A286385 -id:A286385 - cp's OEIS frontend

A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 43, 1, 3, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 19, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 5, 7, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Terms a(n) larger than 1 and equal to A252748(n) occur at n = 6, 28, 69, 91, 496, ..., see A326134. See also A349753.

Records 1, 3, 43, 45, 2005, 79243, ... occur at n = 1, 6, 28, 360, 496, 8128, ...

Cf. A000203, A000360, A003961, A033879, A033880, A252748, A286385, A326134.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326060, A326062, A349753.

Array[GCD[#3 - #1, #3 - #2] & @@ {2 #, DivisorSigma[1, #], Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 78] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A252748(n) = (A003961(n) - (2*n));
A286385(n) = (A003961(n) - sigma(n));
A326057(n) = gcd(A252748(n), A286385(n));

a(n) = gcd(A252748(n), A286385(n)) = gcd(A003961(n) - 2n, A003961(n) - A000203(n)).

a(n) = gcd(A252748(n), A033879(n)) = gcd(A286385(n), A033879(n)). [Also A033880 can be used] - Antti Karttunen, May 06 2024

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)).

A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

It holds that a(n) <= A336836(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
The first 3 occurs at n = 19399380, the first 4 at n = 195534950863140268380. See A336389.
If x and y are relatively prime (i.e., gcd(x,y) = 1), then a(x*y) >= max(a(x),a(y)). Compare to a similar comment in A336915.

			For n = 120, sigma(120) = 360 >= 2*120, thus 120 is not deficient, and we get the next number by applying the prime shift, A003961(120) = 945, and sigma(945) = 1920 >= 945*2, so neither 945 is deficient, so we prime shift once again, and A003961(945) = 9625, which is deficient, as sigma(9625) = 14976 < 2*9625. Thus after two iteration steps we encounter a deficient number, and therefore a(120) = 2.

Cf. A003961, A005100, A005940, A023196, A046523, A047802, A071364, A108951, A286385, A294934, A336834, A337474, A337475.
Cf. A336389 (position of the first occurrence of a term >= n).
Differs from A294936 for the first time at n=120.
Cf. also A246271, A252459, A336836 and A336915 for similar iterations.

Array[-1 + Length@ NestWhileList[If[# == 1, 1, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]] &, #, DivisorSigma[1, #] >= 2 # &] &, 120] (* Michael De Vlieger, Aug 27 2020 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };

If A294934(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A003961(n)).
From Antti Karttunen, Aug 21-Sep 01 2020: (Start)
For all n >= 1,
a(A046523(n)) >= a(n).
a(A071364(n)) >= a(n).
a(A108951(n)) = A337474(n).
a(A025487(n)) = A337475(n).
(End)

A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 25, 26, 28, 33, 46, 55, 57, 69, 91, 93, 496, 1034, 1054, 1558, 2211, 2626, 4825, 8128, 11222, 12046, 12639, 28225, 32043, 68727, 89575, 970225, 1392386, 2245557, 8550146, 12371554, 16322559, 22799825, 33550336, 48980427, 51326726, 55037217, 60406599, 68258725, 142901438, 325422273, 342534446

Offset: 1

Antti Karttunen, Dec 12 2024

nonn

Numbers k such that A252748(k) divides A286385(k).

Conjecture: Apart from a(5)=6, this is a subsequence of A319630, i.e., for all terms k<>6, gcd(k, A003961(k)) = 1. See also A372562, A372566.

Amiram Eldar, Table of n, a(n) for n = 1..69 (terms below 10^11; terms 1..60 from Antti Karttunen)
Index entries for sequences related to prime indices in the factorization of n.
Index entries for sequences related to sigma(n).

Cf. A000203, A003961, A252748, A286385, A319630, A372562, A372566.
Positions of 0's in A378981.
Subsequence of A263837.
Subsequences: A000396, A048674, A348514, A326134, A349753 (odd terms of this sequence).
Cf. also A378983.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[k_] := Module[{fct = FactorInteger[k], m, s}, s = Times @@ f1 @@@ fct; m = Times @@ f2 @@@ fct; Divisible[m - s, m - 2*k]]; q[1] = True; Select[Range[10^5], q] (* Amiram Eldar, Dec 19 2024 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = { my(u=A003961(n)); ((u-sigma(n))%((2*n)-u)); };
isA378980(n) = !A378981(n);

A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).

Original entry on oeis.org

0, 1, 2, 11, 2, 36, 4, 85, 46, 58, 2, 324, 4, 120, 120, 575, 2, 693, 4, 566, 248, 172, 6, 2340, 94, 270, 788, 1176, 2, 1800, 6, 3661, 348, 358, 336, 5967, 4, 492, 548, 4210, 2, 3744, 4, 1820, 2490, 744, 6, 15372, 380, 2271, 720, 2826, 6, 11304, 392, 8760, 992, 946, 2, 15480, 6, 1232, 5164, 22631, 636, 5904, 4, 3866

Offset: 1

Antti Karttunen, Feb 22 2021

Cf. A000203, A000720, A001223, A003961, A003973, A286385, A341528, A341529, A341530.

Cf. Sequences A001359, A029710, A031924 give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.

Array[#2 DivisorSigma[1, #1] - #1 DivisorSigma[1, #2] & @@ {#, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 68] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));

a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).
For all primes p, a(p) = (q*(p+1)) - (p*(q+1)) = (pq + q) - (pq + p) = q - p = A001223(A000720(p)), where q = nextprime(p) = A003961(p).
And in general, a(p^e) = (q^e * (p^(e+1)-1)/(p-1)) - ((p^e) * (q^(e+1)-1)/(q-1)), where q = A003961(p).
Thus, a(p^2) = (p + 1)*q^2 - p^2*q - p^2,
      a(p^3) = (p^2 + p + 1)*q^3 - p^3*q^2 - p^3*q - p^3,
      a(p^4) = (p^3 + p^2 + p + 1)*q^4 - p^4*q^3 - p^4*q^2 - p^4*q - p^4,
      etc.

A326134 Numbers k such that A326057(k) is equal to A252748(k) and A252748(k) is not 1.

Original entry on oeis.org

6, 28, 69, 91, 496, 2211, 4825, 8128, 12639, 22799825, 33550336, 60406599, 68258725, 569173299, 794579511, 984210266, 2830283326, 8589869056, 10759889913, 80295059913, 85871289682

Offset: 1

Antti Karttunen, Jun 11 2019

No other terms below 3221225472.
Numbers k such that A252748(k) [= A003961(k) - 2*k] <> 1 (i.e., k is not in A348514), and A286385(k) [= A003961(k) - A000203(k)] = m*A252748(k) for some positive integer m. Note that this entails that k is nonabundant (A000203(k) <= 2*k) and primeshift-abundant (A252748(k) > 2), thus this is a subsequence of A341614. - revised Dec 13 2024
This is a subsequence of A378980, see further comments there. - Antti Karttunen, Dec 13 2024

			28 is a term as A252748(28) = 43 > 1 and A286385(28) = 43, which is a multiple of 43.
69 is a term as A252748(69) = 7 > 1 and A286385(69) = 49 is a multiple of 7.
91 is a term as A252748(91) = 5 > 1 and A286385(91) = 75 is a multiple of 5.

Cf. A000396 (subsequence), A003961, A252748, A286385, A326057, A337345, A337372, A341611, A348514, A379216, A379217.
Subsequence of the following sequences: A246282, A341614, A378980.
Odd terms form a subsequence of A349753.

Select[Range[10^5], And[#3 - #1 != 1, GCD[#3 - #1, #3 - #2] == #3 - #1] & @@ {2 #, DivisorSigma[1, #], Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
isA326134(n) = { my(s = A003961(n), t = (s-(2*n)), u = s-sigma(n)); ((1!=t)&&!(u%t)&&((u/t)>0)); };

a(18) from Antti Karttunen, Dec 14 2024

a(19)..a(21) from Antti Karttunen (from the b-file of A378980 computed by Amiram Eldar), Dec 20 2024

A337378 Numbers k for which A003961(k) > 2*sigma(k).

Original entry on oeis.org

16, 24, 27, 32, 36, 40, 45, 48, 49, 54, 56, 63, 64, 72, 80, 81, 84, 90, 96, 98, 99, 100, 104, 105, 108, 112, 117, 120, 125, 126, 128, 135, 140, 144, 147, 152, 153, 160, 162, 168, 171, 175, 176, 180, 184, 189, 192, 196, 198, 200, 207, 208, 210, 216, 224, 225, 234, 240, 243, 245, 248, 250, 252, 256, 264, 270, 272, 273

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Note that A003961(n) >= sigma(n) for all n. See A286385.

Subsequence of A246282 and of A337381.
Positions of negative terms in A377984, and in A378751.
Cf. A000203, A003961, A286385.
Cf. A337379 (complement), A337380 (characteristic function).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337378(n) = (A003961(n)>2*sigma(n));

A349753 Odd numbers k for which A003961(k)-2k divides A003961(k)-sigma(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, 3, 7, 25, 33, 55, 57, 69, 91, 93, 2211, 4825, 12639, 28225, 32043, 68727, 89575, 970225, 2245557, 16322559, 22799825, 48980427, 55037217, 60406599, 68258725, 325422273, 414690595, 569173299, 794579511, 10056372275, 10475647197, 10759889913, 11154517557

Offset: 1

Antti Karttunen, Dec 01 2021

nonn

Numbers k for which A326057(k) = gcd(A003961(k)-2k, A003961(k)-sigma(k)) is equal to abs(A252748(k)) = |A003961(k)-2k|.
The odd terms of A326134 form a subsequence of this sequence. Unlike in A326134, here we don't constrain the value of A252748(k) = A003961(k)-2k, thus allowing also values <= +1. Because of that, the odd terms of A048674 and A348514 are all included here, for example 57 and 68727 that occur in A348514, and 1, 3, 25, 33, 93, 970225, 325422273, 414690595 that occur in A048674.
Conjecture (1): This is a subsequence of A319630, in other words, for all terms k, gcd(k, A003961(k)) = 1.
Conjecture (2): Apart from 1, there are no common terms with A349169, which would imply that no odd perfect numbers exist.
None of the 36 initial terms is Zumkeller, in A083207, because all are deficient (in A005100). See also A337372. - Antti Karttunen, Dec 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..36 (terms below 10^11)
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A005100, A083207, A252748, A286385, A319630, A326057, A326134, A337372, A349169.
Cf. A048674, A348514.
Subsequence of A378980 (its odd terms).

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Divisible[(sn = s[n]) - DivisorSigma[1, n], sn - 2*n]; Select[Range[1, 10^6, 2], q] (* Amiram Eldar, Dec 04 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349753(n) = if(!(n%2), 0, my(s = A003961(n), t = (s-(2*n)), u = s-sigma(n)); !(u%t));

A336852 a(n) = sigma(A003961(n)) - sigma(n).

Original entry on oeis.org

0, 1, 2, 6, 2, 12, 4, 25, 18, 14, 2, 50, 4, 24, 24, 90, 2, 85, 4, 62, 40, 20, 6, 180, 26, 30, 116, 100, 2, 120, 6, 301, 36, 26, 48, 312, 4, 36, 52, 230, 2, 192, 4, 98, 170, 48, 6, 602, 76, 135, 48, 136, 6, 504, 40, 360, 64, 38, 2, 456, 6, 56, 268, 966, 60, 192, 4, 134, 84, 240, 2, 1045, 6, 54, 218, 172, 72, 264, 4, 782

Offset: 1

Antti Karttunen, Aug 05 2020

nonn

Inverse Möbius transform of A336853(n) = (A003961(n) - n).

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A003973, A286385, A336851, A336853.

Cf. A001105 (positions of odd terms), A001359 (positions of 2's).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336852(n) = (sigma(A003961(n)) - sigma(n));

PARI
```
A336852(n) = sumdiv(n,d,A003961(d)-d);
```

a(n) = Sum_{d|n} (A003961(d)-d).
a(n) = A003973(n) - A000203(n) = A000203(A003961(n)) - A000203(n).
a(n) = A336851(n) + A286385(n).

A337379 Numbers k for which A003961(k) < 2*sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 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, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 101, 102, 103

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Cf. A000203, A003961, A286385.
Cf. A337378 (complement).
Positions of zeros in A337380.
Cf. also A246281, A337382 (subsequences).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337379(n) = (A003961(n)<2*sigma(n));

cp's OEIS Frontend

A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

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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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A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

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A378980 Numbers k such that (A003961(k)-2*k) divides (A003961(k)-sigma(k)), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)*A003961(n)) - (n*sigma(A003961(n))).

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A326134 Numbers k such that A326057(k) is equal to A252748(k) and A252748(k) is not 1.

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A337378 Numbers k for which A003961(k) > 2*sigma(k).

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A349753 Odd numbers k for which A003961(k)-2k divides A003961(k)-sigma(k), where A003961 shifts the prime factorization one step towards larger primes, and sigma is the sum of divisors function.

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A341512 a(n) = A341529(n) - A341528(n) = (sigma(n)A003961(n)) - (nsigma(A003961(n))).