A344753 -id:A344753 - cp's OEIS frontend

A344755 Numbers k such that A344753(k) is a multiple of A048250(k), and k is a multiple of A344753(k)/A048250(k).

6, 28, 150, 496, 528, 1980, 4560, 8128, 8736, 11400, 19872, 20664, 75840, 82080, 253080, 254880, 741744, 1627290, 5130300, 5607360, 7529760, 19645440, 20718720, 33550336, 35092512, 45643392, 45995040, 56424960, 86944320, 169910136, 174013920, 180442080, 196378992, 242040960, 304577280, 314511360, 326611440, 451344960

Offset: 1

Antti Karttunen, May 29 2021

nonn

Numbers k for which A344753(k)/A048250(k) is a divisor of k.

Perfect numbers (A000396, including also any hypothetical odd terms) are included as only on them A001615 coincides with A344753, and because A001615(n) = A003557(n)*A048250(n), with A003557(n) being a divisor of n.

Cf. A000396 (subsequence), A001065, A001615, A003557, A306927, A048250, A344753.
Subsequence of A344754.
Cf. also A344700.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A344753(n) = sumdiv(n,d,(dA344755(n) = { my(t=A344753(n),u=A048250(n)); ((0==(t%u))&&(0==(n%(t/u)))); };

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).

A344754 Numbers k such that A344753(k) is a multiple of A048250(k).

Original entry on oeis.org

1, 6, 24, 28, 54, 96, 112, 150, 153, 216, 294, 384, 448, 486, 496, 528, 672, 726, 864, 1014, 1080, 1377, 1500, 1536, 1734, 1792, 1944, 1980, 1984, 2112, 2166, 2250, 2376, 2688, 3174, 3456, 3672, 3750, 4320, 4374, 4560, 4753, 5046, 5292, 5766, 6000, 6048, 6144, 6720, 7168, 7776, 7936, 8128, 8214, 8448, 8700, 8736, 9024

Offset: 1

Antti Karttunen, May 29 2021

nonn

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

Cf. A001065, A001615, A306927, A048250, A344753.
Subsequences: A000396, A344755.
Cf. also A344700.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A344753(n) = sumdiv(n,d,(dA344754(n) = (0==(A344753(n)%A048250(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))); };

A344997 a(n) = A173557(n) * A344753(n).

Original entry on oeis.org

0, 2, 4, 5, 8, 24, 12, 11, 14, 64, 20, 56, 24, 120, 144, 23, 32, 78, 36, 152, 264, 280, 44, 120, 44, 384, 44, 288, 56, 672, 60, 47, 600, 640, 624, 182, 72, 792, 816, 328, 80, 1296, 84, 680, 480, 1144, 92, 248, 90, 332, 1344, 936, 104, 240, 1360, 624, 1656, 1792, 116, 1536, 120, 2040, 888, 95, 1824, 3120, 132, 1568, 2376

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Cf. A008966, A173557, A344753.
Cf. also A344996.
Cf. A002117, A013661, A307868.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 0; a[n_] := Module[{fct = FactorInteger[n], p}, p = fct[[;; , 1]]; Times @@ (p - 1)*(Times @@ f @@@ fct + n*Times @@ (1 + 1/p) - 2*n)]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A344753(n) = sumdiv(n,d,(dA344997(n) = (A173557(n)*A344753(n));

a(n) = A173557(n) * A344753(n).
a(n) = Product(p_i - 1) * [Sum_{d|n, dA008966(n/d) * d)], where p_i are distinct primes dividing n.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1/zeta(2) - 2 * A307868 + zeta(2)*zeta(3) * Product_{p prime} (1 - 2/p^2 - 1/p^3 + 1/p^4 + 3/p^5 - 2/p^6) = 0.283799589272... . - Amiram Eldar, Dec 08 2023

A051709 a(n) = sigma(n) + phi(n) - 2n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 2, 0, 8, 0, 2, 2, 7, 0, 9, 0, 10, 2, 2, 0, 20, 1, 2, 4, 12, 0, 20, 0, 15, 2, 2, 2, 31, 0, 2, 2, 26, 0, 24, 0, 16, 12, 2, 0, 44, 1, 13, 2, 18, 0, 30, 2, 32, 2, 2, 0, 64, 0, 2, 14, 31, 2, 32, 0, 22, 2, 28, 0, 75, 0, 2, 14, 24, 2, 36, 0

Offset: 1

Jud McCranie and Carlos Rivera

nonn

Sigma is the sum of divisors (A000203), and phi is the Euler totient function (A000010). - Michael B. Porter, Jul 05 2013
Because sigma and phi are multiplicative functions, it is easy to show that (1) if a(n)=0, then n is prime or 1 and (2) if a(n)=2, then n is the product of two distinct prime numbers. Note that a(n) is the n-th term of the Dirichlet series whose generating function is given below. Using the generating function, it is theoretically possible to compute a(n). Hence a(n)=0 could be used as a primality test and a(n)=2 could be used as a test for membership in P2 (A006881). - T. D. Noe, Aug 01 2002
It appears that a(n) - A002033(n) = zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)) + 1/(zeta(s)-2). - Eric Desbiaux, Jul 04 2013
a(n) = 1 if and only if n = prime(k)^2 (n is in A001248). It seems that a(n) = k has only finitely many solutions for k >= 3. - Jianing Song, Jun 27 2021

			a(5) = sigma(5) + phi(5) - 2*5 = 6 + 4 - 10 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (First 1000 terms from T. D. Noe.)
Carlos Rivera, Puzzle 76. z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.

Cf. A000010, A000203, A001065, A001248, A005843, A006881, A051612, A051953, A065387, A072780, A228498 (= a(n^2)), A297159, A324048, A344994, A344995, A344996, A345048, A345054.
Cf. A278373 (range of this sequence), A056996 (numbers not present).
Cf. also A344753, A345001 (analogous sequences).

Table[DivisorSigma[1,n]+EulerPhi[n]-2n,{n,80}] (* Harvey P. Dale, Apr 08 2015 *)

a(n)=sigma(n)+eulerphi(n)-2*n \\ Charles R Greathouse IV, Jul 05 2013

A051709(n) = -sumdiv(n,d,(dAntti Karttunen, Mar 02 2018

Dirichlet g.f.: zeta(s-1) * (zeta(s) - 2 + 1/zeta(s)). - T. D. Noe, Aug 01 2002
From Antti Karttunen, Mar 02 2018: (Start)
a(n) = A001065(n) - A051953(n).  [Difference between the sum of proper divisors of n and their Moebius-transform.]
a(n) = -Sum_{d|n, dA008683(n/d)*A001065(d). (End)
Sum_{k=1..n} a(k) = (3/(Pi^2) + Pi^2/12 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 03 2023

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

Original entry on oeis.org

-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));

A344700 Numbers k for which A306927(k) [= A001615(k)-k] is a multiple of A344705(k) [= A001615(k)-A001065(k)], and their quotient is nonnegative.

Original entry on oeis.org

1, 6, 24, 28, 168, 496, 864, 1080, 1520, 1836, 2016, 2088, 2112, 2520, 2912, 2976, 3000, 3024, 3240, 3800, 8128, 9000, 11088, 11232, 11448, 14160, 14688, 16920, 17028, 18360, 19872, 20520, 20880, 25280, 25488, 27552, 29376, 30800, 31200, 31320, 31968, 35400, 39240, 44064, 48768, 49896, 50760, 51480, 51660, 52200, 55680

Offset: 1

Antti Karttunen, May 28 2021

nonn

Numbers k for which A344704(k) = A344705(k), i.e., numbers k such that gcd(A001615(k)-k, A001615(k)-A001065(k)) = A001615(k) - A001065(k).
Note that A306927(k) is always nonnegative, but A344705(k) = A033879(k) + A306927(k) gets also negative values. Number k is perfect only when A033879(k) = A344705(k) - A306927(k) = 0, that is, when A344705(k) = A306927(k), which necessitates that A306927(k) should be a multiple of A344705(k), and their quotient should be nonnegative (actually = +1).
In the range 1 .. 2^31 there are 782 such numbers, of which only the initial 1 is odd.

Cf. A000203, A001065, A001615, A033879, A244963, A306927, A344704, A344705, A344752 (gives the quotient A306927(k)/A344705(k) computed for these terms), A344753.
Cf. A000396 (subsequence).
Cf. also A344754, A344755.

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
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); (gcd(t-n,u)==u); };
\\ Alternatively as:
isA344700(n) = { my(t=A001615(n), s=sigma(n), u = (n+t)-s); ((u>0)&&(0==((t-n)%u))); };

A344705 a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 5, 8, 10, 11, 8, 13, 14, 15, 9, 17, 15, 19, 14, 21, 22, 23, 12, 24, 26, 23, 20, 29, 30, 31, 17, 33, 34, 35, 17, 37, 38, 39, 22, 41, 42, 43, 32, 39, 46, 47, 20, 48, 47, 51, 38, 53, 42, 55, 32, 57, 58, 59, 36, 61, 62, 55, 33, 65, 66, 67, 50, 69, 70, 71, 21, 73, 74, 71, 56, 77, 78, 79, 38, 68, 82

Offset: 1

Antti Karttunen, May 28 2021

First negative term occurs as a(1440) = -18.

Antti Karttunen, Table of n, a(n) for n = 1..16560
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.

Cf. A013661, A082020.

f[p_, e_] := (p^(e+1) - 1)/(p-1); a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, n * (Times @@ (1 + 1/fct[[;; , 1]]) + 1) - Times @@ f @@@ fct]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
A344705(n) = ((n + A001615(n)) - sigma(n));

a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).
a(n) = A033879(n) + A306927(n).
a(n) = n + A344753(n) - 2*A001065(n).
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 + 1 - Pi^2/6 = 0.874883... . - Amiram Eldar, Dec 08 2023

cp's OEIS Frontend

A344755 Numbers k such that A344753(k) is a multiple of A048250(k), and k is a multiple of A344753(k)/A048250(k).

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

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A344754 Numbers k such that A344753(k) is a multiple of A048250(k).

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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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A344997 a(n) = A173557(n) * A344753(n).

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A051709 a(n) = sigma(n) + phi(n) - 2n.

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