A173441 -id:A173441 - cp's OEIS frontend

A326697 a(n) is the sum of divisors d of n such that sigma(d) divides n.

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 5, 1, 8, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 40, 1, 1, 1, 17, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 28

Offset: 1

Jaroslav Krizek, Jul 19 2019

nonn

a(A097603(n)) > 1.
See A173441 and A326698 for number and product of such divisors.
From Bernard Schott, Aug 13 2019: (Start)
a(n) = 1 if n is in A000961,
a(n) = 1 if n is in A006881 \ {6},
a(n) = 1 if n is in A001749 \ {12, 28}. (End)

			For n = 12, divisors d of 12: 1, 2, 3, 4, 6, 12;
corresponding sigma(d): 1, 3, 4, 7, 12, 28;
sigma(d) divides n for 4 divisors d: 1, 2, 3, 6;
a(12) = 1 + 2 + 3 + 6 = 12.

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

Cf. A000203, A000961, A173441, A326698.

[&+[d: d in Divisors(n) | IsIntegral(n / SumOfDivisors(d))]: n in [1..100]];

a[n_] := DivisorSum[n, # &, Divisible[n, DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, Jul 21 2019 *)

a(n) = sumdiv(n, d, d*(!(n % sigma(d)))); \\ Michel Marcus, Jul 19 2019

A326698 a(n) is the product of divisors d of n such that sigma(d) divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 10, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 784, 1, 1, 1, 180, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Jaroslav Krizek, Jul 19 2019

nonn

a(A097603(n)) > 1.

See A173441 and A326697 for number and sum such divisors.

			For n = 12, divisors d of 12: 1, 2, 3, 4, 6, 12;
corresponding sigma(d): 1, 3, 4, 7, 12, 28;
sigma(d) divides n for 4 divisors d: 1, 2, 3, 6;
a(12) = 1 * 2 * 3 * 6 = 36.

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

Cf. A000203, A000961, A173441, A326697.

[&*[d: d in Divisors(n) | IsIntegral(n / SumOfDivisors(d))]: n in [1..100]];

a[n_] := Times @@ Select[Divisors[n], Divisible[n, DivisorSigma[1, #] &]]; Array[a, 100] (* Amiram Eldar, Jul 21 2019 *)

a(n) = my(p=1); fordiv(n, d, if (!(n % sigma(d)), p *= d)); p; \\ Michel Marcus, Jul 19 2019

A327156 a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where A286561(n,x) gives the highest exponent of x dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 5, 1, 8, 1, 1, 1, 16, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A173441.

Cf. also A293514, A322312, A323155, A327154, A327155.

A327156(n) = { my(m=1,v); fordiv(n,d,if((d>1) && ((v = valuation(n,sigma(d)))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A173441(n).

A173442 Number of divisors d of number n such that sigma(d) does not divide n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 3, 4, 1, 4, 1, 5, 3, 3, 1, 4, 2, 3, 3, 4, 1, 5, 1, 5, 3, 3, 3, 5, 1, 3, 3, 7, 1, 6, 1, 5, 5, 3, 1, 6, 2, 5, 3, 5, 1, 6, 3, 4, 3, 3, 1, 7, 1, 3, 5, 6, 3, 6, 1, 5, 3, 7, 1, 8, 1, 3, 5, 5, 3, 6, 1, 9, 4, 3, 1, 6, 3, 3, 3, 7, 1, 8, 3, 5, 3, 3, 3, 8, 1, 5, 5, 8

Offset: 1

Jaroslav Krizek, Feb 18 2010

nonn

Sigma(n) = A000203(n). a(n) = A000005(n) - A173441(n).

a(n) >= 1 for n >= 2, with equality if and only if n is prime. - Robert Israel, Oct 10 2017

			For n = 12, a(12) = 2. We see that the divisors of 12 are 1, 2, 3, 4, 6, 12. The corresponding sigma(d) are 1, 3, 4, 7, 12, 28. The sigma(d) which do not divide n for 2 divisors d are 4 and 12.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A173441.

f:= n -> nops(select(t -> n mod numtheory:-sigma(t) <> 0, numtheory:-divisors(n))):
map(f, [$1..100]); # Robert Israel, Oct 10 2017

Table[Length[Select[Divisors[n], Not[Divisible[n, DivisorSigma[1, #]]], &]], {n, 100}] (* Alonso del Arte, Oct 10 2017 *)

a(n) = sumdiv(n, d, (n % sigma(d)) != 0); \\ Michel Marcus, Oct 11 2017

More terms from Robert Israel, Oct 10 2017

A309253 a(n) is the smallest number m with exactly n such divisors d that sigma(d) divides m.

Original entry on oeis.org

1, 6, 30, 12, 60, 84, 1140, 120, 168, 2340, 1848, 360, 2184, 1080, 4368, 840, 10440, 1680, 7920, 2520, 6552, 3360, 7560, 5040, 13104, 27720, 73440, 36960, 21840, 15120, 72072, 10080, 95760, 26208, 63840, 20160, 146160, 144144, 87360, 174720, 1071360, 166320

Offset: 1

Jaroslav Krizek, Aug 08 2019

nonn

			For n = 3; a(3) = 30 because 30 is the smallest number with exactly 3 divisors d that sigma(d) is also its divisor, namely 1, 2 and 5 as sigma(1) = 1, sigma(2) = 3 and sigma(5) = 6, and all these (1, 3 and 6) are divisors of 30.

Cf. A173441, A326696, A326697, A326698.

A309253:=func; [A309253(n): n in[1..40]];

f[n_] := Count[Divisors[n], ?(Divisible[n, DivisorSigma[1, #]] &)]; m = 42; s = Table[0, {m}]; c = 0; n = 1; While[c < m, v = f[n]; If[v <=m && s[[v]] == 0, c++; s[[v]] = n]; n++]; s (* _Amiram Eldar, Aug 08 2019 *)

a(n) = my(m=1); while(sumdiv(m, d, !(m%sigma(d))) != n, m++); m; \\ Jinyuan Wang, Aug 08 2019

A323880 Number of divisors d > 1 of n such that A003415(d) divides n, where A003415 gives the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 3, 2, 2, 1, 5, 1, 2, 2, 3, 2, 3, 1, 3, 2, 4, 1, 5, 1, 2, 2, 3, 2, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 2, 3, 2, 2, 2, 6, 1, 3, 2, 4, 1, 3, 1, 3, 3

Offset: 1

Antti Karttunen, Feb 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415.

Cf. also A173441, A323878, A323879.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A323880(n) = sumdiv(n,d,(d>1)&&!(n%A003415(d)));

a(n) = Sum_{d|n, d>1} [A003415(d)|n], where [ ] is the Iverson bracket, and A003415 gives the arithmetic derivative of n.

A326696 Numbers m with at least one divisor d > 1 such that sigma(d) divides m.

Original entry on oeis.org

6, 12, 18, 24, 28, 30, 36, 42, 48, 54, 56, 60, 66, 72, 78, 84, 90, 96, 102, 108, 112, 114, 117, 120, 126, 132, 138, 140, 144, 150, 156, 162, 168, 174, 180, 182, 186, 192, 196, 198, 204, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282

Offset: 1

Jaroslav Krizek, Aug 30 2019

nonn

All integers m contain at least one divisor d (number 1) such that sigma(d) divides m.
See A309253 for the smallest numbers m with n divisors d such that sigma(d) divides m for n >= 1.
Supersequence of A097603 (multiples of perfect numbers).
From Bernard Schott, Sep 04 2019: (Start)
If m = 6 * k with k >= 1, then 2 divides m and sigma(2) = 3 also divides m; so, the positive multiples of 6 belong to this sequence.
This sequence is generated by the primitive terms. A primitive term m is necessarily of the form d * sigma(d) where 1 < d < m is a divisor of m. The first few primitives are: 6, 28, 117, 182, ...
Some subsequences of such primitives, not exhaustive list:
1) d is prime p and m = p * sigma(p) = p * (p+1) is oblong.
For p = 2, 13, 19, 37, ..., we get 6, 182, 380, 1406, ...
2) d = p^2 with p prime, and m = p^2 * (p^2 + p + 1).
For p = 2, 3, 5, 7, ..., we get m = 28, 117, 775, 2793, ...
3) d = 2^(q-1) and m = 2^(q-1) * (2^q -1), with q prime in A000043 and 2^q - 1 is a Mersenne prime in A000668, then m is a perfect number in A000039.
For q prime = 2, 3, 5, 7, 13, ..., we get m = 6, 28, 496, 8128, 33550336, ... (End)

			Divisors d of 12: 1, 2, 3, 4, 6, 12; corresponding sigma(d):1, 3, 4, 7, 12, 28; sigma(d) divides 12 for 4 divisors d > 1: 2, 3 and 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A173441, A309253, A323652, A326697, A326698.

Subsequences: A008588 \ {0}, A097603.

[m: m in [1..10^5] | #[d: d in Divisors(m) | IsIntegral(m / SumOfDivisors(d) ) and d gt 1] gt 0];

filter:= proc(n) local d;
  uses numtheory;
  ormap(t -> n mod sigma(t) = 0, divisors(n) minus {1})
end proc:
select(filter, [$2..1000]); # Robert Israel, Oct 07 2019

aQ[n_] := AnyTrue[Rest @ Divisors[n], Divisible[n, DivisorSigma[1, #]] &]; Select[Range[282], aQ] (* Amiram Eldar, Aug 31 2019 *)

isok(m) = fordiv(m, d, if ((d>1) && (!(m % sigma(d))), return(1))); \\ Michel Marcus, Sep 03 2019

A173441(a(n)) > 1; A326697(a(n)) > 1; A326697(a(n)) > 1.

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A326697 a(n) is the sum of divisors d of n such that sigma(d) divides n.

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A326698 a(n) is the product of divisors d of n such that sigma(d) divides n.

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A327156 a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where A286561(n,x) gives the highest exponent of x dividing n.

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A173442 Number of divisors d of number n such that sigma(d) does not divide n.

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A309253 a(n) is the smallest number m with exactly n such divisors d that sigma(d) divides m.

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A323880 Number of divisors d > 1 of n such that A003415(d) divides n, where A003415 gives the arithmetic derivative of n.

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A326696 Numbers m with at least one divisor d > 1 such that sigma(d) divides m.

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