A341529 -id:A341529 - cp's OEIS frontend

A336849 a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

1, 3, 5, 9, 7, 5, 11, 27, 25, 21, 13, 15, 17, 11, 35, 81, 19, 75, 23, 63, 55, 39, 29, 9, 49, 17, 125, 33, 31, 35, 37, 243, 65, 57, 77, 225, 41, 23, 85, 189, 43, 55, 47, 9, 175, 29, 53, 135, 121, 49, 19, 17, 59, 125, 13, 99, 115, 93, 61, 105, 67, 111, 275, 729, 119, 65, 71, 171, 29, 77, 73, 135, 79, 41, 245, 69, 143

Offset: 1

Antti Karttunen, Aug 06 2020

If there are no more 1's in this sequence after the initial one, then there are no odd terms of A007691 (multiply perfect numbers) larger than one.

Denominator of the ratio A003973(n) / A003961(n), also denominator of the ratio (A341528(n)/A341529(n)) / (n / sigma(n)). - Antti Karttunen, Feb 16 2021

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. A003961, A003973, A007691, A017666, A336848, A336850, A341528, A341529, A341530.

Cf. A341525 (numerators).

f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := (gn = g[n])/GCD[gn, DivisorSigma[1, gn]]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
\\ Or alternatively as:
A336849(n) = { my(u=A003961(n)); denominator(sigma(u)/u); };

a(n) = A003961(n) / A336850(n) = A003961(n) / gcd(A003961(n), A003973(n)).

a(n) = A017666(A003961(n)).

A349745 Numbers k for which k * gcd(sigma(k), A003961(k)) is equal to 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, 120, 216, 672, 2464, 22176, 228480, 523776, 640640, 837760, 5581440, 5765760, 7539840, 12999168, 19603584, 33860736, 38342304, 71344000, 95472000, 102136320, 197308800, 220093440, 345080736, 459818240, 807009280, 975576960, 1476304896, 1510831360, 1773584640

Offset: 1

Antti Karttunen, Nov 29 2021

nonn

Numbers k for which k * A342671(k) = A000203(k) * A322361(k).
Numbers k such that gcd(A064987(k), A191002(k)) = gcd(A064987(k), A341529(k)).
Obviously, all odd terms in this sequence must be squares.
All the terms k of A005820 that satisfy A007949(k) < A007814(k) [i.e., whose 3-adic valuation is strictly less than their 2-adic valuation] are also terms of this sequence. Incidentally, the first six known terms of A005820 satisfy this condition, while on the other hand, any hypothetical odd 3-perfect number would be excluded from this sequence. Also, as a corollary, any hypothetical 3-perfect numbers of the form 4u+2 must not be multiples of 3 if they are to appear here. Similarly for any k which occurs in A349169, for 2*k to occur in this sequence, it shouldn't be a multiple of 3 and k should also be a term of A191218. See question 2 and its partial answer in A349169.
From Antti Karttunen, Feb 13-20 2022: (Start)
Question: Are all terms/2 (A351548) abundant, from n > 1 onward?
Note that of the 65 known 5-multiperfect numbers, all others except these three 1245087725796543283200, 1940351499647188992000, 4010059765937523916800 are also included in this sequence. The three exceptions are distinguished by the fact that their 3 and 5-adic valuations are equal. In 62 others the former is larger.
If k satisfying the condition were of the form 4u+2, then it should be one of the terms of A191218 doubled as only then both k and sigma(k) are of the form 4u+2, with equal 2-adic valuations for both. More precisely, one of the terms of A351538.
(End)

Amiram Eldar, Table of n, a(n) for n = 1..52 (terms below 10^11)
Antti Karttunen, Ratio A342671(n)/A322361(n) plotted with OEIS Plot2 tool
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A005101, A005820, A007814, A007949, A046060, A064987, A191002, A191218, A248150, A322361, A341529, A342671, A347870, A351534, A351536, A351538.
Cf. also A349169, A349746, A351458, A351549 for other variants.
Subsequence of A351554 and also of its subsequence A351551.
Cf. A351459 (subsequence, intersection with A351458), A351548 (terms halved).

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := NextPrime[p]^e; q[1] = True; q[n_] := n * GCD[(s = Times @@ f1 @@@ (f = FactorInteger[n])), (r = Times @@ f2 @@@ f)] == s*GCD[n, r]; Select[Range[10^6], q] (* Amiram Eldar, Nov 29 2021 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349745(n) = { my(s=sigma(n),u=A003961(n)); (n*gcd(s,u) == (s*gcd(n,u))); };

For all n >= 1, A007814(A000203(a(n))) = A007814(a(n)). [sigma preserves the 2-adic valuation of the terms of this sequence]

A341530 a(n) = gcd(nsigma(A003961(n)), sigma(n)A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 36, 4, 5, 1, 2, 2, 36, 2, 24, 120, 1, 2, 9, 4, 2, 8, 4, 6, 180, 1, 18, 4, 168, 2, 360, 2, 7, 12, 2, 336, 117, 2, 12, 4, 10, 2, 288, 4, 364, 30, 24, 6, 36, 19, 3, 360, 18, 6, 72, 56, 120, 16, 2, 2, 360, 2, 16, 4, 1, 12, 144, 4, 2, 60, 336, 2, 45, 2, 6, 10, 12, 264, 72, 4, 2, 11, 2, 6, 2016, 4, 12, 24

Offset: 1

Antti Karttunen, Feb 16 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191
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, A028982 (positions of odd terms), A341512, A341526, A341527, A341528, A341529, A342670.

Cf. A342674 (same sequence applied onto prime shift array A246278).

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341530(n) = { my(t=A003961(n), s=sigma(t)); gcd((n*s), sigma(n)*t); };

a(n) = gcd(A341528(n), A341529(n)) = gcd(n*A003973(n), A000203(n)*A003961(n)).

a(n) = gcd(A341512(n), A341528(n)) = gcd(A341512(n), A341529(n)) = A342670(A003961(n)). - Antti Karttunen, Mar 24 2021

A341527 Denominator of ratio nsigma(A003961(n)) / sigma(n)A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

1, 9, 10, 63, 21, 5, 22, 81, 325, 189, 78, 35, 119, 33, 7, 2511, 171, 325, 115, 1323, 220, 351, 116, 45, 1519, 119, 1250, 33, 465, 21, 592, 2187, 260, 1539, 11, 175, 779, 345, 1190, 1701, 903, 55, 517, 27, 455, 261, 424, 1395, 363, 4557, 19, 833, 531, 625, 117, 297, 575, 4185, 1830, 147, 2077, 666, 7150, 92583, 833, 195

Offset: 1

Antti Karttunen, Feb 16 2021

Denominator of ratio A341528(n)/A341529(n). A341526 gives the numerator, see comments there.

Cf. A000203, A003961, A003973, A017665, A017666, A336849, A341525, A341528, A341529, A341530.
Cf. A341526 (numerators).
Cf. A341627 (same sequence as applied onto prime shift array A246278).

f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Denominator[n*DivisorSigma[1, (gn = g[n])]/(DivisorSigma[1, n] * gn)]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341527(n) = { my(s=A003961(n)); denominator((sigma(s)*n)/(sigma(n)*s)); };

a(n) = A341529(n) / A341530(n) = A341529(n) / gcd(A341528(n), A341529(n)).

For all n > 1, a(n) > A341526(n).

A337381 Numbers k for which A003973(k) >= 2*sigma(k).

Original entry on oeis.org

6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Note that A003973(n) >= sigma(n) for all n. See A336852.
Like the abundancy index (ratio A000203(n)/n), and ratio A003961(n)/n, the ratio A003973(n)/sigma(n) is also multiplicative and > 1 for all n > 1. Thus if the number has a proper divisor that is in this sequence, then the number itself is also. See A337543 for those terms included here, but which have no proper divisor in this sequence. - Antti Karttunen, Aug 31 2020
All terms are in A246282 because A341528(n) < A341529(n) for all n > 1. - Antti Karttunen, Feb 22 2021

Cf. A000203, A003961, A003973, A336852, A337541, A337542, A341528, A341529.
Cf. A337382 (complement), A337383 (characteristic function).
Subsequences: A337378, A337384, A337386, A337543 (primitive terms).
Subsequence of A246282.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
isA337381(n) = (A003973(n)>=2*sigma(n));

A341526 Numerator of ratio nsigma(A003961(n)) / sigma(n)A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

1, 8, 9, 52, 20, 4, 21, 64, 279, 160, 77, 26, 117, 28, 6, 1936, 170, 248, 114, 1040, 189, 308, 115, 32, 1425, 104, 1053, 26, 464, 16, 589, 1664, 231, 1360, 10, 124, 777, 304, 1053, 1280, 902, 42, 516, 22, 372, 230, 423, 968, 343, 3800, 17, 676, 530, 468, 110, 224, 513, 3712, 1829, 104, 2074, 589, 5859, 69952, 780, 154

Offset: 1

Antti Karttunen, Feb 16 2021

Like the ratios sigma(n)/n, A003973(n)/A003961(n) and A003961(n)/n, also the ratio r(n) = A341528(n)/A341529(n) is multiplicative: if gcd(x,y) = 1, r(x*y) = r(x)*r(y).

Cf. A000203, A003961, A003973, A017665, A017666, A336849, A341525, A341528, A341529, A341530.
Cf. A341527 (denominators).
Cf. A341626 (same sequence as applied onto prime shift array A246278).

f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := Numerator[n*DivisorSigma[1, (gn = g[n])]/(DivisorSigma[1, n] * gn)]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341526(n) = { my(s=A003961(n)); numerator((sigma(s)*n)/(sigma(n)*s)); };

a(n) = A341528(n) / A341530(n) = A341528(n) / gcd(A341528(n), A341529(n)).

For all n > 1, a(n) < A341527(n).

A341627 Square array A(n,k) = A341527(A246278(n,k)), read by falling antidiagonals; denominators of the columnwise first quotients of A341605/A341606.

Original entry on oeis.org

9, 63, 10, 5, 325, 21, 81, 7, 1519, 22, 189, 1250, 11, 363, 78, 35, 220, 13377, 52, 22477, 119, 33, 455, 117, 66550, 34, 52887, 171, 2511, 260, 4774, 374, 804102, 133, 110827, 115, 325, 6875, 833, 2574, 6669, 584647, 69, 201549, 116, 1323, 3038, 1875181, 627, 205751, 13685, 1790199, 58, 465073, 465

Offset: 1

Antti Karttunen, Feb 16 2021

			The top left corner of the array:
   n =  1       2    3        4      5        6      7             8        9
  2n =  2       4    6        8     10       12     14            16       18
----+--------------------------------------------------------------------------
  1 |   9,     63,   5,      81,   189,      35,    33,         2511,     325,
  2 |  10,    325,   7,    1250,   220,     455,   260,         6875,    3038,
  3 |  21,   1519,  11,   13377,   117,    4774,   833,      1875181,    1089,
  4 |  22,    363,  52,   66550,   374,    2574,   627,     41009441,    6422,
  5 |  78,  22477,  34,  804102,  6669,  205751,  1495,    459974905,  317322,
  6 | 119,  52887, 133,  584647, 13685,  531981, 13804,   2584223261,  775789,
  7 | 171, 110827,  69, 1790199,  9918,  670795, 15903,  11564815861, 1813941,
  8 | 115, 201549,  58, 2202227, 17825, 1016508, 34040,  38495207801, 2325365,
  9 | 116, 465073,  93, 5170468, 68672, 7457205, 90364, 206922836641, 3348124,
etc.

Cf. A246278, A341605, A341606, A341527.

Cf. A341626 (numerators), A341628 (the greatest prime factor of these terms).

up_to = 105;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A341627sq(row,col) = A341527(A246278sq(row,col));
A341627list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341627sq(col,(a-(col-1))))); (v); };
v341627 = A341627list(up_to);
A341627(n) = v341627[n];

A(n,k) = A341527(A246278(n,k)), where A341527(n) is the denominator of the ratio (n * sigma(A003961(n))) / (sigma(n) * A003961(n)), i.e., of A341528(n)/A341529(n).

For all n, k, A(n,k) > A341626(n, k).

A341525 Numerator of A003973(n) / A003961(n).

Original entry on oeis.org

1, 4, 6, 13, 8, 8, 12, 40, 31, 32, 14, 26, 18, 16, 48, 121, 20, 124, 24, 104, 72, 56, 30, 16, 57, 24, 156, 52, 32, 64, 38, 364, 84, 80, 96, 403, 42, 32, 108, 320, 44, 96, 48, 14, 248, 40, 54, 242, 133, 76, 24, 26, 60, 208, 16, 160, 144, 128, 62, 208, 68, 152, 372, 1093, 144, 112, 72, 260, 36, 128, 74, 248, 80, 56

Offset: 1

Antti Karttunen, Feb 16 2021

Also numerator of the ratio (A341528(n)/A341529(n)) / (n/sigma(n)).

Cf. A000203, A000290 (positions of odd terms), A003961, A003973, A017665, A151800, A336850, A341526, A341527, A341528, A341529, A341530.

Cf. A336849 (denominators).

f[p_, e_] := (p^(e + 1) - 1)/((p - 1)*p^e); g[p_, e_] := f[NextPrime[p], e]; a[1] = 1; a[n_] := Numerator[Times @@ g @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341525(n) = { my(u=A003961(n), s=sigma(u)); (s/gcd(u, s)); };

a(n) = A017665(A003961(n)).
a(n) = A003973(n) / A336850(n) = A003973(n) / gcd(A003961(n), A003973(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A341525(k)/A336849(k) = 1 / Product_{p prime} (1 - 1/(p*nextprime(p))) = 1.3766054560..., where nextprime(p) = A151800(p). - Amiram Eldar, Dec 28 2024

A342661 a(n) = n * sigma(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma gives the sum of the divisors of its argument.

Original entry on oeis.org

1, 2, 9, 4, 20, 18, 42, 8, 63, 40, 88, 36, 156, 84, 180, 16, 238, 126, 342, 80, 378, 176, 460, 72, 325, 312, 405, 168, 696, 360, 930, 32, 792, 476, 840, 252, 1184, 684, 1404, 160, 1558, 756, 1806, 352, 1260, 920, 2068, 144, 1519, 650, 2142, 624, 2544, 810, 1760, 336, 3078, 1392, 3186, 720, 3660, 1860, 2646, 64, 3120

Offset: 1

Antti Karttunen, Mar 23 2021

Cf. A000203, A064989, A151799, A326041, A341528, A341529 [= a(A003961(n))], A342662, A342663, A342664.

f[p_, e_] := If[p == 2, 2^e, Module[{q = NextPrime[p, -1]}, p^e*(q^(e + 1) - 1)/(q - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

A064989(n) = { my(f); 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) };
A326041(n) = sigma(A064989(n));
A342661(n) = (n*A326041(n));

Multiplicative with a(p^e) = (p^e) * (q^(e+1)-1)/(q-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.
a(n) = n * A326041(n) = n * A000203(A064989(n)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/9) * Product_{p prime > 2} (p^3/((p+1)*(p^2-prevprime(p)))) = 0.1815217..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

A342662 a(n) = sigma(n) * A064989(n), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 3, 8, 7, 18, 24, 40, 15, 52, 54, 84, 56, 154, 120, 144, 31, 234, 156, 340, 126, 320, 252, 456, 120, 279, 462, 320, 280, 690, 432, 928, 63, 672, 702, 720, 364, 1178, 1020, 1232, 270, 1554, 960, 1804, 588, 936, 1368, 2064, 248, 1425, 837, 1872, 1078, 2538, 960, 1512, 600, 2720, 2070, 3180, 1008, 3658, 2784, 2080, 127

Offset: 1

Antti Karttunen, Mar 23 2021

Cf. A000203, A064989, A151799, A341528 [= a(A003961(n))], A341529, A342661, A342663, A342664.

f[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]*(p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

A064989(n) = { my(f); 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) };
A342662(n) = (sigma(n)*A064989(n));

Multiplicative with a(p^e) = q^e * (p^(e+1)-1)/(p-1), where q = 1 for p = 2, and for odd primes p, q = A151799(p), i.e., the previous prime.
a(n) = A000203(n) * A064989(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (16/63) * Product_{p prime > 2} p^4*(p-1)/((p^3-prevprime(p))*(p^2-prevprime(p))) = 0.1935405..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022

cp's OEIS Frontend

A336849 a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

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A349745 Numbers k for which k * gcd(sigma(k), A003961(k)) is equal to 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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A341530 a(n) = gcd(n*sigma(A003961(n)), sigma(n)*A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors of n.

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A341527 Denominator of ratio n*sigma(A003961(n)) / sigma(n)*A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.

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A337381 Numbers k for which A003973(k) >= 2*sigma(k).

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A341526 Numerator of ratio n*sigma(A003961(n)) / sigma(n)*A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.

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A341627 Square array A(n,k) = A341527(A246278(n,k)), read by falling antidiagonals; denominators of the columnwise first quotients of A341605/A341606.

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A341525 Numerator of A003973(n) / A003961(n).

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A341530 a(n) = gcd(nsigma(A003961(n)), sigma(n)A003961(n)), where A003961 shifts the prime factorization of n one step towards larger primes, and sigma is the sum of divisors of n.

A341527 Denominator of ratio nsigma(A003961(n)) / sigma(n)A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.

A341526 Numerator of ratio nsigma(A003961(n)) / sigma(n)A003961(n), where sigma is the sum of divisors of n, and A003961 shifts the prime factorization of n one step towards larger primes.