A070039 -id:A070039 - cp's OEIS frontend

A347142 Sum of 4th powers of divisors of n that are < sqrt(n).

0, 1, 1, 1, 1, 17, 1, 17, 1, 17, 1, 98, 1, 17, 82, 17, 1, 98, 1, 273, 82, 17, 1, 354, 1, 17, 82, 273, 1, 723, 1, 273, 82, 17, 626, 354, 1, 17, 82, 898, 1, 1394, 1, 273, 707, 17, 1, 1650, 1, 642, 82, 273, 1, 1394, 626, 2674, 82, 17, 1, 2275, 1, 17, 2483, 273, 626

Offset: 1

Ilya Gutkovskiy, Aug 19 2021

nonn

Cf. A001159, A056924, A070039, A279363, A339353, A339354, A347143.

Table[DivisorSum[n, #^4 &, # < Sqrt[n] &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[k^4 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A347142(n) = { my(s=0); fordiv(n,d,if((d^2)>=n,return(s)); s += (d^4)); }; \\ Antti Karttunen, Aug 19 2021

G.f.: Sum_{k>=1} k^4 * x^(k*(k + 1)) / (1 - x^k).

A348954 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d) * d.

Original entry on oeis.org

0, 1, -1, 1, -1, -1, -1, 3, -1, -1, -1, 6, -1, -1, -4, 3, -1, 2, -1, -1, -4, -1, -1, 10, -1, -1, -4, -1, -1, 7, -1, 7, -4, -1, -6, 2, -1, -1, -4, 12, -1, -4, -1, -1, -9, -1, -1, 16, -1, 4, -4, -1, -1, -4, -6, 14, -4, -1, -1, 13, -1, -1, -11, 7, -6, -4, -1, -1, -4, 11, -1, 8, -1, -1, -9, -1, -8, -4, -1, 20

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000593, A070039, A109506, A333810, A348660, A348951, A348952, A348953, A348955, A348956.

Table[DivisorSum[n, (-1)^(n/#) # &, # < Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k (k + 1))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A348954(n) = sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^(k*(k + 1)) / (1 + x^k).

A033832 Sum of odd divisors of n < sqrt(n) = sum of even divisors of n < sqrt(n).

Original entry on oeis.org

1, 40, 100, 208, 928, 1044, 3904, 10692, 17444, 29524, 36652, 45980, 87604, 91044, 136808, 158652, 161564, 171028, 187068, 218652, 230044, 260608, 287868, 406812, 438124, 450492, 583110, 665684, 719550, 731850, 736648, 865444, 1045504

Offset: 1

Naohiro Nomoto

All terms except first one appear to be even. - Michel Marcus, Jul 15 2013

Amiram Eldar, Table of n, a(n) for n = 1..500
kaleidoscopic numbers [broken link]

Cf. A070039, A113184.

aQ[n_] := DivisorSum[n, # * (-1)^# &, # < Sqrt[n] & ] == 0; Select[Range[10^4], aQ] (* Amiram Eldar, Sep 23 2019 *)

isok(n) = {so = 0; se = 0; fordiv (n, d, if (d < sqrt(n), if (d % 2, so += d, se += d))); return (so == se);} \\ Michel Marcus, Jul 14 2013

Prepended a(1)=1, Michel Marcus, Jul 15 2013

A102883 Terms which share a divisor greater than 1 with the value formed by adding them to the sum of their divisors less than or equal to their square roots.

Original entry on oeis.org

6, 12, 18, 24, 28, 36, 40, 42, 45, 48, 50, 54, 56, 60, 66, 70, 72, 75, 78, 80, 90, 96, 98, 100, 102, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 144, 150, 154, 156, 160, 162, 165, 170, 174, 176, 180, 186, 190, 192, 196, 198, 200, 204, 208, 210, 216, 220

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Jan 15 2005

nonn

At first glance it appears that all multiples of 6 are included in this sequence, but a number of them are excluded (such as 5*6 and 14*6.) See A102884 for a list of multiples of 6 not included in this sequence.

			a(3) = 18 because the divisors of 18 less than or equal to its square root are 1, 2 and 3. 18+1+2+3 = 24 and GCD(18,24) = 6 which is greater than 1. 18 is the third term which satisfies this requirement.

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

Cf. A070039, A102884.

compute := proc (n) local m, a; m := n; for a to n^.5 do if `mod`(n,a) = 0 then m := m+a end if end do; m end proc L:=[];for i from 1 to 2500 do; if i>0 then; x:=compute(i); if gcd(x,i) > 1 then L:=[op(L),i]; fi; fi; od;L;

aQ[n_] := GCD[n, DivisorSum[n, # &, # < Sqrt[n] &]] > 1; Select[Range[220], aQ] (* Amiram Eldar, Aug 28 2019 *)

A088345 n is divisible by the sum of all divisors of n which are less than the square root of n (values of n where 1 is the only divisor less than sqrt(n) are excluded as trivial cases.).

Original entry on oeis.org

6, 12, 18, 28, 45, 48, 56, 72, 80, 96, 117, 196, 396, 475, 496, 702, 704, 775, 992, 1100, 1326, 1568, 1792, 2009, 2150, 2622, 2952, 3042, 3321, 3672, 4140, 5328, 5852, 6750, 6860, 7154, 7605, 7680, 8128, 9102, 10575, 11008, 12126, 12168, 12384, 12810

Offset: 1

Chuck Seggelin, Nov 07 2003

nonn

If values of n where only the divisor 1 is < sqrt(n) were not excluded, then this sequence would include the primes and the squares of primes.

			a(4)=28 because sqrt(28)=5.291502622 and the divisors of 28 which are less than 5.291502622 are 1, 2 and 4. These divisors sum to 7 which divides 28.

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

Cf. A070039.

j := {}; for i to 1000 do; d := divisors(i) minus {i}; if d<>{1} then v := 0; s := evalf(sqrt(i)); for f in d do; if f1 then if i mod v = 0 then print(i,v,i/v); j := j union {i} fi; fi; fi; od; j;

Mathematica

ds[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; aQ[n_] := (d = ds[n]) > 1 && Divisible[n, d]; Select[Range[12810], aQ] (* Amiram Eldar, Aug 28 2019 *)

A176314 Sum of remainders of mod(n, k), for 1 <= k <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 3, 0, 2, 2, 1, 1, 4, 2, 5, 2, 2, 3, 6, 0, 3, 5, 6, 4, 8, 2, 6, 4, 5, 7, 6, 1, 6, 9, 11, 5, 10, 4, 9, 8, 5, 8, 13, 3, 8, 7, 10, 10, 16, 11, 12, 5, 8, 12, 18, 4, 10, 14, 10, 10, 12, 8, 15, 16, 20, 13, 20, 4, 11, 16, 15, 16, 16, 12, 19, 7, 11, 17, 25, 11, 14, 20, 25

Offset: 1

Franklin T. Adams-Watters, Apr 15 2010

nonn

It appears, as one would expect, that a(n) is asymptotically 1/4 n.

24 is the last n for which a(n) = 0.

Cf. A004125, A070039.

Total/@Table[Mod[n,k],{n,90},{k,Sqrt[n]}] (* Harvey P. Dale, Nov 20 2013 *)

PARI
```
a(n) = sum(k=2,sqrtint(n),n%k)
```

a(n+1) = a(n) + floor(sqrt(n)) - A070039(n+1).

A363520 Product of the divisors of n that are < sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 8, 3, 2, 1, 24, 1, 2, 3, 8, 1, 30, 1, 8, 3, 2, 5, 24, 1, 2, 3, 40, 1, 36, 1, 8, 15, 2, 1, 144, 1, 10, 3, 8, 1, 36, 5, 56, 3, 2, 1, 720, 1, 2, 21, 8, 5, 36, 1, 8, 3, 70, 1, 1152, 1, 2, 15, 8, 7, 36, 1, 320, 3, 2, 1

Offset: 1

Wesley Ivan Hurt, Jun 07 2023

			The product of divisors of 16 that are < sqrt(16) = 4 is 1*2 = 2, so a(16) = 2.

Cf. A000196, A010052, A072499, A363521.

Cf. A070039 (sum of those divisors).

a[n_] := Times @@ Select[Divisors[n], #^2 < n &]; Array[a, 100]

a(n) = vecprod(select(x->(x^2Michel Marcus, Jun 08 2023

a(n) = Product_{d|n, d

a(n) = Product_{k=1..floor(sqrt(n-1))} k^c(n/k), where c(m) = 1-ceiling(m)+floor(m).

a(n) = A072499(n)/A000196(n)^A010052(n) for n>=1.

A372834 a(n) is the numerator of Sum_{d|n, d < sqrt(n)} 1/d.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 11, 1, 3, 4, 3, 1, 11, 1, 7, 4, 3, 1, 25, 1, 3, 4, 7, 1, 61, 1, 7, 4, 3, 6, 25, 1, 3, 4, 39, 1, 2, 1, 7, 23, 3, 1, 9, 1, 17, 4, 7, 1, 2, 6, 53, 4, 3, 1, 49, 1, 3, 31, 7, 6, 2, 1, 7, 4, 129, 1, 19, 1, 3, 23, 7, 8, 2, 1, 83

Offset: 1

Ilya Gutkovskiy, May 14 2024

			0, 1, 1, 1, 1, 3/2, 1, 3/2, 1, 3/2, 1, 11/6, 1, 3/2, 4/3, 3/2, 1, 11/6, ...

Cf. A017665, A070039, A372835 (denominators).

nmax = 80; CoefficientList[Series[Sum[x^(k (k + 1))/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = numerator(sumdiv(n, d, if (d^2 < n, 1/d))); \\ Michel Marcus, May 14 2024

Numerators of coefficients in expansion of Sum_{k>=1} x^(k*(k+1)) / (k * (1 - x^k)).

A372835 a(n) is the denominator of Sum_{d|n, d < sqrt(n)} 1/d.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4, 1, 30, 1, 4, 3, 2, 5, 12, 1, 2, 3, 20, 1, 1, 1, 4, 15, 2, 1, 4, 1, 10, 3, 4, 1, 1, 5, 28, 3, 2, 1, 20, 1, 2, 21, 4, 5, 1, 1, 4, 3, 70, 1, 8, 1, 2, 15, 4, 7, 1, 1, 40

Offset: 1

Ilya Gutkovskiy, May 14 2024

			0, 1, 1, 1, 1, 3/2, 1, 3/2, 1, 3/2, 1, 11/6, 1, 3/2, 4/3, 3/2, 1, 11/6, ...

Cf. A017666, A070039, A372834 (numerators).

nmax = 80; CoefficientList[Series[Sum[x^(k (k + 1))/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest

a(n) = denominator(sumdiv(n, d, if (d^2 < n, 1/d))); \\ Michel Marcus, May 14 2024

Denominators of coefficients in expansion of Sum_{k>=1} x^(k*(k+1)) / (k * (1 - x^k)).

A373032 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k*(k+1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, -3, 1, -3, 1, -3, 1, 6, 1, -3, 10, -3, 1, 6, 1, -19, 10, -3, 1, -10, 1, -3, 10, -19, 1, 31, 1, -19, 10, -3, 26, -10, 1, -3, 10, 6, 1, -30, 1, -19, 35, -3, 1, -46, 1, 22, 10, -19, 1, -30, 26, 30, 10, -3, 1, -21, 1, -3, 59, -19, 26, -30, 1, -19, 10, 71

Offset: 1

Ilya Gutkovskiy, May 20 2024

sign

Cf. A056924, A064027, A070039, A321543, A333809, A333810, A339353, A344300, A372625, A373031.

nmax = 70; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(d+1) * d^2.

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cp's OEIS Frontend

A347142 Sum of 4th powers of divisors of n that are < sqrt(n).

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A348954 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d) * d.

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A033832 Sum of odd divisors of n < sqrt(n) = sum of even divisors of n < sqrt(n).

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A102883 Terms which share a divisor greater than 1 with the value formed by adding them to the sum of their divisors less than or equal to their square roots.

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A088345 n is divisible by the sum of all divisors of n which are less than the square root of n (values of n where 1 is the only divisor less than sqrt(n) are excluded as trivial cases.).

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A176314 Sum of remainders of mod(n, k), for 1 <= k <= sqrt(n).

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A363520 Product of the divisors of n that are < sqrt(n).

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A372834 a(n) is the numerator of Sum_{d|n, d < sqrt(n)} 1/d.

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