A326144 -id:A326144 - cp's OEIS frontend

A336555 Numbers k for which A326144(k) <> A336566(k).

45, 63, 75, 99, 117, 126, 144, 147, 150, 153, 171, 175, 198, 207, 216, 242, 245, 250, 252, 261, 275, 279, 294, 300, 315, 325, 333, 342, 350, 363, 369, 387, 396, 405, 414, 423, 425, 475, 477, 495, 504, 507, 525, 531, 539, 549, 550, 558, 567, 575, 585, 588, 600, 603, 605, 612, 630, 637, 639, 657, 675, 684, 693, 700, 711

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Question: Is A228058 a subsequence of this one?

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A228058, A326144, A336566.

\\ Needs also code from A326144 and A336566:
isA336555(n) = (A326144(n) != A336566(n));

A336646 a(n) = n - A326144(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 39, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 61, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 73, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007947, A066503, A326143, A326144.
Cf. A326145 (positions where coincides with A007947).
Cf. A336555 (positions where differs from A336647).
Cf. also A336645, A336647.

A007947(n) = factorback(factorint(n)[, 1]);
A066503(n) = (n - A007947(n));
A326143(n) = (sigma(n)-A007947(n)-n);
A326144(n) = gcd(A066503(n), A326143(n));
A336646(n) = (n - A326144(n));

A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 4, 12, 2, 1, 1, 8, 1, 3, 1, 5, 2, 1, 4, 1, 5, 1, 24, 28, 6, 15, 1, 1, 1, 1, 1, 36, 2, 1, 1, 40, 2, 3, 4, 4, 10, 1, 4, 1, 7, 15, 3, 4, 2, 19, 4, 1, 1, 1, 8, 60, 2, 1, 1, 1, 6, 3, 1, 1, 2, 35, 1, 72, 1, 1, 12, 1, 2, 3, 1, 1, 1, 1, 4, 1, 2, 1, 4, 8, 27, 5, 8, 29, 2, 7, 60, 48, 1, 1, 1, 100, 6, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

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

Cf. A000203, A008833, A326039, A326040, A326044, A326045.

Cf. also A009194, A325385, A325813, A325975, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140, A326144.

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326040(n) = (sigma(n)-A008833(sigma(n)));
A326046(n) = gcd(n-A326039(n), A326040(n)-n);

a(n) = gcd(A326044(n), A326045(n)) = gcd(n-A326039(n), A326040(n)-n).

A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing n.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 6, 1, 5, 1, 10, 4, 12, 1, 2, 1, 16, 3, 18, 2, 10, 1, 22, 4, 19, 5, 2, 24, 28, 1, 30, 1, 2, 1, 2, 19, 36, 1, 2, 2, 40, 1, 42, 4, 12, 5, 46, 4, 41, 1, 10, 6, 52, 3, 2, 4, 2, 1, 58, 8, 60, 1, 2, 1, 2, 1, 66, 2, 2, 1, 70, 3, 72, 1, 2, 12, 2, 1, 78, 2, 41, 1, 82, 8, 2, 5, 2, 4, 88, 27, 10, 8, 2, 1, 2, 20, 96, 1, 6

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Composite numbers n such that a(n) = A326055(n) start as: 6, 28, 336, 496, 792, 8128, 31968, 3606912, ...

Nonsquare odd numbers n such that a(n) = abs(A326054(n)) start as: 21, 153, 301, 697, 1333, 1909, 1917, 2041, 3901, 4753, 24601, 24957, 26977, 29161, 29637, 56953, 67077, 96361, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326053, A326054, A326055.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326057, A326060, A326062, A326129, A326130, A326140, A326144.

A008833(n) = (n/core(n));
A326053(n) = (sigma(n)-A008833(n));
A326054(n) = (A326053(n)-n);
A326055(n) = (n-A008833(n));
A326056(n) = gcd(A326054(n), A326055(n));

a(n) = gcd(A326054(n), A326055(n)) = gcd((A000203(n)-A008833(n))-n, n-A008833(n)).

A326129 a(n) = gcd(A326127(n), A326128(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 1, 12, 4, 6, 1, 16, 1, 18, 1, 10, 8, 22, 6, 1, 10, 2, 21, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 1, 4, 20, 46, 1, 1, 1, 30, 3, 52, 12, 38, 2, 34, 26, 58, 3, 60, 28, 2, 1, 46, 12, 66, 1, 42, 4, 70, 1, 72, 34, 2, 3, 58, 12, 78, 1, 1, 38, 82, 7, 62, 40, 54, 2, 88, 2, 70, 1, 58, 44, 70, 30

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Question: Are there any other numbers than those in A000396 that satisfy a(k) = A326128(k)?

See also comments in A336641, where all such k should reside. - Antti Karttunen, Jul 29 2020

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

Cf. A000396, A007913, A326126, A326127, A326128.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326130, A326140, A326144, A336641, A336645.

A326127(n) = (sigma(n)-core(n)-n);
A326128(n) = (n-core(n));
A326129(n) = gcd(A326127(n), A326128(n));

a(n) = n - A336645(n). - Antti Karttunen, Jul 29 2020

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 4, 5, 1, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 1, 4, 4, 1, 2, 1, 1, 2, 3, 4, 4, 1, 1, 2, 2, 1, 4, 5, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 3, 1, 2, 3, 2, 4, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 4, 2, 5, 4, 1, 1, 2, 2, 3, 1, 2, 1, 4, 4, 1, 1, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Cf. A000120, A000203, A005187, A033879, A326131.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326140, A326144.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A326130(n) = gcd(hammingweight(n), A294898(n));

A326130(n) = gcd(hammingweight(n), sigma(n)-A005187(n));

a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), A000203(n)-A005187(n)).

A326143 a(n) = A326142(n) - n, where A326142 gives the sum of all other divisors of n except its largest squarefree divisor.

Original entry on oeis.org

-1, -1, -2, 1, -4, 0, -6, 5, 1, -2, -10, 10, -12, -4, -6, 13, -16, 15, -18, 12, -10, -8, -22, 30, 1, -10, 10, 14, -28, 12, -30, 29, -18, -14, -22, 49, -36, -16, -22, 40, -40, 12, -42, 18, 18, -20, -46, 70, 1, 33, -30, 20, -52, 60, -38, 50, -34, -26, -58, 78, -60, -28, 20, 61, -46, 12, -66, 24, -42, 4, -70, 117, -72, -34, 34, 26, -58, 12

Offset: 1

Antti Karttunen, Jun 09 2019

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000203, A001065, A007947, A033879, A066503, A326142, A326144.
Cf. also A326054, A326127.
Cf. A013661, A065463.

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; a[n_] := DivisorSigma[1, n] - rad[n] - n; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A326143(n) = (sigma(n)-A007947(n)-n);

a(n) = A326142(n) - n = (A000203(n)-A007947(n)) - n = A001065(n) - A007947(n).
a(n) = A066503(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065463 - 1 = -0.0595081... . - Amiram Eldar, Dec 05 2023

A326140 a(n) = gcd(A318878(n), A318879(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 2, 12, 2, 6, 1, 16, 1, 18, 2, 10, 2, 22, 2, 19, 2, 14, 6, 28, 6, 30, 1, 18, 2, 22, 1, 36, 2, 22, 2, 40, 2, 42, 2, 12, 2, 46, 2, 41, 1, 30, 6, 52, 2, 38, 2, 34, 2, 58, 6, 60, 2, 22, 1, 46, 6, 66, 2, 42, 2, 70, 1, 72, 2, 26, 6, 58, 2, 78, 2, 41, 2, 82, 2, 62, 2, 54, 2, 88, 6, 70, 2, 58, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

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

Cf. A033879, A083254, A318878, A318879, A326141.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326144.

A326140(n) = { my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); gcd(t,u); };

A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);
A318879(n) = sumdiv(n,d,d=d-(2*eulerphi(d)); (d>0)*d);
A326140(n) = gcd(A318878(n), A318879(n));

A336566 a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 6, 1, 10, 2, 14, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 2, 3, 20, 46, 14, 1, 1, 30, 2, 52, 12, 38, 2, 34, 26, 58, 6, 60, 28, 1, 1, 46, 12, 66, 2, 42, 4, 70, 3, 72, 34, 1, 2, 58, 12, 78, 2, 1, 38, 82, 14, 62, 40, 54, 2, 88

Offset: 1

Antti Karttunen, Jul 27 2020

nonn

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

Cf. A057723, A308135, A336563, A336564, A336565.

Differs from A326144 at the positions given by A336555, for the first time at n=45, where a(45) = 3, while A326144(45) = 6.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336563(n) = (A057723(n)-n);
A336564(n) = (n - A308135(n));
A336566(n) = gcd(A336563(n), A336564(n));

a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(n));

A326145 Numbers n for which n - A007947(n) is equal to gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Original entry on oeis.org

6, 28, 496, 936, 1638, 8128, 33550336

Offset: 1

Antti Karttunen, Jun 09 2019

Numbers n such that either A066503(n) and A326143(n) are both zero or A066503(n) is not zero and divides A326143(n).
Question: Are there any odd terms?
No other terms < 2^31.

Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A000396 (a subsequence), A007947, A066503, A228058, A326142, A326143, A326144.

A007947(n) = factorback(factorint(n)[, 1]);
isA326145(n) = { my(b=A007947(n), t=n-b, u = (sigma(n)-b)-n); (gcd(t,u)==t); };
\\ Or alternatively as:
isA326145(n) = { my(t=A326143(n), u=A066503(n)); ((!u && !t)||(u && !(t%u))); };

cp's OEIS Frontend

A336555 Numbers k for which A326144(k) <> A336566(k).

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A336646 a(n) = n - A326144(n).

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A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

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A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing n.

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A326129 a(n) = gcd(A326127(n), A326128(n)).

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A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

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A326143 a(n) = A326142(n) - n, where A326142 gives the sum of all other divisors of n except its largest squarefree divisor.

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Mathematica

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A326140 a(n) = gcd(A318878(n), A318879(n)).

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A336566 a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(n)).

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