A064605
Numbers k such that A064602(k) is divisible by k.
Original entry on oeis.org
1, 2, 8, 74, 146, 150, 158, 307, 526, 541, 16157, 20289, 271343, 953614, 1002122, 2233204, 3015123, 15988923, 48033767, 85110518238
Offset: 1
Summing divisor-square sums for j = 1..8 gives 1+5+10+21+26+50+50+85 = 248, which is divisible by 8, so 8 is a term and the integer quotient is 31.
Cf.
A001157,
A064602,
A050226,
A056650,
A064606,
A064607,
A064610,
A064611,
A064612,
A048290,
A062982,
A045345.
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k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[2, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Apr 25 2011 *)
A064610
Places k where A064608(k) (partial sums of unitary tau) is divisible by k.
Original entry on oeis.org
1, 35, 37, 1015, 27417, 27421, 27449, 27453, 19774739, 530743781, 530743799, 530743807, 530743813
Offset: 1
For n = 37, the sum A064608(37) = 1+2+2+2+2+4+2+...+4+4+4+2 = 111 = 3*37, so 37 is in the sequence.
Analogous "integer-mean" sequences for various arithmetical functions are
A050226,
A056650,
A064605,
A064606,
A064607,
A048290,
A063986,
A063971,
A064911,
A062982,
A045345.
-
s[1] = 1; s[n_] := s[n] = s[n - 1] + 2^PrimeNu[n]; Select[Range[30000], Divisible[s[#], #] &] (* Amiram Eldar, Jun 04 2021 *)
A064611
Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).
Original entry on oeis.org
1, 2, 8, 11, 12, 174, 212, 524, 1567, 14096, 19795, 38466, 42114, 55575, 338809, 498001, 1175281, 2424880, 3994532, 7908519, 48453784, 696840720, 5497869355, 7479239685
Offset: 1
udivisor sums[=usigma(j) values] from 1 to 8 are added: 1+3+4+5+6+12+8+9=48; it is divisible by 8, thus 8 is here.
Cf.
A034448,
A064609,
A050226,
A056550,
A064605,
A064606,
A064607,
A064610,
A064612,
A048290,
A062982,
A045345.
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s = Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 10^6}]; Module[{a = First@ s, b = {First@ s}}, Do[a += s[[i]]; If[Divisible[a, i], AppendTo[b, i]], {i, 2, Length@ s}]; b] (* Michael De Vlieger, Mar 18 2017 *)
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.
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s = 0; k = 1; Do[ While[s = s + DivisorSigma[0, k]; s < k*n, k++ ]; Print[k]; k++, {n, 1, 20}]
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A085829(n) = {local(s,k);s=1;k=1;while(sMichael B. Porter, Oct 23 2009
A064607
Numbers k such that A064604(k) is divisible by k.
Original entry on oeis.org
1, 2, 7, 151, 257, 1823, 3048, 5588, 6875, 7201, 8973, 24099, 5249801, 9177919, 18926164, 70079434, 78647747, 705686794, 2530414370, 3557744074, 25364328389, 32487653727, 66843959963
Offset: 1
Adding 4th-power divisor-sums for j = 1..7 gives 1+17+82+273+626+1394+2402 = 4795 which is divisible by 7, so 7 is a term and the integer quotient is 655.
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k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[4, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G.Wilson v, Aug 25 2011 *)
A064612
Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).
Original entry on oeis.org
1, 4, 5, 2178, 416417176, 416417184, 416417185, 416417186, 416417194, 416417204, 416417206, 416417208, 416417213, 416417214, 416417231, 416417271, 416417318, 416417319, 416417326, 416417335, 416417336, 416417338, 416417339, 416417374
Offset: 1
Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.
A057494
a(n) = Sum_{k = 1..10^n} d(k) where d(n) = number of divisors of n (A000005).
Original entry on oeis.org
1, 27, 482, 7069, 93668, 1166750, 13970034, 162725364, 1857511568, 20877697634, 231802823220, 2548286736297, 27785452449086, 300880375389757, 3239062263181054, 34693207724724246, 369957928177109416, 3929837791070240368, 41600963003695964400, 439035480966899467508
Offset: 0
- Henri Lifchitz, Table of n, a(n) for n = 0..36
- Terence Tao, Ernest Croot III, and Harald Helfgott, Deterministic methods to find primes, Mathematics of Computation, 81 (2012), 1233-1246. arXiv:1009.3956, [math.NT], 2010-2012.
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k = s = 0; Do[ While[ k < 10^n, k++; s = s + DivisorSigma[ 0, k ] ]; Print[s], {n, 0, 8} ]
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a(n) = sum(k=1, 10^n, numdiv(k)); \\ Michel Marcus, Feb 19 2017
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from math import isqrt
def A057494(n): return -(s:=isqrt(m:=10**n))**2+(sum(m//k for k in range(1,s+1))<<1) # Chai Wah Wu, Oct 23 2023
A064606
Numbers k such that A064603(k) is divisible by k.
Original entry on oeis.org
1, 2, 7, 45, 184, 210, 267, 732, 1282, 3487, 98374, 137620, 159597, 645174, 3949726, 7867343, 13215333, 14153570, 14262845, 317186286, 337222295, 2788845412, 10937683400, 72836157215, 95250594634
Offset: 1
Adding divisor-cube sums for j = 1..7 gives 1+9+28+73+126+252+344 = 833 = 7*119, which is divisible by 7, so 7 is a term and the integer quotient is 119.
A063986
Numbers k that divide Sum_{j=1..k} A051953(j) where A051953(j) = j - Phi(j). Arithmetic mean of first k cototient values is an integer.
Original entry on oeis.org
1, 4, 5, 24, 25, 249, 600, 617, 12272, 13763, 21332, 25228, 783665, 15748051, 41846733, 195853251, 2488541984, 14399065016, 21119309213, 22430204140, 43787603128, 157825075944, 206651865067, 271605149320, 374049315076, 650288309748
Offset: 1
k=5: (1 + 1 + 2 + 2 + 4)/5 = 2.
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s = 0; Do[s = s + n - EulerPhi[n]; If[ IntegerQ[s/n], Print[n]], {n, 1, 10^7} ]
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
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k = s = 0; Do[ While[ k < 2^n, k++; s = s + DivisorSigma[ 0, k ]]; Print[s], {n, 0, 29} ]
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a(n) = sum(k=1, 2^n, numdiv(k)); \\ Michel Marcus, Oct 10 2021
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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
Showing 1-10 of 31 results.
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