A319683 -id:A319683 - cp's OEIS frontend

A319684 Sum of A003415(d) over the divisors d of n, where A003415 is arithmetic derivative.

0, 1, 1, 5, 1, 7, 1, 17, 7, 9, 1, 27, 1, 11, 10, 49, 1, 34, 1, 37, 12, 15, 1, 83, 11, 17, 34, 47, 1, 54, 1, 129, 16, 21, 14, 114, 1, 23, 18, 117, 1, 68, 1, 67, 55, 27, 1, 227, 15, 64, 22, 77, 1, 142, 18, 151, 24, 33, 1, 190, 1, 35, 69, 321, 20, 96, 1, 97, 28, 90, 1, 326, 1, 41, 75, 107, 20, 110, 1, 325, 142, 45, 1, 244, 24, 47, 34, 219, 1, 243, 22

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Inverse Möbius transform of A003415.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A319683.

Cf. also A300251, A300252, A305809, A317835.

Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] &, 91]] (* Michael De Vlieger, May 24 2021 *)

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
A319684(n) = sumdiv(n,d,A003415(d));

a(n) = Sum_{d|n} A003415(d).

a(n) = A319683(n) + A003415(n).

A319356 a(n) = Product_{d|n, dA003415(d)), where A003415(d) gives arithmetic derivative of d.

Original entry on oeis.org

1, 2, 2, 6, 2, 18, 2, 66, 6, 18, 2, 2574, 2, 18, 18, 2706, 2, 3978, 2, 3762, 18, 18, 2, 6226506, 6, 18, 102, 5742, 2, 306774, 2, 370722, 18, 18, 18, 203956038, 2, 18, 18, 14961474, 2, 631098, 2, 8514, 7038, 18, 2, 168047170434, 6, 10602, 18, 10494, 2, 33626034, 18, 32252814, 18, 18, 2, 2529917014482, 2, 18, 9486, 155332518, 18, 1418742, 2, 14058, 18, 1219914, 2

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A003415, A319357 (rgs-transform).

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
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };

a(n) = Product_{d|n, dA000040(1+A003415(d)).
A001221(a(n)) = A319685(n).
A056239(A064989(a(n))) = A319683(n).

A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 8, 2, 9, 4, 4, 2, 10, 3, 4, 11, 12, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 16, 2, 17, 2, 18, 19, 4, 2, 20, 3, 21, 4, 22, 2, 23, 4, 24, 4, 4, 2, 25, 2, 4, 26, 27, 4, 28, 2, 29, 4, 30, 2, 31, 2, 4, 32, 33, 4, 34, 2, 35, 36, 4, 2, 37, 4, 4, 4, 38, 2, 39, 4, 40, 4, 4, 4, 41, 2, 42, 43, 44, 2, 45, 2, 46, 47

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Restricted growth sequence transform of A319356.
The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, then also the last implication given below is valid.
For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j),
  a(i) = a(j) => A319683(i) = A319683(j),
  a(i) = a(j) => A319686(i) = A319686(j),
  a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]

			Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A319356.
Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),
Cf. also A300245, A300249, A319353, A319693.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };
v319357 = rgs_transform(vector(up_to,n,A319356(n)));
A319357(n) = v319357[n];

A344584 Difference between the inverse Möbius transform of the arithmetic derivative of n and the sum of the proper divisors of n: a(n) = A319684(n) - A001065(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 10, 3, 1, 0, 11, 0, 1, 1, 34, 0, 13, 0, 15, 1, 1, 0, 47, 5, 1, 21, 19, 0, 12, 0, 98, 1, 1, 1, 59, 0, 1, 1, 67, 0, 14, 0, 27, 22, 1, 0, 151, 7, 21, 1, 31, 0, 76, 1, 87, 1, 1, 0, 82, 0, 1, 28, 258, 1, 18, 0, 39, 1, 16, 0, 203, 0, 1, 26, 43, 1, 20, 0, 219, 102, 1, 0, 104, 1, 1, 1, 127, 0, 99, 1, 51, 1, 1, 1, 423

Offset: 1

Antti Karttunen, May 24 2021

nonn

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

Cf. A000203, A001065, A003415, A168036, A211991, A319683, A319684.

Inverse Möbius transform of A344178.

Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] - DivisorSigma[1, #] + # &, 96]] (* Michael De Vlieger, May 24 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A319684(n) = sumdiv(n, d, A003415(d));
A344584(n) = (A319684(n) - (sigma(n)-n));

a(n) = A319684(n) - A001065(n) = A211991(n) + A319683(n).
a(n) = Sum_{d|n} A344178(d).
a(n) = n + Sum_{d|n} A168036(d).

A259924 Numbers n such that sigma(n) - n = sum_{k divides n, k < n} k', where sigma(n) is the sum of the divisors of n and k' is the arithmetic derivative of k.

Original entry on oeis.org

1, 780, 1064, 1289560, 1428228, 18107748, 186000889725, 680691912588

Offset: 1

Paolo P. Lava, Jul 09 2015

a(7) > 10^9. - Giovanni Resta, Jul 15 2015

a(9) > 10^13. - Hiroaki Yamanouchi, Sep 10 2015

			Aliquot parts of 780 are 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390. Their arithmetic derivatives are 0, 1, 1, 4, 1, 5, 7, 16, 1, 8, 24, 15, 31, 16, 56, 92, 18, 71, 101, 220, 119, 332, 433. Their sum is 1572 and sigma(780) - 780 = 2352 - 780 = 1572.
Aliquot parts of 1064 are 1, 2, 4, 7, 8, 14, 19, 28, 38, 56, 76, 133, 152, 266, 532. Their arithmetic derivatives are 0, 1, 4, 1, 12, 9, 1, 32, 21, 92, 80, 26, 236, 185, 636. Their sum is 1336 and sigma(1064) - 1064 = 2400 - 1064 = 1336.

Cf. A001065, A003415, A319683.

with(numtheory): P:=proc(q) local a,k,n,p;
for n from 3 to q do a:=sort([op(divisors(n))]);
a:=add(a[k]*add(op(2,p)/op(1,p),p=ifactors(a[k])[2]),k=2..nops(a)-1);
if sigma(n)-n=a then print(n); fi; od; end: P(10^9);

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; Select[Range@ 1500000, DivisorSigma[1, #] - # == Total[f /@ Most@ Divisors@ #] &] (* Michael De Vlieger, Jul 16 2015, after Michael Somos at A003415 *)

a(6) from Giovanni Resta, Jul 15 2015

a(1) inserted and a(7)-a(8) added by Hiroaki Yamanouchi, Sep 10 2015

cp's OEIS Frontend

A319684 Sum of A003415(d) over the divisors d of n, where A003415 is arithmetic derivative.

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A319356 a(n) = Product_{d|n, dA003415(d)), where A003415(d) gives arithmetic derivative of d.

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A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

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A344584 Difference between the inverse Möbius transform of the arithmetic derivative of n and the sum of the proper divisors of n: a(n) = A319684(n) - A001065(n).

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A259924 Numbers n such that sigma(n) - n = sum_{k divides n, k < n} k', where sigma(n) is the sum of the divisors of n and k' is the arithmetic derivative of k.

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