A341528 -id:A341528 - cp's OEIS frontend

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

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

A341628 Square array A(n,k) = A006530(A341527(A246278(n,k))), read by falling antidiagonals.

Original entry on oeis.org

3, 7, 5, 5, 13, 7, 3, 7, 31, 11, 7, 5, 11, 11, 13, 7, 11, 13, 13, 19, 17, 11, 13, 13, 11, 17, 61, 19, 31, 13, 31, 17, 61, 19, 307, 23, 13, 11, 17, 13, 19, 17, 23, 127, 29, 7, 31, 71, 19, 19, 23, 29, 29, 79, 31, 13, 13, 11, 2801, 23, 61, 29, 181, 31, 67, 37, 5, 17, 31, 19, 3221, 29, 307, 31, 53, 37, 331, 41

Offset: 1

Antti Karttunen, Feb 16 2021

			The top left corner of the array:
   n=   1     2   3     4   5     6   7        8     9    10  11    12  13    14
  2n=   2     4   6     8  10    12  14       16    18    20  22    24  26    28
-----+---------------------------------------------------------------------------
   1 |  3,    7,  5,    3,  7,    7, 11,      31,   13,    7, 13,    5, 17,   11,
   2 |  5,   13,  7,    5, 11,   13, 13,      11,   31,   13, 17,    7, 19,   13,
   3 |  7,   31, 11,   13, 13,   31, 17,      71,   11,   31, 19,   13, 23,   31,
   4 | 11,   11, 13,   11, 17,   13, 19,    2801,   19,   17, 23,   13, 29,   19,
   5 | 13,   19, 17,   61, 19,   19, 23,    3221,   61,   19, 29,   61, 31,   23,
   6 | 17,   61, 19,   17, 23,   61, 29,   30941,  307,   61, 31,   19, 37,   61,
   7 | 19,  307, 23,   29, 29,  307, 31,   88741,  127,  307, 37,   29, 41,  307,
   8 | 23,  127, 29,  181, 31,  127, 37,     911,   79,  127, 41,  181, 43,  127,
   9 | 29,   79, 31,   53, 37,   79, 41,  292561,   67,   79, 43,   53, 47,   79,
  10 | 31,   67, 37,  421, 41,   67, 43,  732541,  331,   67, 47,  421, 53,   67,
  11 | 37,  331, 41,   37, 43,  331, 47,   17351,   67,  331, 53,   41, 59,  331,
  12 | 41,   67, 43,  137, 47,   67, 53,    4271, 1723,   67, 59,  137, 61,   67,
  13 | 43, 1723, 47,   43, 53, 1723, 59,  579281,  631, 1723, 61,   47, 67, 1723,
  14 | 47,  631, 53,   47, 59,  631, 61, 3500201,   61,  631, 67,   53, 71,  631,
  15 | 53,   61, 59,   53, 61,   61, 67,   14621,  409,   61, 71,   59, 73,   67,
  16 | 59,  409, 61,  281, 67,  409, 71,    5581, 3541,  409, 73,  281, 79,  409,
  17 | 61, 3541, 67, 1741, 71, 3541, 73,     181,   97, 3541, 79, 1741, 83, 3541,
  18 | 67,   97, 71, 1861, 73,   97, 79,   21491,   71,   97, 83, 1861, 89,   97,
  19 | 71,   71, 73,  449, 79,   73, 83,   26881, 5113,   79, 89,  449, 97,   83,

Cf. A006530, A246278, A341527, A341627.

Cf. also A341607.

up_to = 105;
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341527(n) = denominator(A341528(n) / A341529(n));
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));
A341628sq(row,col) = A006530(A341527(A246278sq(row,col)));
A341628list(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] = A341628sq(col,(a-(col-1))))); (v); };
v341628 = A341628list(up_to);
A341628(n) = v341628[n];

A(n,k) = A006530(A341627(n,k)) = A006530(A341527(A246278(n,k))).

A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).

Original entry on oeis.org

0, 1, 2, 10, 2, 33, 4, 74, 44, 55, 2, 278, 4, 115, 116, 490, 2, 613, 4, 498, 242, 169, 6, 1942, 92, 265, 742, 1046, 2, 1591, 6, 3086, 344, 355, 330, 4986, 4, 487, 542, 3570, 2, 3347, 4, 1638, 2326, 737, 6, 12542, 376, 2121, 716, 2546, 6, 9869, 388, 7510, 986, 943, 2, 12894, 6, 1225, 4872, 18970, 630, 5353, 4, 3498, 1492

Offset: 1

Antti Karttunen, Jul 13 2021

nonn

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346240, A347096, A347124, A347125.

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

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));
A346239(n) = sumdiv(n,d,moebius(n/d)*A341512(d));

a(n) = Sum_{d|n} A008683(n/d) * A341512(d).
a(n) = A341512(n) - A346240(n).
a(n) = A347125(n) - A347124(n). - Antti Karttunen, Aug 25 2021

A342673 a(n) = gcd(n, sigma(A003961(n))), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 2, 3, 1, 1, 6, 1, 8, 1, 2, 1, 6, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 24, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 2, 3, 40, 1, 6, 1, 2, 1, 2, 1, 6, 7, 2, 3, 26, 1, 6, 1, 8, 3, 2, 1, 12, 1, 2, 3, 1, 1, 6, 1, 4, 3, 2, 1, 8, 1, 2, 3, 4, 7, 6, 1, 8, 1, 2, 1, 12, 5, 2, 3, 8, 1, 2, 1, 2, 3, 2, 1, 24, 1, 14, 1, 1, 1, 6, 1, 8, 3

Offset: 1

Antti Karttunen, Mar 20 2021

nonn

Cf. A000203, A003961, A003973, A342671.

Cf. also A336850, A341528.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342673(n) = gcd(n,sigma(A003961(n)));

a(n) = gcd(n, A003973(n)) = gcd(n, A000203(A003961(n))).

A346240 Difference between A341512 and its Möbius transform.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 11, 2, 3, 0, 46, 0, 5, 4, 85, 0, 80, 0, 68, 6, 3, 0, 398, 2, 5, 46, 130, 0, 209, 0, 575, 4, 3, 6, 981, 0, 5, 6, 640, 0, 397, 0, 182, 164, 7, 0, 2830, 4, 150, 4, 280, 0, 1435, 4, 1250, 6, 3, 0, 2586, 0, 7, 292, 3661, 6, 551, 0, 368, 8, 507, 0, 7983, 0, 5, 212, 502, 6, 847, 0, 4700, 788, 3, 0, 5078, 4, 5, 4, 1894

Offset: 1

Antti Karttunen, Jul 13 2021

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346239.

Cf. also A347097.

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));
A346240(n) = -sumdiv(n,d,(dA341512(d));

a(n) = -Sum_{d|n, dA008683(n/d) * A341512(d).

a(n) = A341512(n) - A346239(n).

cp's OEIS Frontend

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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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.

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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.

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A341628 Square array A(n,k) = A006530(A341527(A246278(n,k))), read by falling antidiagonals.

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A346239 Möbius transform of A341512, sigma(n)*A003961(n) - n*sigma(A003961(n)).

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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.

A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).