A230164 -id:A230164 - cp's OEIS frontend

A342021 Numbers k such that A003415(sigma(k)) = k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

5, 8, 41, 47057

Offset: 1

Antti Karttunen, Apr 08 2021

a(5) > 2^33, if it exists.

Terms of this sequence and A230165 group into pairs (m, sigma(m)), where m is a term of this sequence and sigma(m) is a term of A230165. - Max Alekseyev, Feb 13 2025

Fixed points of A342925.
Positions of 0's in A342926.
Subsequence of A343217.
Cf. A000203, A003415, A229342, A230164, A230165, A347889.

a(n) = A003415(A230165(n)). - Max Alekseyev, Feb 13 2025

A230165 Numbers k such that k = sigma(k'), where k' is the arithmetic derivative of k.

Original entry on oeis.org

6, 15, 42, 47058

Offset: 1

Paolo P. Lava, Oct 14 2013

a(5) > 10^10. - Giovanni Resta, Oct 14 2013

Terms of this sequence and A342021 group into pairs (m, m'), where m is a term of this sequence and m' is a term of A342021. - Max Alekseyev, Feb 13 2025

			Arithmetic derivative of 15 is 8 and sigma(8) = 15.

Cf. A003415, A229342, A230164, A342021.

with(numtheory); P:=proc(q) local n; for n from 1 to q do
if n=sigma(n*add(op(2,p)/op(1,p),p=ifactors(n)[2])) then print(n);
fi; od; end: P(10^9);

a(n) = sigma(A342021(n)). - Max Alekseyev, Feb 13 2025

A321182 Composite numbers k such that sigma(k)/k' is an integer, where k' is the arithmetic derivative of k.

Original entry on oeis.org

15, 35, 45, 95, 119, 143, 209, 287, 319, 323, 377, 527, 559, 693, 779, 899, 923, 989, 1007, 1189, 1199, 1343, 1349, 1763, 1919, 2159, 2507, 2759, 2911, 3239, 3599, 3827, 4031, 4607, 5183, 5207, 5249, 5459, 5543, 6439, 6811, 6887, 7067, 7279, 7739, 8159, 8639, 9179

Offset: 1

Paolo P. Lava, Oct 29 2018

Alternative definition: Composite numbers such that the ratio between the sum of the reciprocal of their divisors and the sum of the reciprocal of their prime factors, counted with multiplicity, is an integer.
Mainly squarefree numbers, which are a subset of A242152, apart from some sporadic terms: 45, 693, 6811, 17296, 24016, 71753, 1165669, etc.
A230164 is a subsequence (ratio equal to 1).

			Divisors of 45 are 1, 3, 5, 9, 15, 45 and prime factors 3^2, 5: (1/1 + 1/3 + 1/5 + 1/9 + 1/15 + 1 /45)/(1/3 + 1/3 + 1/5) = 2
Divisors of 119 are 1, 7, 17, 119 and prime factors 7, 17: (1/1 + 1/7 + 1/17 + 1 /119)/(1/7 + 1/17) = 6.
Divisors of 552521 are 1, 37, 109, 137, 4033, 5069, 14933, 552521 and prime factors 37, 109, 137: (1/1 + 1/37 + 1/109 + 1/137 + 1 /4033 + 1/5069 + 1/14933 + 1/552521)/(1/37 + 1/109 + 1/137) = 24.

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

Cf. A000203, A003415, A230164, A242152.

with(numtheory): P:=proc(q) local a,b,c,k,n; for n from 2 to q do if not isprime(n) then a:=add(1/a,a=divisors(n)); b:=ifactors(n)[2]; c:=add(b[k][2]/b[k][1],k=1..nops(b)); if frac(a/c)=0 then print(n); fi; fi; od; end: P(10^7);

Select[Range[4, 10^4], And[CompositeQ@ #, IntegerQ[DivisorSigma[1, #]/If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger[Abs@ #]]]]] &] (* Michael De Vlieger, Oct 31 2018 *)

ard(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]); \\ A003415
isok(n) = (n>1) && !isprime(n) && (frac(sigma(n)/ard(n)) == 0); \\ Michel Marcus, Oct 30 2018

A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

Original entry on oeis.org

1, 12, 18, 220, 396, 17296, 24016, 287532, 4661056, 64288512, 334144656, 358585488, 555192576, 568719616, 2172649216, 2451538112, 2645953344, 2955423888, 6704333824, 26996772032, 88734733632, 147861504000, 311063879024, 371226582848, 429391876096

Offset: 1

Paolo P. Lava, Mar 07 2014

A212128 and A230164 are subsets of this sequence.

a(26) > 10^12. - Giovanni Resta, Mar 11 2014

			Divisors of 12 are 1, 2, 3, 4, 6, 12 and 1/1 + 1/2 + 1/3 +1/4 + 1/6 + 1/12 = 7/3. Prime factors of 12 are 2^2, 3 and 1/2 + 1/2 + 1/3 = 4/3. Finally 7/3 - 4/3 = 1 that is an integer.

Cf. A003415, A212128, A224346, A230164.

with(numtheory); P:=proc(q) local a,b,c,k,n;
for n from 1 to q do if not isprime(n) then b:=sigma(n)/n;
a:=ifactors(n)[2]; c:=add(a[k][2]/a[k][1],k=1..nops(a));
if type(b-c,integer) then lprint(n,b-c); fi; fi; od; end: P(10^6);

a(9)-a(10), a(13)-a(17), a(19)-a(25) from Giovanni Resta, Mar 11 2014

A239274 Numbers k such that k' = sigma(k) - phi(k), where k' = A003415(k) is the arithmetic derivative of k.

Original entry on oeis.org

1, 88, 1104, 492624, 30818736, 200827760, 1598249556, 4575586644, 8491046375, 8554970196, 29917317500, 54481242640, 202195886375, 3201891498000

Offset: 1

Giovanni Resta, Mar 13 2014

nonn

a(15) > 10^13.

Cf. A000010, A000203, A003415, A166374, A230164.

ad[1]=0; ad[n_]:=n*Total[(#1[[2]] / #1[[1]]&) /@ FactorInteger[n]]; Select[Range[500000], ad[#] == DivisorSigma[1,#] - EulerPhi[#] &]

A248816 Numbers that are equal to the arithmetic derivative of the sum of their aliquot parts.

Original entry on oeis.org

152, 284, 4316, 18632, 25484, 2657259, 8394752, 12186976, 17702756, 1172473731, 2147581952, 13716855652, 63831498112

Offset: 1

Paolo P. Lava, Oct 15 2014

Solutions of the equations n = (sigma(n)-n)'.
a(12) > 5*10^9. - Michel Marcus, Nov 01 2014
There could be a relation with terms in A125246 and A228450, since some terms of these sequences are here also. - Michel Marcus, Oct 30 2014
a(14) > 10^11. - Giovanni Resta, May 29 2016

			Sum of the aliquot parts of 284 is sigma(284) - 284 = 220 and the arithmetic derivative of 220 is 284.~

Cf. A000203, A003415, A230164.

with(numtheory): P:= proc(q) local a,n,p; for n from 1 to q do
a:=(sigma(n)-n)*add(op(2,p)/op(1,p),p=ifactors(sigma(n)-n)[2]);
if n=a then print(n); fi; od; end: P(10^9);

ad(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
isok(n) = ad(sigma(n) - n) == n; \\ Michel Marcus, Oct 28 2014

a(6)-a(11) from Michel Marcus, Oct 28 2014

a(12)-a(13) from Giovanni Resta, May 29 2016

cp's OEIS Frontend

A342021 Numbers k such that A003415(sigma(k)) = k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

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A230165 Numbers k such that k = sigma(k'), where k' is the arithmetic derivative of k.

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A321182 Composite numbers k such that sigma(k)/k' is an integer, where k' is the arithmetic derivative of k.

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A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

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A239274 Numbers k such that k' = sigma(k) - phi(k), where k' = A003415(k) is the arithmetic derivative of k.

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Mathematica

A248816 Numbers that are equal to the arithmetic derivative of the sum of their aliquot parts.

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Maple

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