A349164 -id:A349164 - cp's OEIS frontend

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1

Offset: 1

Antti Karttunen, Mar 20 2021

nonn

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 15, 105, 3003, 3465, 13923, 45045, 264537, 459459, 745875, 1541475, 5221125, 8729721, 10790325, 14171625, 29288025, 34563375, 57034575, 71430975, 99201375, 109643625, 144729585, 205016175, 255835125, 295708875, 356080725, 399242025, 419159475, 449323875, 928602675, 939495375, 1083656925, 1941623775, 1962350685, 2083228875

Offset: 1

Antti Karttunen, Nov 10 2021

Numbers k such that A348990(k) [= k/gcd(k, A003961(k))] is equal to A348992(k), which is the odd part of A349162(k), thus all terms must be odd, as A348990 preserves the parity of its argument.
Equally, numbers k for which gcd(A064987(k), A191002(k)) is equal to A000265(gcd(A064987(k), A341529(k))).
Also odd numbers k for which A348993(k) = A319627(k).
Odd terms of A336702 are given by the intersection of this sequence and A349174.
Conjectures:
  (1) After 1, all terms are multiples of 3. (Why?)
  (2) After 1, all terms are in A104210, in other words, for all n > 1, gcd(a(n), A003961(a(n))) > 1. Note that if we encountered a term k with gcd(k, A003961(k)) = 1, then we would have discovered an odd multiperfect number.
  (3) Apart from 1, 15, 105, 3003, 13923, 264537, all other terms are abundant. [These apparently are also the only terms that are not Zumkeller, A083207. Note added Dec 05 2024]
  (4) After 1, all terms are in A248150. (Cf. also A386430).
  (5) After 1, all terms are in A348748.
  (6) Apart from 1, there are no common terms with A349753.
Note: If any of the last four conjectures could be proved, it would refute the existence of odd perfect numbers at once. Note that it seems that gcd(sigma(k), A003961(k)) < k, for all k except these four: 1, 2, 20, 160.
Questions:
  (1) For any term x here, can 2*x be in A349745? (Partial answer: at least x should be in A191218 and should not be a multiple of 3). Would this then imply that x is an odd perfect number? (Which could explain the points (1) and (4) in above, assuming the nonexistence of opn's).

Cf. A000203, A003961, A007949, A064989, A083207, A104210, A161942, A191002, A191218, A228058, A319627, A322361, A326134, A329963, A342671, A348748, A348990, A348992, A348993, A349162, A349164, A349174.
Cf. also A349745, A349746, A349753.
Subsequence of A349749, and of A386430.

Select[Range[10^6], #1/GCD[#1, #3] == #2/(2^IntegerExponent[#2, 2]*GCD[#2, #3]) & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A000265(n) = (n >> valuation(n, 2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349169(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == A000265(s)*gcd(n,u)); }; \\ (Program simplified Nov 30 2021)

For all n >= 1, A007949(A000203(a(n))) = A007949(a(n)). [sigma preserves the 3-adic valuation of the terms of this sequence] - Antti Karttunen, Nov 29 2021

Name changed and comment section rewritten by Antti Karttunen, Nov 29 2021

A349161 a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 5, 9, 7, 5, 11, 9, 25, 7, 13, 45, 17, 11, 35, 81, 19, 25, 23, 3, 55, 13, 29, 9, 49, 17, 25, 99, 31, 35, 37, 27, 65, 19, 77, 225, 41, 23, 85, 21, 43, 55, 47, 39, 175, 29, 53, 405, 121, 49, 95, 153, 59, 25, 91, 99, 23, 31, 61, 15, 67, 37, 275, 729, 17, 65, 71, 19, 145, 77, 73, 45, 79, 41, 245, 207, 143, 85, 83

Offset: 1

Antti Karttunen, Nov 09 2021

Numerator of ratio A003961(n) / A000203(n). Sequence A349162 gives the denominators.
Numerator of ratio A003961(n) / A161942(n). Sequence A348992 gives the denominators.
Both ratios are multiplicative because the constituent sequences are.
No 1's occur as terms after a(2), because for n > 2, sigma(n) < A003961(n). (See A286385).

Cf. A000203, A003961, A161942, A286385, A319626, A319627, A326042, A336702, A342671, A349163, A349164, A349627, A349628.

Cf. A348992, A349162 (denominators).

Array[#2/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 79] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349161(n) = { my(u=A003961(n)); (u/gcd(u,sigma(n))); };

from math import prod, gcd
from sympy import nextprime, factorint
def A349161(n):
    f = factorint(n).items()
    a = prod(nextprime(p)**e for p, e in f)
    b = prod((p**(e+1)-1)//(p-1) for p, e in f)
    return a//gcd(a,b) # Chai Wah Wu, Mar 17 2023

a(n) = A003961(n) / A342671(n) = A003961(n) / gcd(A000203(n), A003961(n)).

a(n) = A003961(A349164(n)).

A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 5, 13, 6, 12, 28, 14, 8, 24, 31, 18, 13, 20, 2, 32, 12, 24, 4, 31, 14, 8, 56, 30, 24, 32, 7, 48, 18, 48, 91, 38, 20, 56, 10, 42, 32, 44, 28, 78, 24, 48, 124, 57, 31, 72, 98, 54, 8, 72, 40, 16, 30, 60, 8, 62, 32, 104, 127, 12, 48, 68, 14, 96, 48, 72, 13, 74, 38, 124, 140, 96, 56, 80, 62, 121, 42

Offset: 1

Antti Karttunen, Nov 09 2021

Denominator of ratio A003961(n) / A000203(n).
Small values are rare, but are not limited to the beginning. For example in range 1 .. 2^25, a(n) = 4 at n = 3, 6, 24, 792, 2720, 122944, 31307472.
Question: Would it be possible to prove that a(n) > 1 for all n > 2?
Obviously, 1's may occur only on squares & twice squares (A028982). See also comments in A350072. - Antti Karttunen, Feb 16 2022

Cf. A000203, A003961, A028982 (positions of odd terms), A319630, A336702, A342671, A348992 (the odd part), A348993, A349161 (numerators), A349163, A349164, A349627, A349628, A350072 [= a(n^2)].

Cf. also A349745, A351551, A351554.

Array[#1/GCD[##] & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &, 82] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349162(n) = { my(s=sigma(n)); (s/gcd(s,A003961(n))); };

a(n) = A000203(n) / A342671(n) = A000203(n) / gcd(A000203(n), A003961(n)).

A349163 a(n) = A064989(gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 10, 1, 2, 1, 6, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 10, 1, 2, 1, 1, 5, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 11, 5, 1, 2, 1, 2, 1

Offset: 1

Antti Karttunen, Nov 09 2021

nonn

Cf. A000203, A003961, A342671, A349161, A349162, A349165 (positions of 1's), A349166 (of terms > 1).

Cf. A349144 and A349168 [positions where a(n) is / is not relatively prime with A349164(n) = n/a(n)].

Array[Times @@ Map[If[#1 <= 2, 1, NextPrime[#1, -1]]^#2 & @@ # &, FactorInteger@ GCD[##]] & @@ {DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &, 105] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A349163(n) = A064989(gcd(sigma(n),A003961(n)));

a(n) = A064989(A342671(n)).

a(n) = n / A349164(n).

A349174 Odd numbers k for which gcd(k, A003961(k)) is equal to gcd(sigma(k), A003961(k)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169

Offset: 1

Antti Karttunen, Nov 10 2021

nonn

Odd numbers k for which A322361(k) = A342671(k).
Odd numbers k for which A348994(k) = A349161(k).
Odd numbers k such that A319626(k) = A349164(k).
Odd terms of A336702 form a subsequence of this sequence. See also A349169.
Ratio of odd numbers residing in this sequence, vs. in A349175 seems to slowly decrease, but still apparently stays > 2 for a long time. E.g., for range 2 .. 2^28, it is 95302074/38915653 = 2.4489...

Cf. A000203, A003961, A319626, A322361, A336702, A342671, A348994, A349161, A349164, A349169.
Cf. A349175 (complement among the odd numbers).
Union of A349176 and A349177.

Select[Range[1, 169, 2], GCD[#1, #3] == GCD[#2, #3] & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349174(n) = if(!(n%2),0,my(u=A003961(n)); gcd(u,sigma(n))==gcd(u,n));

A349176 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

135, 285, 435, 455, 855, 885, 1185, 1287, 1305, 1335, 1425, 1435, 1485, 1635, 2235, 2275, 2295, 2655, 2685, 2905, 2985, 3105, 3135, 3185, 3311, 3395, 3435, 3555, 3585, 4005, 4035, 4185, 4425, 4785, 4865, 4905, 4995, 5385, 5685, 5805, 5835, 5845, 5925, 6135, 6237, 6335, 6345, 6585, 6675, 6735, 7125, 7155, 7175, 7185

Offset: 1

Antti Karttunen, Nov 11 2021

nonn

			For n = 135 = 3^3 * 5, sigma(135) = 240 = 2^4 * 3 * 5, A003961(135) = 5^3 * 7 = 875, and gcd(135,875) = gcd(240,875) = 5, which is larger than 1, therefore 135 is included in the sequence.

Intersection of A104210 and A349174, or equally, intersection of A349166 and A349174.
Subsequence of A372567.
Cf. A000203, A003961, A319626, A348994, A349161, A349164, A349169.

Select[Range[1, 7200, 2], And[#1/#2 == #1/#3, #2 > 1] & @@ {#3, GCD[#1, #3], GCD[#2, #3]} & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));

A349168 Numbers k such that sigma(k) and A003961(k) share a prime power divisor that is not a unitary divisor of A003961(k). Here A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

8, 20, 24, 27, 32, 40, 44, 54, 56, 60, 72, 80, 88, 92, 96, 100, 104, 108, 116, 120, 128, 132, 135, 140, 152, 160, 164, 168, 171, 176, 180, 184, 188, 189, 196, 200, 216, 224, 232, 236, 240, 248, 260, 261, 264, 270, 272, 276, 280, 288, 296, 297, 300, 308, 312, 320, 325, 328, 332, 342, 344, 348, 351, 352, 360, 368, 376

Offset: 1

Antti Karttunen, Nov 10 2021

nonn

Numbers k such that A342671(k) [= gcd(sigma(k), A003961(k))] and A349161(k) [= A003961(k)/A342671(k)] share a prime factor.

Numbers k for which A349163(k) and A349164(k) are not relatively prime.

			For n = 8 = 2^3, sigma(8) = 15 = 3*5, while A003961(8) = 3^3 = 27. These share the prime power divisor 3, which however is not a unitary divisor of 27, therefore 8 is included in this sequence.
For n = 32 = 2^5, sigma(32) = 63 = 3^2 * 7, while A003961(32) = 3^5 = 243. These share the prime power divisor 3^2, which however is not a unitary divisor of 243, therefore 32 is included.
For n = 40 = 2^3 * 5, sigma(40) = 90 = 2 * 3^2 * 5, while A003961(40) = 3^3 * 7 = 189. These share the prime power divisor 3^2, which however is not a unitary divisor of 189, therefore 40 is included.

Cf. A000203, A003961, A342671, A349161, A349163, A349164.

Subsequence of A349166.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349168(n) = { my(u=A003961(n), x=gcd(u,sigma(n))); (1!=gcd(x,u/x)); };

A387164 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)), and that satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

117, 153, 333, 369, 425, 477, 549, 637, 657, 845, 873, 909, 925, 1017, 1053, 1233, 1325, 1377, 1413, 1421, 1445, 1525, 1557, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2097, 2169, 2225, 2313, 2493, 2525, 2529, 2597, 2637, 2725, 2817, 2825, 2853, 2989, 2997, 3033, 3177, 3321, 3357, 3425, 3509, 3573, 3577, 3609, 3725

Offset: 1

Antti Karttunen, Aug 28 2025

Terms k of A228058 for which A322361(k) = A342671(k), or equally, such that A319626(k) = A349164(k).

Intersection of A228058 and A349174.
Union of A387166 and A387167.
Differs from its subsequence A387167 for the first time at n=201, where a(201) = 14157, while A387167(201) = 14225.
Cf. A000203, A003961, A319626, A322361, A342671, A349164.
Cf. also A371082.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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));
isA349174(n) = if(!(n%2), 0, my(u=A003961(n)); gcd(u, sigma(n))==gcd(u, n));
isA387164(n) = (isA228058(n) && isA349174(n));

A349144 Numbers k for which A342671(k) [= gcd(sigma(k), A003961(k))] and A349161(k) [= A003961(k)/A342671(k)] are relatively prime, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95

Offset: 1

Antti Karttunen, Nov 11 2021

nonn

Numbers k for which A349163(k) and A349164(k) are coprime, i.e., k such that A349163(k) and A349164(k) are unitary divisors of k.

Cf. A000203, A003961, A342671, A349161, A349163, A349164.
Complement of A349168.
Cf. A349165 (subsequence).

Select[Range[95], GCD[#2, #1/#2] == 1 & @@ {#2, #2/GCD[##]} & @@ {DivisorSigma[1, #], If[# == 1, 1, Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]]} &] (* Michael De Vlieger, Nov 11 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349144(n) = { my(u=A003961(n), x=gcd(u,sigma(n))); (1==gcd(x,u/x)); };

cp's OEIS Frontend

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

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A349169 Numbers k such that k * gcd(sigma(k), A003961(k)) is equal to the odd part of {sigma(k) * gcd(k, A003961(k))}, where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A349161 a(n) = A003961(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A349162 a(n) = sigma(n) / gcd(sigma(n), A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors function.

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A349163 a(n) = A064989(gcd(sigma(n), A003961(n))), where A003961 shifts the prime factorization of n one step towards larger primes, while A064989 shifts it back towards smaller primes, and sigma is the sum of divisors function.

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A349174 Odd numbers k for which gcd(k, A003961(k)) is equal to gcd(sigma(k), A003961(k)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

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A349176 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

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A349168 Numbers k such that sigma(k) and A003961(k) share a prime power divisor that is not a unitary divisor of A003961(k). Here A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

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