A325975 -id:A325975 - cp's OEIS frontend

A325977 a(n) = (1/2)*(A034460(n) + A325313(n)).

0, 1, 1, 0, 1, 6, 1, -2, -2, 8, 1, 4, 1, 10, 9, -6, 1, 3, 1, 4, 11, 14, 1, 0, -9, 16, -11, 4, 1, 42, 1, -14, 15, 20, 13, -5, 1, 22, 17, -4, 1, 54, 1, 4, -3, 26, 1, -8, -20, -2, 21, 4, 1, -6, 17, -8, 23, 32, 1, 36, 1, 34, -7, -30, 19, 78, 1, 4, 27, 74, 1, -21, 1, 40, -11, 4, 19, 90, 1, -20, -38, 44, 1, 44, 23, 46, 33, -16, 1, 36, 21, 4

Offset: 1

Antti Karttunen, Jun 02 2019

sign

Question: Are n = 1, 4, 24, 240, 349440 (A325963) the only positions of zeros in this sequence?

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

Cf. A033879, A034448, A034460, A048250, A306633, A325313, A325963, A325973, A325974, A325975, A325978, A325979, A325981.

Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #2) - 2 #1 + Times @@ (1 + #2[[;; , 1]]) & @@ {#, FactorInteger[#]}] &, 90] (* Michael De Vlieger, Jun 06 2019, after Giovanni Resta at A034448 and Amiram Eldar at A048250. *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);

a(n) = (1/2)*(A034460(n) + A325313(n)).
a(n) = A325973(n) - n.
a(n) = A325978(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) - 1)/4 = 0.0921081944... . - Amiram Eldar, Feb 22 2024

A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, -1, 3, 10, 11, 0, 13, 14, 15, -5, 17, 0, 19, 2, 21, 22, 23, -12, 10, 26, 3, 4, 29, 30, 31, -13, 33, 34, 35, -24, 37, 38, 39, -14, 41, 42, 43, 8, 9, 46, 47, -36, 21, 5, 51, 10, 53, -18, 55, -16, 57, 58, 59, -12, 61, 62, 15, -29, 65, 66, 67, 14, 69, 70, 71, -72, 73, 74, 15, 16, 77, 78, 79, -46, 3, 82, 83, -12, 85

Offset: 1

Antti Karttunen, Jun 02 2019

sign

Question: Are a(12) = 0 and a(18) = 0 the only zeros in this sequence?

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

Cf. A000203, A033879, A034448, A048146, A162296, A325314, A325814, A325973, A325974, A325975, A325977, A325979, A325981.

Cf. A002117, A013661.

Table[(1/2) If[n == 1, 2, 2 n - DivisorSigma[1, n] + Times @@ (1 + FactorInteger[n][[;; , 1]]) - DivisorSum[n, # &, ! CoprimeQ[#, n/#] &]], {n, 85}] (* Michael De Vlieger, Jun 06 2019 *)

A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);

a(n) = (1/2)*(A325314(n) + A325814(n)).
a(n) = n - A325974(n).
a(n) = A033879(n) + A325977(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/4 - zeta(2)*(1/2 - 1/(4*zeta(3))) = 0.2696411609... . - Amiram Eldar, Feb 22 2024

A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 9, 6, 0, 0, 12, 0, 0, 0, 21, 0, 18, 0, 18, 0, 0, 0, 36, 15, 0, 24, 24, 0, 0, 0, 45, 0, 0, 0, 60, 0, 0, 0, 54, 0, 0, 0, 36, 36, 0, 0, 84, 28, 45, 0, 42, 0, 72, 0, 72, 0, 0, 0, 72, 0, 0, 48, 93, 0, 0, 0, 54, 0, 0, 0, 144, 0, 0, 60, 60, 0, 0, 0, 126, 78, 0, 0, 96, 0, 0, 0, 108, 0, 108, 0, 72, 0, 0, 0, 180, 0, 84, 72

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, non-unitary divisors are 2, 3, 6, 12 and 18 so A048146(36) = 2+3+6+12+18 = 41, while the nonsquarefree divisors are 4, 9, 12, 18 and 36, so A162296(36) = 4+9+12+18+36 = 79, thus a(36) = (41+79)/2 = 60.

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

Cf. A000203, A005117, A034448, A048146, A162296, A325973, A325975, A325978.

Cf. A002117, A013661.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325974(n) = ((A048146(n)+A162296(n))/2);

a(n) = (1/2)*(A048146(n) + A162296(n)).
a(n) = A000203(n) - A325973(n).
a(n) = n - A325978(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)*(1/2 - 1/(4*zeta(3))) - 1/4 = 0.2303588390... . - Amiram Eldar, Feb 22 2024

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)).

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));

A325979 Odd numbers k for which gcd(A325977(k), A325978(k)) is equal to abs(A325978(k)).

Original entry on oeis.org

1, 3465, 72981, 78651, 80937, 152703, 199341, 201771, 241605, 253287, 492507, 631881, 880821, 933147, 985473, 1063755, 1209285, 1244133, 1292445, 1313235, 1327095, 1347885, 1360881, 1451835, 1521135, 1597365, 1620375, 1814373, 2015475, 2664585, 6058233, 6676371, 8186751, 11119761, 17496243, 18379935, 28695627

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

Provided that A325977(k) and A325978(k) are never zero for the same k, these are odd numbers k such that A325978(k) is not zero and divides A325977(k).

Of the first 281 terms, only a(5) = 80937, a(51) = 86086881, a(175) = 43024468437, and a(262) = 564858541521 are in A228058. - Updated Jul 20 2025

Giovanni Resta, Table of n, a(n) for n = 1..281 (terms < 10^12; first 65 terms from Antti Karttunen)
Index entries for sequences where any odd perfect numbers must occur.

Cf. A228058, A325975, A325977, A325978, A325981.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
isA325979(n) = ((n%2)&&(A325975(n)==abs(A325978(n))));
\\ Or alternatively as:
isA325979(n) = if(!(n%2),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(y&&!(x%y)));

A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 5, 2, 10, 4, 12, 2, 3, 1, 16, 1, 18, 2, 1, 2, 22, 4, 19, 2, 14, 4, 28, 6, 30, 1, 3, 2, 1, 1, 36, 2, 1, 2, 40, 2, 42, 4, 3, 2, 46, 4, 41, 1, 3, 2, 52, 2, 1, 8, 1, 2, 58, 12, 60, 2, 1, 1, 1, 6, 66, 2, 3, 2, 70, 1, 72, 2, 2, 4, 1, 2, 78, 2, 41, 2, 82, 4, 1, 2, 3, 4, 88, 6, 7, 4, 1, 2, 5, 4, 96, 1, 3, 1, 100, 6, 102, 2, 3

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A032742, A326065, A326066, A326067, A326068.

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

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326066(n) = (sigma(n) - sigma(A032742(n)));
A326067(n) = (A326066(n) - n);
A326065(n) = sigma(A032742(n));
A326068(n) = (n - A326065(n));
A326069(n) = gcd(A326067(n), A326068(n));

a(n) = gcd(A326067(n), A326068(n)) = gcd(A326066(n) - n, n - A326065(n)).

A326071 Numbers k such that A325977(k) has a different sign as A325978(k).

Original entry on oeis.org

1, 4, 9, 12, 18, 24, 25, 27, 45, 49, 50, 60, 63, 75, 81, 84, 90, 98, 99, 117, 120, 121, 125, 126, 132, 140, 147, 150, 153, 156, 168, 169, 171, 175, 180, 198, 204, 207, 228, 234, 240, 242, 243, 245, 261, 264, 270, 275, 276, 279, 280, 289, 294, 297, 306, 312, 325, 333, 338, 342, 343, 348, 350, 351, 361, 363, 369, 372

Offset: 1

Antti Karttunen, Jun 07 2019

nonn

See comments in A326070.

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

Cf. A325963, A325975, A325977, A325978, A325981.

Cf. A326070 (complement), A326072 (characteristic function).

isA326071(n) = (sign(A325977(n))!=sign(A325978(n))); \\ For other needed code, see under A325975.

A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 1, 18, 2, 2, 4, 22, 2, 1, 2, 2, 26, 28, 4, 30, 1, 6, 2, 2, 1, 36, 4, 2, 2, 40, 4, 42, 2, 6, 4, 46, 2, 1, 1, 6, 2, 52, 4, 2, 2, 2, 2, 58, 2, 60, 4, 2, 1, 2, 4, 66, 2, 6, 4, 70, 1, 72, 2, 2, 2, 2, 4, 78, 26, 1, 2, 82, 2, 2, 4, 6, 2, 88, 2, 14, 2, 2, 4, 10, 2, 96, 1, 6, 1, 100, 4, 102, 2, 6

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

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

Cf. A000203, A020639, A046666, A326146, A326148.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326073, A326129, A326130, A326140, A326144.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326147(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n);

a(n) = gcd(n-A020639(n), A000203(n)-A020639(n)-n).

For n > 1, a(n) = gcd(A046666(n), A326146(n)).

A326070 Numbers k such that A325977(k) has the same sign as A325978(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 100

Offset: 1

Antti Karttunen, Jun 07 2019

nonn

Here A325977(k) = A325973(k) - k and A325978(k) = k - A325974(k), where A325973(k) is the average of {sum of unitary divisors} and {sum of squarefree divisors} = (1/2) * (A034448(k) + A048250(k)) while A325974(k) is the average of {sum of non-unitary divisors} and {sum of nonsquarefree divisors} = (1/2)*(A048146(k) + A162296(k)). Only if signs of A325977(k) and A325978(k) are equal can their difference A325978(k) - A325977(k) = (k - A325974(k)) - (A325973(k) - k) = 2k - (A325973(k) + A325974(k)) = 2k - A000203(k) = A033879(k) be zero, which happens when k is a perfect number (in A000396).

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

Cf. A000203, A033879, A325963, A325973, A325974, A325975, A325977, A325978, A325981.

Cf. A326071 (complement), A326072, A000396 (a subsequence).

isA326070(n) = (sign(A325977(n))==sign(A325978(n))); \\ For other needed code, see under A325975.

cp's OEIS Frontend

A325977 a(n) = (1/2)*(A034460(n) + A325313(n)).

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A325978 a(n) = (1/2)*(A325314(n) + A325814(n)).

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A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

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

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

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A325979 Odd numbers k for which gcd(A325977(k), A325978(k)) is equal to abs(A325978(k)).

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A326069 a(n) = gcd((sigma(n) - sigma(A032742(n))) - n, n - sigma(A032742(n))), where A032742 gives the largest proper divisor of n.

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A326071 Numbers k such that A325977(k) has a different sign as A325978(k).

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A326147 a(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.