A325973 -id:A325973 - cp's OEIS frontend

A348984 a(n) = gcd(sigma(n), A325973(n)), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 8, 8, 30, 72, 32, 9, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 24, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 4, although a(4) = 1 and a(27) = 8.

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

Cf. A000203, A048250, A034448, A325973, A325974, A348985, A348986.
Differs from A348047 for the first time at n=108, where a(108) = 4, while A348047(108) = 8.
Cf. also A348733, A348946.

f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := GCD[s[n], DivisorSigma[1, n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A348984(n) = gcd(sigma(n), A325973(n));

a(n) = gcd(A000203(n), A325973(n)).
a(n) = gcd(A000203(n), A325974(n)) = gcd(A325973(n), A325974(n)).
a(n) = A000203(n) / A348985(n) = A325973(n) / A348986(n).

A348985 Numerator of ratio sigma(n) / A325973(n), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 5, 7, 1, 1, 1, 7, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 7, 1, 57, 13

Offset: 1

Antti Karttunen, Nov 06 2021

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 70 <> 35 = 7*5 = a(4)*(27).

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

Cf. A000203, A048250, A034448, A325973, A325974, A348984, A348986 (denominators).
Differs from A348048 for the first time at n=108, where a(108) = 70, while A348048(108) = 35.
Cf. also A348948.

f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := Numerator[DivisorSigma[1, n]/s[n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A348985(n) = { my(s=sigma(n)); (s/gcd(s, A325973(n))); };

a(n) = A000203(n) / A348984(n) = sigma(n) / gcd(sigma(n), A325973(n)).

A348986 Denominator of ratio sigma(n) / A325973(n), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 2, 7, 1, 1, 4, 1, 1, 1, 10, 1, 7, 1, 4, 1, 1, 1, 2, 16, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 31, 1, 1, 1, 2, 1, 1, 1, 4, 7, 1, 1, 10, 29, 16, 1, 4, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 7, 34, 1, 1, 1, 4, 1, 1, 1, 17, 1, 1, 16, 4, 1, 1, 1, 10, 43, 1, 1, 4, 1, 1, 1, 2, 1, 7, 1, 4, 1, 1, 1, 2, 1, 29, 7, 74, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 06 2021

This is not multiplicative: a(4) = 4 and a(9) = 7, but a(36) = 31, not 28.

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

Cf. A000203, A048250, A034448, A325973, A325974, A348984, A348985 (numerators).

Cf. also A348947.

f1[p_, e_] := p + 1; f2[p_, e_] := p^e + 1; s[1] = 1; s[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f)/2; a[n_] := Denominator[DivisorSigma[1, n]/s[n]]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A348986(n) = { my(am=A325973(n)); (am/gcd(sigma(n),am)); };

a(n) = A325973(n) / A348984(n) = A325973(n) / gcd(A000203(n), A325973(n)).

A378433 Dirichlet inverse of A325973, where A325973 is the arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, -3, -4, 5, -6, 12, -8, -9, 9, 18, -12, -20, -14, 24, 24, 15, -18, -27, -20, -30, 32, 36, -24, 36, 20, 42, -24, -40, -30, -72, -32, -27, 48, 54, 48, 42, -38, 60, 56, 54, -42, -96, -44, -60, -54, 72, -48, -60, 35, -60, 72, -70, -54, 72, 72, 72, 80, 90, -60, 120, -62, 96, -72, 45, 84, -144, -68, -90, 96, -144, -72, -72

Offset: 1

Antti Karttunen, Nov 26 2024

sign

Apparently differs from A378434 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

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

Cf. A034448, A048111, A048250, A325973, A378434.

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
memoA378433 = Map();
A378433(n) = if(1==n,1,my(v); if(mapisdefined(memoA378433,n,&v), v, v = -sumdiv(n,d,if(dA325973(n/d)*A378433(d),0)); mapput(memoA378433,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA325973(n/d) * a(d).

A359431 a(n) = A325973(n) - A326043(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 6, 0, 8, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 10, 0, 0, 0, 14, 0

Offset: 1

Antti Karttunen, Jan 04 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A048107 (positions of 0's), A325973, A326043, A359471.

Cf. also comments in A325981.

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A326043(n) = if(1==n, n, my(f = factor(n)); prod(i=1, #f~, floor((1/f[i, 2]) * ((f[i, 2]-1) + (((f[i, 1]^(1+f[i, 2])) - 1)/(f[i, 1]-1))))));
A359431(n) = (A325973(n)-A326043(n));

A325975 a(n) = gcd(A325977(n), A325978(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

See comments in A325979 and A325981.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034460, A048107, A325313, A325314, A325814, A325973, A325974, A325977, A325978, A325979, A325981.

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

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325313(n) = (A048250(n) - n);
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
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325977(n) = ((A034460(n)+A325313(n))/2);
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
\\ Or alternatively, as:
A325975(n) = (1/2)*gcd((A034460(n)+A325313(n)),(A325814(n)+A325314(n)));

a(n) = gcd(A325977(n), A325978(n)).

a(n) = (1/2)*gcd(A034460(n)+A325313(n), A325814(n)+A325314(n)).

A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

Original entry on oeis.org

45, 495, 585, 765, 855, 1305, 18837, 21525, 31635, 38295, 45315, 50445, 51255, 60435, 63495, 68085, 77265, 96615, 1403115, 2446353, 3411975, 3999465, 4091745, 4233537, 4287255, 4631319, 10813425, 10967085, 11490345, 15578199, 16143309, 16329645, 16633071, 17179515, 17311203, 17355915, 21159075, 21933975, 22579725

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

Provided that A325977 and A325978 are never zero on same n, these are odd composite numbers n such that A325977(n) is not zero and divides A325978(n).
Based on the first 147 terms it seems that this sequence is a subsequence of A072357, that is each term has exactly one prime factor with exponent 2, with one or more other prime factors that are all unitary (i.e., each term satisfies A001222(x) - A001221(x) = 1). On the other hand, it has been proved that no odd perfect number, if such numbers exist at all, can have such a factorization (see A326137 and a link to P. P. Nielsen's paper there).
Nineteen initial terms factorize as:
= 3^2 * 5^1,
= 3^2 * 5^1 * 11^1,
= 3^2 * 5^1 * 13^1,
= 3^2 * 5^1 * 17^1,
= 3^2 * 5^1 * 19^1,
= 3^2 * 5^1 * 29^1,
= 3^2 * 7^1 * 13^1 * 23^1,
= 3^1 * 5^2 * 7^1 * 41^1,
= 3^2 * 5^1 * 19^1 * 37^1,
= 3^2 * 5^1 * 23^1 * 37^1,
= 3^2 * 5^1 * 19^1 * 53^1,
= 3^2 * 5^1 * 19^1 * 59^1,
= 3^2 * 5^1 * 17^1 * 67^1,
= 3^2 * 5^1 * 17^1 * 79^1,
= 3^2 * 5^1 * 17^1 * 83^1,
= 3^2 * 5^1 * 17^1 * 89^1,
= 3^2 * 5^1 * 17^1 * 101^1,
= 3^2 * 5^1 * 19^1 * 113^1,
  1403115 = 3^1 * 5^1 * 7^2 * 23^1 * 83^1,
and the 62nd term as a(62) = 2919199437 = 3^2 * 7^1 * 11^1 * 43^1 * 163^1 * 601^1.
If we select a subsequence of terms for which the quotient A325978(n)/A325977(n) is positive, then we are left with the following numbers: 495, 585, 31635, 38295, 45315, 51255, 60435, 63495, 1403115, 3999465, etc. which is a subsequence of A326070.

Antti Karttunen (terms 1-62) & Giovanni Resta, Table of n, a(n) for n = 1..147
Index entries for sequences where any odd perfect numbers must occur

Cf. A072357, A228058, A325973, A325974, A325975, A325977, A325978, A325979, A326064, A326070, A326071, A326137.

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));
isA325981(n) = ((n%2)&&!isprime(n)&&(A325975(n)==abs(A325977(n))));
\\ Or alternatively as:
isA325981(n) = if(!(n%2)||isprime(n),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(x&&!(y%x)));

A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86

Offset: 1

Labos Elemer

nonn

Numbers with at most one 2 and no 3s or higher in their prime exponents. - Charles R Greathouse IV, Aug 25 2016

A disjoint union of A005117 and A060687. The asymptotic density of this sequence is (6/Pi^2) * (1 + Sum_{p prime} 1/(p*(p+1))) = A059956 * (1 + A179119) = A059956 + A271971 = 0.8086828238... - Amiram Eldar, Nov 07 2020

			n = 420 = 2*2*3*5*7, 4 distinct prime factors, 24 divisors of which 16 are unitary and 8 are not; ud(n) > nud(n) and 2^(4+1) = 32 is larger than d, the number of divisors.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Complement of A048108.
A072357 is a subsequence.
Cf. A000005, A001221, A005117, A034444, A048105, A059956, A060687, A179119, A271971, A325973.

Select[Range[500], 2^(1 + PrimeNu[#]) > DivisorSigma[0, #] &] (* G. C. Greubel, May 05 2017 *)

is(n)=my(f=factor(n)[, 2], t); for(i=1, #f, if(f[i]>1, if(t||f[i]>2, return(0), t=1))); 1 \\ Charles R Greathouse IV, Sep 17 2015

is(n)=n==1 || factorback(factor(n)[,2])<3 \\ Charles R Greathouse IV, Aug 25 2016

Numbers for which 2^(r(n)+1) > d(n), where r = A001221, d = A000005.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A348984 a(n) = gcd(sigma(n), A325973(n)), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

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A348985 Numerator of ratio sigma(n) / A325973(n), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

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A348986 Denominator of ratio sigma(n) / A325973(n), where A325973 is the arithmetic mean of {sum of squarefree divisors} and {sum of unitary divisors}.

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A378433 Dirichlet inverse of A325973, where A325973 is the arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}.

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A359431 a(n) = A325973(n) - A326043(n).

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A325975 a(n) = gcd(A325977(n), A325978(n)).

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A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

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A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

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