044, 41600963003695964039, 439035480966899467108
Offset: 1
We have the following a(1)=23 two-digit numbers not ending in zero: 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, 99; each is divisible by its tens digit.
-
k = s = 0; Do[ While[ k < 10^n - 1, k++; s = s + DivisorSigma[ 0, k ]]; Print[s], {n, 9}] (* Robert G. Wilson v, Jun 05 2004 *)
-
from math import isqrt
def A095256(n): return -(s:=isqrt(m:=10**n))**2+(sum(m//k for k in range(1,s+1))<<1)-(n+1)**2 # Chai Wah Wu, Oct 23 2023
A136364
a(n) = Sum_{ composite k, 1 <= k <= 10^n} (Number of divisors d of k with 1 < d < k).
Original entry on oeis.org
8, 283, 5070, 73669, 966751, 11970035, 142725365, 1657511569, 18877697635, 211802823221, 2348286736298, 25785452449087, 280880375389758, 3039062263181055, 32693207724724247, 349957928177109417, 3729837791070240369, 39600963003695964401, 419035480966899467509
Offset: 1
a(1) = 8 because there are 8 distinct divisors of composite numbers k through 10^1; these are for 4(1); 6(2); 8(2); 9(1); 10(2); and 1 + 2 + 2 + 1 + 2 = 8.
A174425
Total number of divisors of all n-digit numbers.
Original entry on oeis.org
23, 450, 6580, 86590, 1073071, 12803271, 148755315, 1694786187, 19020186047, 210925125565, 2316483913054, 25237165712764, 273094922940644, 2938181887791268, 31454145461543161, 335264720452385137, 3559879862893130917, 37671125212625723995, 397434517963203503069
Offset: 1
For n = 1; a(1) = 23 because tau (r) of 1-digit numbers r = 1 to 9: {1, 2, 2, 3, 2, 4, 2, 4, 3}. Sum is 23.
-
\\ too slow for n > 20; here b(n) is A006218(n).
b(n)={sum(k=1, sqrtint(n), n\k)*2 - sqrtint(n)^2}
a(n)={b(10^n-1)-b(10^(n-1)-1)} \\ Andrew Howroyd, Jan 13 2020
A189020
a(n) = Sum_{k=1..10^n} tau_4(k), where tau_4 is the number of ordered factorizations into 4 factors (A007426).
Original entry on oeis.org
1, 89, 3575, 93237, 1951526, 35270969, 578262093, 8840109380, 128217432396, 1784942188189, 24045237260214, 315312623543840, 4042957241191810, 50862246063060180, 629513636928477232, 7681900592647818929, 92587253467765253144, 1103781870246459696784, 13031388731053572679450, 152516435040764735691556, 1771079109308495896176156
Offset: 0
Showing 1-8 of 8 results.
Comments