A345003 -id:A345003 - 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

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).

A345051 Numbers k such that A345048(k) is equal to A345049(k).

Original entry on oeis.org

2, 6, 9, 15, 28, 496, 625, 1225, 3993, 8128, 117649, 218491, 857375, 3788435, 4259571, 33550336, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 8589869056, 10214476341

Offset: 1

Antti Karttunen, Jun 06 2021

Numbers k for which A342001(n) * A051709(n) = A173557(n) * A345001(n).

Conjecture: Sequence is a disjoint union of A000396 and A166374, i.e., there are no terms of any other kind.

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

Cf. A051709, A173557, A342001, A345001, A345048, A345049.
Positions of zeros in A345050.
Cf. A000396, A166374 (subsequences).
Cf. also A345003.

PARI
```
isA345051(n) = (0==A345050(n));
```

a(21)-a(27) from Amiram Eldar, Dec 08 2023

A345005 Odd numbers whose arithmetic derivative (A003415) is equal to Dedekind psi (A001615) applied to the same number.

Original entry on oeis.org

81, 3375, 15625, 231525, 713097, 1058841, 1500625, 4348377, 5764801, 16891497, 163555875, 209548647, 239010993, 239160735, 254205875, 267651475, 405189675, 451699875, 958403475, 1050284375, 1213014231, 1501534375, 1695809375, 1809323971, 1942143291

Offset: 1

Antti Karttunen, Jun 05 2021

Conjectured to be also the odd numbers k for which A344998(k) = A344999(k).

Odd terms in A301939, but see also A345003.

Cf. A001615, A003415, A344998, A344999.

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA345005(n) = ((n%2)&&(A003415(n)==A001615(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).

A345004 Numbers k such that A345001(k)A048250(k) is equal to A342001(k)A344753(k) and A345001(n)/A342001(n) is an integer.

Original entry on oeis.org

6, 28, 496, 5292, 8128, 33550336

Offset: 1

Antti Karttunen, Jun 05 2021

Equally, we may require that A344753(k)/A048250(k) is an integer, thus this is a subsequence of A344754.

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

Cf. A000396 (subsequence), A003415, A003557, A048250, A342001, A344753, A345001.

Intersection of A344754 and A345003.

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));
isA345004(n) = { my(u=A345001(n),v=A342001(n)); (v>0&&1==denominator(u/v)&&(u*A048250(n) == v*A344753(n))); };

A345043 a(n) = A344998(n) - A344999(n).

Original entry on oeis.org

1, 2, 6, 1, 20, 0, 42, 0, 10, 22, 110, -16, 156, 60, 96, -1, 272, -15, 342, -12, 220, 184, 506, -12, 76, 270, 14, 0, 812, -492, 930, -2, 612, 490, 792, -38, 1332, 624, 880, -10, 1640, -660, 1806, 48, 132, 940, 2162, 56, 246, 63, 1560, 84, 2756, -36, 2128, -8, 1972, 1534, 3422, -1248, 3660, 1764, 330, -3, 3036, -996, 4422

Offset: 1

Antti Karttunen, Jun 06 2021

sign

Each term a(n) seems to be a multiple of A033879(n). Factor (quotient) is negative at least for 16, 32, 48, 64, 72, 80, 96, 112, 128, 144, 160, 176, 192, 200, 208, 216, 224, 240, 243, 256, 272, 288, 304, 320, 324, 336, 352, 368, 384, 392, 400, 416, 432, 448, 464, 480, 486, ...

Cf. A033879, A344998, A344999, A345003 (positions of zeros).

a(n) = A344998(n) - A344999(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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A344998 a(n) = A342001(n) * A344753(n).

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

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A345051 Numbers k such that A345048(k) is equal to A345049(k).

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A345005 Odd numbers whose arithmetic derivative (A003415) is equal to Dedekind psi (A001615) applied to the same number.

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

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A345004 Numbers k such that A345001(k)*A048250(k) is equal to A342001(k)*A344753(k) and A345001(n)/A342001(n) is an integer.

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A345043 a(n) = A344998(n) - A344999(n).

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A345004 Numbers k such that A345001(k)A048250(k) is equal to A342001(k)A344753(k) and A345001(n)/A342001(n) is an integer.