A075708 -id:A075708 - cp's OEIS frontend

A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

1, 1, 1, 4, -1, 1, 0, 4, -5, 1, 24, -16, 4, -13, 1, -8, 40, -48, 4, -29, 1, 104, -88, 72, -112, 4, -61, 1, -48, 184, -248, 136, -240, 4, -125, 1, 352, -400, 344, -568, 264, -496, 4, -253, 1, 80, 544, -1104, 664, -1208, 520, -1008, 4, -509, 1, 1424, -784, 928, -2512, 1304, -2488, 1032, -2032, 4, -1021, 1

Offset: 1

Antti Karttunen, Nov 22 2019

			The top left corner of the array:
   n   p_n |k=1,     2, 3,      4,     5,      6,     7,       8,      9,      10
  ---------+----------------------------------------------------------------------
   1 ->  2 |  1,     1, 4,      0,    24,     -8,   104,     -48,    352,      80,
   2 ->  3 |  1,    -1, 4,    -16,    40,    -88,   184,    -400,    544,    -784,
   3 ->  5 |  1,    -5, 4,    -48,    72,   -248,   344,   -1104,    928,   -2512,
   4 ->  7 |  1,   -13, 4,   -112,   136,   -568,   664,   -2512,   1696,   -5968,
   5 -> 11 |  1,   -29, 4,   -240,   264,  -1208,  1304,   -5328,   3232,  -12880,
   6 -> 13 |  1,   -61, 4,   -496,   520,  -2488,  2584,  -10960,   6304,  -26704,
   7 -> 17 |  1,  -125, 4,  -1008,  1032,  -5048,  5144,  -22224,  12448,  -54352,
   8 -> 19 |  1,  -253, 4,  -2032,  2056, -10168, 10264,  -44752,  24736, -109648,
   9 -> 23 |  1,  -509, 4,  -4080,  4104, -20408, 20504,  -89808,  49312, -220240,
  10 -> 29 |  1, -1021, 4,  -8176,  8200, -40888, 40984, -179920,  98464, -441424,
  11 -> 31 |  1, -2045, 4, -16368, 16392, -81848, 81944, -360144, 196768, -883792,

Cf. A000079, A000203, A000225, A075708, A182944, A329610, A329644, A329890.
Rows 1-2: A329891, A329892 (from n>=1).
Column 1: A000012, Column 2: -A036563(n) (from n>=1), Column 3: A010709.

up_to = 105;
A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329637sq(n,k) = ((2^(n+k-1)) - (((2^n)-1) * A329890(k)));
A329637list(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] = A329637sq(col,(a-(col-1))))); (v); };
v329637 = A329637list(up_to);
A329637(n) = v329637[n];

A(n, k) = A329644(A182944(n, k)).

A(n, k) = A000079(n+k-1) - (A000225(n) * A329890(k)).

A096853 a(n) = A062401(2^n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 48, 64, 144, 288, 512, 576, 2304, 4096, 10240, 18432, 36288, 65536, 184320, 262144, 552960, 718848, 1492992, 2822400, 9123840, 13418496, 44695552, 68762880, 106168320, 109486080, 580386816, 1073741824, 2155507200, 2366668800, 6920601600, 12081954816

Offset: 1

Labos Elemer, Jul 19 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..392

Cf. A000010, A000203, A000225, A062401, A062402, A053287, A075708, A096853, A096854, A096855, A096856.

Table[ EulerPhi[ DivisorSigma[1, 2^n - 1]], {n, 33}]

a(n) = A000010(A000203(A000225(n))). - Michel Marcus, Dec 19 2013

a(n) = A000010(A075708(n)). - Amiram Eldar, Jun 04 2024

Edited and extended by Robert G. Wilson v, Jul 23 2004

a(33)-a(35) from Amiram Eldar, Jun 04 2024

A103291 Numbers k such that sigma(2^k-1) >= 2*(2^k-1)-1, i.e., the number 2^k-1 is perfect, abundant, or least deficient.

Original entry on oeis.org

1, 12, 24, 36, 40, 48, 60, 72, 80, 84, 90, 96, 108, 120, 132, 140, 144, 156, 160, 168, 180, 192, 200, 204, 210, 216, 220, 228, 240, 252, 264, 270, 276, 280, 288, 300, 312, 320, 324, 330, 336, 348, 360, 372, 384, 396, 400, 408, 420, 432, 440, 444, 450, 456, 468

Offset: 1

Max Alekseyev, Jan 28 2005

Is there an odd term besides 1? Numbers 2^a(i)-1 form set difference of sequences A103289 and A096399.

Odd terms > 1 exist, but there are none < 10^7. If k > 1 is an odd term, then 2^k-1 must have more than 900000 distinct prime factors and all of them must be members of A014663. - David Wasserman, Apr 15 2008

Amiram Eldar, Table of n, a(n) for n = 1..136

Cf. A000203, A075708, A103288, A103289, A103292, A023196.

for(i=1,1000,n=2^i-1;if(sigma(n)>=2*n-1,print1(i, ", ")));

Numbers k such that 2^k-1 is in A103288.

More terms from David Wasserman, Apr 15 2008

A329890 a(1) = 1; for n > 1, a(n) = sigma((2^n)-1) - sigma((2^(n-1))-1), where sigma is A000203, the sum of divisors.

Original entry on oeis.org

1, 3, 4, 16, 8, 72, 24, 304, 160, 944, 624, 6576, -544, 14336, 16384, 72544, 19616, 342528, 50688, 1475584, 466176, 3443712, 2657376, 29486880, -3340288, 54808448, 65971360, 306781024, 77647680, 1475408064, 132153344, 5157119680, 3054411072, 12548176896, 13343981568, 130039259136, -28235160128, 228451400256, 269821673472

Offset: 1

Antti Karttunen, Dec 08 2019

sign

Index entries for sequences related to sigma(n)

From second term onwards, the first differences of A075708.

Cf. A000203, A329637, A329891, A329892.

Join[{1},Table[DivisorSigma[1,2^n-1]-DivisorSigma[1,2^(n-1)-1],{n,2,40}]] (* Harvey P. Dale, Aug 18 2024 *)

A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));

a(1) = 1; and for n > 1, a(n) = A075708(n) - A075708(n-1).

A329891 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^n - (sigma((2^n)-1) - sigma((2^(n-1))-1)).

Original entry on oeis.org

0, 1, 1, 4, 0, 24, -8, 104, -48, 352, 80, 1424, -2480, 8736, 2048, 16384, -7008, 111456, -80384, 473600, -427008, 1630976, 750592, 5731232, -12709664, 36894720, 12300416, 68246368, -38345568, 459223232, -401666240, 2015330304, -862152384, 5535523520, 4631692288, 21015756800, -61319782400, 165674113600, 46426506688, 279934140416, -484569911296

Offset: 0

Antti Karttunen, Nov 23 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..255
Index entries for sequences related to sigma(n)

Row 1 of A329637.

Cf. A000203, A000225, A075708, A156552, A323243, A323244, A329644, A329890, A329892.

A329891(n) = if(n<=1,n,(2^n - sigma((2^n)-1)) + sigma((2^(n-1))-1));

a(0) = 0; for n >= 1, a(n) = A323244(2^n) - A323244(2^(n-1)) = 2^n - A329890(n).

a(n) = A329644(2^n).

A329892 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^(n+1) - 3*(sigma((2^n)-1) - sigma((2^(n-1))-1)).

Original entry on oeis.org

0, 1, -1, 4, -16, 40, -88, 184, -400, 544, -784, 2224, -11536, 18016, -10240, 16384, -86560, 203296, -503296, 896512, -2329600, 2795776, -1942528, 8805088, -54906208, 77129728, -30207616, 70521376, -383472160, 840798784, -2278740544, 3898507264, -6881424448, 8016635968, -3284792320, 28687532032, -252678823936, 359583387328, -135598386880

Offset: 0

Antti Karttunen, Nov 23 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..255
Index entries for sequences related to sigma(n)

Row 2 of A329637.

Cf. A000203, A000225, A075708, A156552, A323243, A329644, A329890, A329891.

A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329892(n) = if(!n,n,2^(n+1) - 3*A329890(n));

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323243(n) = if(1==n, 0, sigma(A156552(n)));
A329644(n) = sumdiv(n, d, moebius(n/d)*((2*A156552(d))-A323243(d)));
A329892(n) = A329644(3^n);

a(n) = A329644(3^n).

a(0) = 0; for n >= 1, a(n) = 2^(n+1) - 3*A329890(n).

A152057 Sum of the distinct prime factors of 2^n-1.

Original entry on oeis.org

0, 0, 3, 7, 8, 31, 10, 127, 25, 80, 45, 112, 28, 8191, 173, 189, 282, 131071, 102, 524287, 91, 471, 798, 178528, 286, 2433, 10925, 262737, 320, 3425, 534, 2147483647, 65819, 599598, 174765, 123150, 266, 616318400, 699053, 129646, 61789, 164524720, 5936

Offset: 0

Roger L. Bagula, Nov 22 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1206 (terms 0..500 from Robert Israel)

Cf. A000225, A008472, A046800, A075708.

Row sums of A060443.

sopf:= n -> convert(numtheory:-factorset(n),`+`):
seq(sopf(2^n-1),n=0..100); # Robert Israel, Jan 14 2021

Table[Sum[FactorInteger[2^n - 1][[m]][[1]], {m, 1, Length[FactorInteger[2^n - 1]]}], {n, 0, 50}]

a(n) = A008472(A000225(n)). - Robert Israel, Jan 14 2021

Edited by N. J. A. Sloane, Nov 26 2008

A247938 Sum of divisors of 2^prime(n)-1.

Original entry on oeis.org

4, 8, 32, 128, 2160, 8192, 131072, 524288, 8567136, 539922240, 2147483648, 138055271872, 2199187780272, 8817412930560, 140828559963840, 9008745449302368, 576463955735383776, 2305843009213693952, 147573953351708377936, 2361193635521975063040

Offset: 1

Vincenzo Librandi, Sep 27 2014

nonn

b-file computed with factorizations in Wagstaff link. a(167) corresponding to 2^991-1 is currently the first unknown term. - Jens Kruse Andersen, Sep 28 2014

Conjecture: a(n)/2^prime(n) reaches its maximum value 135/128 at n = 5. - Jianing Song, Dec 31 2022

R. Bojanić, Asymptotic evaluations of the sum of divisors of certain numbers (in Serbo-Croatian), Bull. Soc. Math.-Phys, R.P. Macédoine, Vol. 5 (1954), pp. 5-15.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 96.

Jens Kruse Andersen and Amiram Eldar, Table of n, a(n) for n = 1..197 (terms 1..166 Jens Kruse Andersen)
Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 271, exercise 22.
Sam Wagstaff, The Cunningham Project.

Subsequence of A075708.

Cf. A000203, A001348, A034785.

[SumOfDivisors(2^p-1): p in PrimesUpTo(100)];

with(numtheory): A247938:=n->sigma(2^ithprime(n)-1): seq(A247938(n), n=1..20); # Wesley Ivan Hurt, Sep 27 2014

Table[DivisorSigma[1, 2^Prime[n]-1], {n, 30}]

vector(50,n,sigma(2^prime(n)-1)) \\ Derek Orr, Sep 27 2014

a(n) = A000203(A001348(n)). - Michel Marcus, Sep 27 2014

Limit_{n->oo} a(n)/A001348(n) = 1 (Bojanić, 1954). - Amiram Eldar, Mar 04 2021

cp's OEIS Frontend

A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

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A096853 a(n) = A062401(2^n-1).

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A103291 Numbers k such that sigma(2^k-1) >= 2*(2^k-1)-1, i.e., the number 2^k-1 is perfect, abundant, or least deficient.

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A329890 a(1) = 1; for n > 1, a(n) = sigma((2^n)-1) - sigma((2^(n-1))-1), where sigma is A000203, the sum of divisors.

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A329891 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^n - (sigma((2^n)-1) - sigma((2^(n-1))-1)).

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A329892 a(0) = 0, a(1) = 1, for n > 1, a(n) = 2^(n+1) - 3*(sigma((2^n)-1) - sigma((2^(n-1))-1)).

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A152057 Sum of the distinct prime factors of 2^n-1.

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A247938 Sum of divisors of 2^prime(n)-1.

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