A345005 -id:A345005 - cp's OEIS frontend

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

-1, 0, -1, 3, -3, 5, -5, 11, 1, 5, -9, 20, -11, 5, 2, 31, -15, 24, -17, 26, 0, 5, -21, 56, -9, 5, 13, 32, -27, 43, -29, 79, -4, 5, -10, 79, -35, 5, -6, 78, -39, 53, -41, 44, 27, 5, -45, 140, -27, 38, -10, 50, -51, 93, -22, 100, -12, 5, -57, 140, -59, 5, 29, 191, -28, 73, -65, 62, -16, 63, -69, 207, -71, 5, 29, 68

Offset: 1

Antti Karttunen, Jun 05 2021

Coincides with A003415 only on perfect numbers (A000396).

Cf. A000203, A000396, A001065, A003415, A033879, A168036, A344999, A345002, A345003, A345004, A345005, A345049.
Cf. also A051709, A344753 (analogous sequences).
Cf. A013661, A136141.

A003415[n_] := If[n < 2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
a[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
Array[a, 80] (* Jean-François Alcover, Jun 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));

a(n) = A003415(n) - A033879(n) = A000203(n) + A003415(n) - 2*n.
a(n) = A001065(n) + A168036(n).
a(n) = A344999(n) / A048250(n) = A345049(n) / A173557(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 + A136141 - 2 = 0.418090735898... . - Amiram Eldar, Dec 08 2023

A345003 Numbers k for which A344998(k) = A344999(k).

Original entry on oeis.org

6, 8, 28, 81, 108, 496, 2500, 2700, 3375, 5292, 8128, 13068, 15625, 18252, 31212, 38988, 57132, 67228, 90828, 94500, 103788, 147852, 181548, 199692, 231525, 238572, 303372, 375948, 401868, 484812, 544428, 575532, 674028, 713097, 744012, 855468, 1016172, 1058841, 1101708, 1145772, 1236492, 1283148, 1379052, 1500625

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Numbers k such that A345001(k)*A048250(k) is equal to A342001(k)*A344753(k).

Conjecture: Sequence is a disjoint union of A000396 and A301939.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Positions of zeros in A345043.
Cf. A001615, A003415, A003557, A048250, A342001, A344753, A344998, A344999, A345001.
Cf. A000396, A301939, A345004, A345005 (subsequences).
Cf. also A345051.

A003415[n_] := If[n<2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
A003557[n_] := n * Times @@ (1/FactorInteger[n][[All, 1]]);
A048250[n_] := Select[Divisors[n], SquareFreeQ] // Total;
A344753[n_] := Sum[d + If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
A342001[n_] := A003415[n]/A003557[n];
A345001[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
A344998[n_] := A342001[n]*A344753[n];
A344999[n_] := A048250[n]*A345001[n];
Reap[For[k = 1, k <= 2*10^6, k++, If[A344998[k] == A344999[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jun 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A344753(n) = sumdiv(n,d,(dA342001(n) = (A003415(n) / A003557(n));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
isA345003(n) = (A345001(n)*A048250(n) == A342001(n)*A344753(n));

A344998 a(n) = A342001(n) * A344753(n).

Original entry on oeis.org

0, 2, 2, 10, 2, 60, 2, 33, 14, 112, 2, 224, 2, 180, 144, 92, 2, 273, 2, 456, 220, 364, 2, 660, 22, 480, 66, 768, 2, 2604, 2, 235, 420, 760, 312, 910, 2, 924, 544, 1394, 2, 4428, 2, 1632, 780, 1300, 2, 1736, 30, 747, 840, 2184, 2, 1080, 544, 2392, 1012, 1984, 2, 8832, 2, 2244, 1258, 570, 684, 9516, 2, 3528, 1404, 8732

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

From Antti Karttunen, Jan 30 2022: (Start)
In addition to 2's that occur on primes, there are also other duplicates, for example, a(39) = a(55) = 544, a(51) = a(91) = 840, a(65) = a(77) = 684, a(343) = a(6241) = 318, a(95) = a(119) = a(143) = 1200 and a(155) = a(203) = a(299) = a(323) = 2664. Note how, apart from 343 = 7^3 and 6241 = 79^2, the duplicate positions in above cases are all squarefree semiprimes, and how the sum of the two prime factors in those cases are equal. E.g. 95 = 5*19, 119 = 7*17, 143 = 11*13, with 5+19 = 7+17 = 11+13 = 24.
Indeed, for squarefree semiprimes pq, A342001(pq) = A003415(pq) = p+q and A344753(pq) = 2*A001065(pq) = 2*(1+p+q), and therefore the product A342001(pq) * A344753(pq) depends only on the sum of p+q.
(End)

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

Cf. A003415, A003557, A342001, A344753, A345043.

Cf. A345003 [gives k for which a(k) = A344999(k)], A345004, A345005.

A003415[n_] := If[n<2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
A003557[n_] := n*Times@@(1/FactorInteger[n][[All, 1]]);
A342001[n_] := A003415[n]/A003557[n];
A344753[n_] := Sum[d+If[SquareFreeQ[n/d], d, 0], {d, Most[Divisors[n]]}];
a[n_] := A342001[n]*A344753[n];
Array[a, 70] (* Jean-François Alcover, Jun 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A344753(n) = sumdiv(n, d, (dA344998(n) = (A342001(n) * A344753(n));

a(n) = A342001(n) * A344753(n).

a(n) = A344999(n) + A345043(n).

A344999 a(n) = A048250(n) * A345001(n).

Original entry on oeis.org

-1, 0, -4, 9, -18, 60, -40, 33, 4, 90, -108, 240, -154, 120, 48, 93, -270, 288, -340, 468, 0, 180, -504, 672, -54, 210, 52, 768, -810, 3096, -928, 237, -192, 270, -480, 948, -1330, 300, -336, 1404, -1638, 5088, -1804, 1584, 648, 360, -2160, 1680, -216, 684, -720, 2100, -2754, 1116, -1584, 2400, -960, 450, -3420, 10080

Offset: 1

Antti Karttunen, Jun 05 2021

sign

Cf. A000203, A003415, A048250, A345001.

Cf. A345003 [gives k for which a(k) = A344998(k)], A345004, A345005.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));
A344999(n) = (A048250(n) * A345001(n));

a(n) = A048250(n) * A345001(n).

a(n) = A344998(n) - A345043(n).

A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

Original entry on oeis.org

8, 81, 108, 2500, 2700, 3375, 5292, 13068, 15625, 18252, 31212, 38988, 57132, 67228, 90828, 94500, 103788, 147852, 181548, 199692, 231525, 238572, 303372, 375948, 401868, 484812, 544428, 575532, 674028, 713097, 744012, 855468, 1016172, 1058841, 1101708, 1145772

Offset: 1

Paolo P. Lava, Mar 29 2018

If n = Product_{k=1..j} p_k ^ i_k with each p_k prime, then psi(n) = n * Product_{k=1..j} (p_k + 1)/p_k and n' = n*Sum_{k=1..j} i_k/p_k.
Thus every number of the form p^(p+1), where p is prime, is in the sequence.
The sequence also contains every number of the form 108*p^2 where p is a prime > 3, or 108*p^3*(p+2) where p > 3 is in A001359. - Robert Israel, Mar 29 2018

			5292 = 2^2 * 3^3 * 7^2.
n' = 5292*(2/2 + 3/3 + 2/7) = 12096,
psi(n) = 5292*(1 + 1/2)*(1 + 1/3)*(1 + 1/7) = 12096.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A001359, A001615, A003415, A166374, A342458. A345005 (gives the odd terms).

Subsequence of A345003.

with(numtheory): P:=proc(n) local a,p; a:=ifactors(n)[2];
if add(op(2,p)/op(1,p),p=a)=mul(1+1/op(1,p),p=a) then n; fi; end:
seq(P(i),i=1..10^6);

selQ[n_] := Module[{f = FactorInteger[n], p, e}, Product[{p, e} = pe; p^e + p^(e-1), {pe, f}] == Sum[{p, e} = pe; (n/p)e, {pe, f}]];
Select[Range[10^6], selQ] (* Jean-François Alcover, Oct 16 2020 *)

dpsi(f) = prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
ader(n, f) = sum(i=1, #f~, n/f[i, 1]*f[i, 2]);
isok(n) = my(f=factor(n)); dpsi(f) == ader(n, f); \\ Michel Marcus, Mar 29 2018

Solutions of the equation n' = psi(n).

cp's OEIS Frontend

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

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A345003 Numbers k for which A344998(k) = A344999(k).

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A344998 a(n) = A342001(n) * A344753(n).

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A344999 a(n) = A048250(n) * A345001(n).

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A301939 Integers whose arithmetic derivative is equal to their Dedekind function.

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