A325981 -id:A325981 - cp's OEIS frontend

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

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)).

A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

Original entry on oeis.org

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 10, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 18, 48, 54, 48, 31, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 40, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 34, 84, 144, 68, 72, 96, 144, 72, 51, 74, 114, 64, 80, 96, 168, 80, 60, 43, 126

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

This is not multiplicative: a(4) = 4, a(9) = 7, but a(36) = 31, not 28. However, the function acts multiplicatively on certain subsequences of natural numbers, like for example when restricted to A048107, where this sequence coincides with A326043.

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, unitary divisors are 1, 4, 9 and 36, so A034448(36) = 1+4+9+36 = 50, while the squarefree divisors are 1, 2, 3 and 6, so A048250(36) = 1+2+3+6 = 12, thus a(36) = (50+12)/2 = 31.
For n = 495, its divisors are 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495. Of these, unitary are 1, 5, 9, 11, 45, 55, 99, 495, whose sum is A034448(495) = 720, while the squarefree divisors are 1, 3, 5, 11, 15, 33, 55, 165, and their sum is A048250(495) = 288. Thus a(495) = (720+288)/2 = 504. Also for 495, whose prime factorization is 3^2 * 5^1 * 11^1 this can be computed faster as the average of ((3^2)+1)*(5+1)*(11+1) and (3+1)*(5+1)*(11+1), thus (1/2)*(3+(3^2)+2)*(5+1)*(11+1) = 504.

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

Cf. A000203, A034448, A048107, A048250, A052548, A289521, A306633, A325974, A325975, A325977, A325978, A325981, A326043.

Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #) + Times @@ (1 + #[[;; , 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
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325973(n) = ((A034448(n)+A048250(n))/2);

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

a(n) = (1/2) * (A034448(n) + A048250(n)).
a(n) = A000203(n) - A325974(n).
a(n) = n + A325977(n).
a(A048107(n)) = A326043(A048107(n)).
For n >= 1, a(2^n) = A052548(n-1) = 2^(n-1) + 2.
For n >= 1, a(3^n) = A289521(n) = (3^n + 5)/2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) + 1)/4 = 0.5921081944... . - Amiram Eldar, Feb 22 2024

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

A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 188, 198, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268, 275, 276, 279, 284, 292, 294, 306, 308

Offset: 1

Reinhard Zumkeller, Jul 18 2002

nonn

Numbers n such that A001222(n) - A001221(n) = 1 and A001221(n)>1.
Numbers with one or more 1's, exactly one 2 and no 3's or higher in their prime exponents. - Antti Karttunen, Sep 19 2019
From Salvador Cerdá, Mar 08 2016: (Start)
12!+1 = 13^2 * 2834329 is in this sequence.
23!+1 = 47^2 * 79 * 148139754736864591 is also in this sequence. (End)
The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} 1/(p*(p+1)) (A271971). - Amiram Eldar, Nov 09 2020

			a(14) = 84 = 7*3*2^2; the following numbers are not terms: 36=6^2, as it is a square; 54=2*3^3, as it is not cubefree; 42=2*3*7, as there is no squared prime; 72=2*6^2, as 72 has two squared prime divisors: 2^2 and 3^2.

Robert Israel, Table of n, a(n) for n = 1..10000 ( 1..100 from Paolo P. Lava)

Cf. A001221, A001222, A054753 (subsequence), A271971, A325981 (conjectured subsequence).

Subsequence of: A004709, A048107, A060687, A067259.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [$2..floor(N^(1/2))]):
SF:= select(numtheory:-issqrfree, [$2..N/4]):
S:= {seq(op(map(p -> p^2*t, select(s -> igcd(s,t)=1 and s^2*t <= N, Primes))), t = SF)}:
sort(convert(S,list)); # Robert Israel, Mar 08 2016

Select[Range@ 308, And[PrimeNu@ # > 1, PrimeOmega@ # - PrimeNu@ # == 1] &] (* Michael De Vlieger, Mar 09 2016 *)

isok(n) = (omega(n) > 1) && (bigomega(n) - omega(n) == 1); \\ Michel Marcus, Jul 16 2015

A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

117, 775, 10309, 56347, 88723, 2896363, 9597529, 12326221, 12654079, 25774633, 29817121, 63455131, 105100903, 203822581, 261019543, 296765173, 422857021, 573332713, 782481673, 900952687, 1129152721, 3350861677, 3703086229, 7395290407, 9347001661, 9350506057

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Nineteen initial terms factored:
   n       a(n)   factorization   A060681(a(n))/A318505(a(n))
      117 =   3^2 *    13,           (3)
      775 =   5^2 *    31,          (10)
    10309 =  13^2 *    61,          (39)
    56347 =  29^2 *    67,          (58)
    88723 =  17^2 *   307,         (136)
  2896363 =  41^2 *  1723,         (820)
  9597529 =  73^2 *  1801,        (1314)
 12326221 =  59^2 *  3541,        (1711)
 12654079 = 113^2 *   991,         (904)
 25774633 =  71^2 *  5113,        (2485)
 29817121 =  97^2 *  3169,        (2328)
 63455131 =  89^2 *  8011,        (3916)
105100903 = 101^2 * 10303,        (5050)
203822581 = 157^2 *  8269,        (6123)
261019543 = 349^2 *  2143,        (2094)
296765173 = 131^2 * 17293,        (8515)
422857021 = 233^2 *  7789,        (6757)
573332713 = 331^2 *  5233,        (4965)
782481673 = 167^2 * 28057,       (13861).
Note how the quotient (in the rightmost column) seems always to be a multiple of non-unitary prime factor and less than the unitary prime factor.
For p, q prime, if p^2+p+1 = kq and k+1|p-1, then p^2*q is in this sequence. - Charlie Neder, Jun 09 2019

Subsequence of A326063.
Cf. A032742, A060681, A246282, A318505.
Cf. also A228058, A325981, A326131, A326141.

Select[Range[15, 10^6 + 1, 2], And[! PrimePowerQ@ #1, Mod[#1 - #2, #2 - #3] == 0] & @@ {#, DivisorSigma[1, #] - #, Divisors[#][[-2]]} &] (* Michael De Vlieger, Jun 22 2019 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326064(n) = if((n%2)&&(2!=isprimepower(n)), my(s=A032742(n), t=sigma(n)-s); (gcd(t-n, n-A032742(n)) == t-n), 0);

More terms from Amiram Eldar, Dec 24 2020

A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 15370304, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056

Offset: 1

Antti Karttunen, Jun 09 2019

No further terms below 2^31.
See also comments in A326133.
The quotients A000120(k)/(sigma(k)-A005187(k)) for these terms are: 1, 1, 5, 1, 3, 1, 5, 9, 2, 2, 1, 2, 2. Ones occur at the positions of perfect numbers.
a(18) > 10^11. - Amiram Eldar, Jan 03 2021

			110 is "1101110" in binary, thus A000120(110) = 5. Sigma(110) = 216, while A005187(110) = 215, thus as 5 = 5*(216-215), 110 is included in this sequence.

Intersection of A326132 and A326133, also of A326132 and A326138.
Cf. A000120, A000396 (subsequence), A005187, A294898, A295296, A295297, A295298, A295299, A326130.
Cf. also A325981, A326141.

q[n_] := Module[{bw = DigitCount[n, 2, 1], ab = DivisorSigma[1, n] - 2*n, sum}, (sum = ab + bw) > 0 && Divisible[bw, sum]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 03 2021 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326131(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == t); };

a(14)-a(17) from Amiram Eldar, Jan 03 2021

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)));

A326137 Numbers with at least five distinct prime factors that satisfy Euler's criterion (A228058) for odd perfect numbers.

Original entry on oeis.org

17342325, 22678425, 31674825, 38686725, 41420925, 45090045, 49358925, 51740325, 54033525, 54695025, 67660425, 68939325, 70703325, 75818925, 76392225, 77106645, 78217425, 81375525, 92400525, 96316605, 97383825, 98750925, 99147825, 102284325, 107694405, 113656725, 115420725, 117890325, 118728225, 120536325, 127766925

Offset: 1

Antti Karttunen, Jun 12 2019

nonn

P. P. Nielsen's 2006 paper shows that any odd perfect number must have at least nine distinct prime factors, thus if such numbers exist at all, they must occur in this sequence.

I conjecture that it is eventually possible to find an easy proof that this sequence has no common terms with A325981, and/or several other sequences (A326064, A326074, A326141, A326148, etc.) listed under index entry "sequences where odd perfect numbers must occur", thus settling the question about the existence of such numbers.

Antti Karttunen, Table of n, a(n) for n = 1..1032; all terms < 2^31
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A228058.

Cf. A001221, A325981, A326064, A326074, A326141, A326148.

isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
isA326137(n) = ((omega(n)>=5)&&isA228058(n));

A326141 Odd numbers n for which A318879(n) is not zero and A318879(n) divides A318878(n); odd numbers such that A326140(n) = A318879(n).

Original entry on oeis.org

105, 195, 4785, 22515, 56865, 228285, 237315, 484245, 671853, 1838145, 1946955, 3446895, 4522695, 12955245, 37730865, 52475055, 53568885, 87612975

Offset: 1

Antti Karttunen, Jun 09 2019

Not a subsequence of A036798, even though many terms are members.

Questions: Are all terms multiples of three? Multiples of 3^(2k+1) but not of 3^(2k)? Are any of the terms included in A228058, A326137?

Index entries for sequences where any odd perfect numbers must occur

Cf. A033879, A083254, A036798, A228058, A318878, A318879, A326137, A326140.

Cf. also A325981, A326131.

isA326141(n) = if(!(n%2),0, my(t=0, u=0); fordiv(n,d, d -= 2*eulerphi(d); if(d<0, t -= d, u += d)); (gcd(t,u)==u));

cp's OEIS Frontend

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

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A325973 Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

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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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A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square.

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A326064 Odd composite numbers n, not squares of primes, such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

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A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

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