A007503 -id:A007503 - cp's OEIS frontend

A272060 Numbers k such that sigma((k-1)/2) + tau((k-1)/2) is prime.

3, 5, 17, 257, 325, 1025, 65537, 82945, 202501, 250001, 2829125, 7496645, 10240001, 13675205, 16000001, 27060805, 48469445, 71402501, 133448705, 150062501, 156250001, 172186885, 182250001, 343064485, 354117125, 453519617, 467943425, 1235663105

Offset: 1

Jaroslav Krizek, Apr 19 2016

nonn

Numbers k such that A000203((k-1)/2) + A000005((k-1)/2) is a prime q.
Corresponding values of primes q are in A055813.
Prime terms are in A272061.
The first 5 known Fermat primes from A019434 are in this sequence.

			sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A000005, A000203, A007503, A019434, A055813, A064205, A272061.

[n: n in [3..1000000] | IsPrime(NumberOfDivisors((n-1) div 2) + SumOfDivisors((n-1) div 2)) and (n-1) mod 2 eq 0];

Select[Range[3, 10^7, 2], PrimeQ[DivisorSigma[1, #] + DivisorSigma[0, #]] &[(# - 1)/2] &] (* Michael De Vlieger, Apr 20 2016 *)

isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
lista(nn) = forstep (n=3, nn, 2, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Apr 19 2016

is(n)=my(f=factor(n\2)); n>2 && isprime(sigma(f)+numdiv(f)) && isprime(n) \\ Charles R Greathouse IV, Apr 29 2016

a(n) = 2*A064205(n) + 1.

A272061 Primes p such that sigma((p-1)/2) + tau((p-1)/2) is prime.

Original entry on oeis.org

3, 5, 17, 257, 65537, 453519617, 1372257937, 1927561217, 21320672257, 76001667857, 138388464037, 1216026685697, 2085136000001, 8503056000001, 30118144000001, 35427446793217, 37015056000001, 83037656250001, 87329473560577, 97850397828097, 222330465562501, 233952748524197

Offset: 1

Jaroslav Krizek, Apr 19 2016

nonn

Primes p such that A007503((p-1)/2) is a prime q.
Corresponding values of primes q: 2, 5, 19, 263, 65551, 496922891, ...
Prime terms from A272060.
The first 5 known Fermat primes from A019434 are in this sequence.
Primes of the form 2*m+1 with m a term of A064205. - Michel Marcus, Apr 25 2016

			sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..235

Cf. A000005, A000203, A007503, A055813, A064205, A272060.

[n: n in [3..1000000] | IsPrime(n) and IsPrime(NumberOfDivisors((n-1) div 2) + SumOfDivisors((n-1) div 2)) and (n-1) mod 2 eq 0];

with(numtheory): A272061:=n->`if`(isprime(n) and isprime(sigma((n-1)/2)+tau((n-1)/2)), n, NULL): seq(A272061(n), n=3..10^5); # Wesley Ivan Hurt, Apr 20 2016

Select[Prime[Range[10000]],PrimeQ[DivisorSigma[1,(#-1)/2] + DivisorSigma[0,(#-1)/2]] & ] (* Robert Price, Apr 21 2016 *)

isok(n) = isprime(sigma((n-1)/2) + numdiv((n-1)/2));
lista(nn) = forprime (p=3, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Apr 19 2016

is(n)=my(f=factor(n\2)); isprime(sigma(f)+numdiv(f)) && isprime(n) \\ Charles R Greathouse IV, Apr 28 2016

a(7)-a(8) from Michel Marcus, Apr 24 2016
a(9) from Charles R Greathouse IV, Apr 29 2016
a(10) from Charles R Greathouse IV, Apr 29 2016
a(11)-a(20), using A064205 bfile, added by Michel Marcus, Nov 23 2022
a(21)-a(22) from Amiram Eldar, Dec 06 2022

A348219 a(n) = tau(n) - omega(n) + n * Sum_{p|n, p prime} 1/p.

Original entry on oeis.org

1, 2, 2, 4, 2, 7, 2, 7, 5, 9, 2, 14, 2, 11, 10, 12, 2, 19, 2, 18, 12, 15, 2, 26, 7, 17, 12, 22, 2, 36, 2, 21, 16, 21, 14, 37, 2, 23, 18, 34, 2, 46, 2, 30, 28, 27, 2, 48, 9, 39, 22, 34, 2, 51, 18, 42, 24, 33, 2, 71, 2, 35, 34, 38, 20, 66, 2, 42, 28, 64, 2, 70, 2, 41, 44, 46

Offset: 1

Wesley Ivan Hurt, Oct 07 2021

nonn

For each divisor d of n, add n/d if d is prime, otherwise add 1. For example, a(9) = 5 can be found using its divisors 1,3,9 to get 1 + 9/3 + 1 = 5.

If p is prime, then a(p) = 2 since we have a(p) = tau(p) - omega(p) + p/p = 2 - 1 + 1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005 (tau), A000010 (phi), A001221 (omega), A005171, A007503, A010051, A069359, A348203, A386438.

A348219(n) = sumdiv(n,d,(n/d)^isprime(d)); \\ Antti Karttunen, Nov 11 2021

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A348219(n) = (A069359(n)+numdiv(n)-omega(n)); \\ Antti Karttunen, Nov 11 2021

a(n) = Sum_{d|n} (n/d)^c(d), where c is the prime characteristic (A010051).
a(n) = A000005(n) - A001221(n) + A069359(n).
a(prime(n)) = 2.
From Wesley Ivan Hurt, Jul 21 2025: (Start)
a(n) = Sum_{d|n} (c(d) + phi(d)*omega(n/d)), where c = A005171.
a(n) = A007503(n) - A386438(n). (End)

A366983 a(n) = Sum_{k=1..n} (k+1) * floor(n/k).

Original entry on oeis.org

2, 7, 13, 23, 31, 47, 57, 76, 92, 114, 128, 162, 178, 206, 234, 270, 290, 335, 357, 405, 441, 481, 507, 575, 609, 655, 699, 761, 793, 873, 907, 976, 1028, 1086, 1138, 1238, 1278, 1342, 1402, 1500, 1544, 1648, 1694, 1784, 1868, 1944, 1994, 2128, 2188, 2287, 2363, 2467

Offset: 1

Seiichi Manyama, Oct 30 2023

Partial sums of A007503.
Cf. A006218, A366984, A366985.
Cf. A024916, A257644.

PARI
```
a(n) = sum(k=1, n, (k+1)*(n\k));
```

from math import isqrt
def A366983(n): return -(s:=isqrt(n))*(s*(s+4)+5)+sum(((q:=n//w)+1)*(q+(w<<1)+4) for w in range(1,s+1))>>1 # Chai Wah Wu, Oct 31 2023

a(n) = A006218(n) + A024916(n).
G.f.: 1/(1-x) * Sum_{k>0} (1/(1-x^k)^2 - 1) = 1/(1-x) * Sum_{k>0} (k+1) * x^k/(1-x^k).
a(n) = A257644(n) - 1. - Hugo Pfoertner, Oct 31 2023

A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d(m-1),m(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 0, 0, 4, 1, 0, 2, 0, 5, 1, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 0, 0, 2, 3, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 2, 0, 4, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 13, 1, 0, 0, 0, 0, 2, 0, 0, 5, 0, 0, 0, 0, 14

Offset: 0

Omar E. Pol, Jun 16 2025

Given a number d whose position is (n,k) so the next number d is in the position (n+d,k+d-1). In other words: we can find the next number d by walking d steps south and then walking d-1 steps east. Hence the Manhattan distance between two nearest numbers d is 2*d - 1.
By definition the m-th number d is in the position (d*(m-1),m*(d-1)) thus all the numbers d are on the same straight line.
The positive terms in the row n >= 1 are the divisors of n in increasing order. Hence the number of positive terms in row n is A000005(n).
The positive terms in the column k >= 1 are 1 plus the divisors of k in decreasing order. Hence the number of positive terms in the column k is A000005(k).
In the row n >= 1, two conjugate divisors of n are equidistant from the position (n,n-1). That position is in the same column that contains to the number n in the row 0.
In the column k >= 1, two conjugate (1 plus divisor)'s of k are equidistant from the position (k,k+1).
On the other hand we can find the divisors of n on a curve which starts at A(n-1,0) = 1 and ends at A(0,n-1) = n. Hence the number of positive terms in the curve (n-1) is A000005(n). Here that curve is called "curve of divisors of n".
Note that every divisor of n less to n on the curve is also a divisor of a number less to n in a row. Hence every divisor of n in a row is also a divisor on the curve of divisors of a number greater than n.
The position of two conjugate divisors of n on the curve is orthogonal and equidistant to the main diagonal of the array.
Curves never touch each other. See the curve and the row both with the divisors of 12 in the Example section.
Drawing all the curves the 2D structure is compatible with a 3D model where in orthogonal position is the arc diagram of A000005. See Links section.
The main diagonal is an irregular triangle read by rows in which row r lists r together with 2*r - 1 zeros, r >= 1.

			The corner 9 X 9 of the square array is as shown below:
.
   \k   0 1 2 3 4 5 6 7 8
   n\ _ _ _ _ _ _ _ _ _ _
     |
   0 |  1 2 3 4 5 6 7 8 9
   1 |  1 0 0 0 0 0 0 0 0
   2 |  1 0 2 0 0 0 0 0 0
   3 |  1 0 0 0 3 0 0 0 0
   4 |  1 0 0 2 0 0 4 0 0
   5 |  1 0 0 0 0 0 0 0 5
   6 |  1 0 0 0 2 0 3 0 0
   7 |  1 0 0 0 0 0 0 0 0
   8 |  1 0 0 0 0 2 0 0 0
  ...
.
The corner 25 X 25 of the square array without the zeros is as shown below:
.
                            1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
   \k   0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
   n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   0 |  1 2 3 4 5 6 7 8 9 ...
   1 |  1
   2 |  1   2
   3 |  1       3
   4 |  1     2     4
   5 |  1               5
   6 |  1       2   3       6
   7 |  1                       7
   8 |  1         2       4         8
   9 |  1               3               9
  10 |  1           2           5           10
  11 |  1                                       11
  12 |  1             2     3   4     6             12
  13 |  1                                               13
  14 |  1               2                   7
  15 |  1                       3       5
  16 |  1                 2           4           8
  17 |  1
  18 |  1                   2       3           6       9
  19 |  1
  20 |  1                     2             4   5
  21 |  1                               3               7
  22 |  1                       2
  23 |  1
  24 |  1                         2         3     4
  ...
.
The corner 25 X 25 of the square array without the zeros with the row and the curve of the divisors of 12 is as shown below:
.
                            1 1 1 1 1 1 1 1 1 1 2 2 2 2 2
   \k   0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
   n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
     |
   0 |                        12
   1 |
   2 |
   3 |
   4 |
   5 |
   6 |                      6
   7 |
   8 |                    4
   9 |                  3
  10 |              2
  11 |  1
  12 |  1             2     3   4     6             12
  ...
.
The position of the conjugate divisors of 12 on the curve is symmetric respect to the main diagonal of the square array.
The position of the conjugate divisors of 12 in row 12 is symmetric respect the position (12,11). That position is in the same column that contains to the number 12 in the row 0.

Omar E. Pol, Illustration of initial terms of A000005 (arc diagram)

Row sums give A000203, n >= 1.
Column sums give A007503 = A000203 + A000005, k >= 1.
Row 0 gives A000027.
Column 0 gives A000012.
Cf. A027750, A056538, A208460.

A087801 Greatest common divisor of tau(n)+sigma(n) and tau(n)*sigma(n), where tau = A000005 and sigma = A000203.

Original entry on oeis.org

1, 1, 2, 1, 4, 16, 2, 1, 1, 2, 2, 2, 4, 4, 4, 1, 4, 9, 2, 12, 4, 8, 2, 4, 1, 2, 4, 2, 4, 16, 2, 3, 4, 2, 4, 1, 4, 16, 4, 2, 4, 8, 2, 18, 12, 4, 2, 2, 3, 9, 4, 4, 4, 64, 4, 64, 4, 2, 2, 36, 4, 4, 2, 1, 8, 8, 2, 12, 4, 8, 2, 9, 4, 2, 2, 2, 4, 16, 2, 4, 1, 2, 2, 4, 16, 8, 4, 4, 4, 6, 4, 6, 4, 4, 4, 24

Offset: 1

Reinhard Zumkeller, Oct 11 2003

nonn

a(n) = GCD(A007503(n), A064840(n)).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A009191, A009194, A009205, A007503, A064840, A286360.

GCD[Total[#],Times@@#]&/@Table[{DivisorSigma[0,n],DivisorSigma[1,n]},{n,100}] (* Harvey P. Dale, Jul 17 2018 *)

A087801(n) = gcd(sigma(n)+numdiv(n), sigma(n)*numdiv(n)); \\ Antti Karttunen, May 22 2017

A163163 a(n) = sigma(n) + tau(n) - n.

Original entry on oeis.org

1, 3, 3, 6, 3, 10, 3, 11, 7, 12, 3, 22, 3, 14, 13, 20, 3, 27, 3, 28, 15, 18, 3, 44, 9, 20, 17, 34, 3, 50, 3, 37, 19, 24, 17, 64, 3, 26, 21, 58, 3, 62, 3, 46, 39, 30, 3, 86, 11, 49, 25, 52, 3, 74, 21, 72, 27, 36, 3, 120, 3, 38, 47, 70, 23, 86, 3, 64, 31, 82, 3, 135, 3, 44, 55, 70, 23, 98, 3

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2009

			a(10) = A007503(10) - 10 = 22 - 10 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Antti Karttunen, Sequence plotted up to n=10000, showing the details better

Cf. A000005, A000027, A000203, A001065, A007503.

Table[Apply[Plus, DivisorSigma[#, n] & /@ {0, 1}] - n, {n, 79}] (* Michael De Vlieger, Feb 25 2018 *)

A163163(n) = (sigma(n)+numdiv(n)-n); \\ Antti Karttunen, Feb 25 2018

a(n) = A007503(n) - n = A000203(n) + A000005(n) - n.

a(n) = A000005(n) + A001065(n). - Antti Karttunen, Feb 25 2018

47 replaced with 37 by R. J. Mathar, Jul 25 2009

A320457 Lesser members of dihedral amicable pairs: numbers (m, k) such that t(m) = t(k) = m + k, where t(k) = sigma(k) + d(k).

Original entry on oeis.org

144, 300, 10434, 15774, 17034, 21032, 22088, 35394, 36872, 65324, 67628, 153868, 188468, 254526, 379026, 483812, 492414, 905212, 1090528, 1198180, 1514212, 1634262, 1886046, 1898420, 2013414, 2184860, 2191588, 2316546, 2596448, 2816156, 3340024, 3854886

Offset: 1

Amiram Eldar, Dec 01 2018

nonn

Jensen and Bussian suggested the calculation of this sequence as a part of a student research project.

			144 is in the sequence since (144, 274) is a pair of dihedral amicable numbers: sigma(144) + d(144) = 403 + 15 = 418, sigma(274) + d(274) = 414 + 4 = 418, and 144 + 274 = 418.

Amiram Eldar, Table of n, a(n) for n = 1..420
David W. Jensen and Michael K. Keane, A Number-Theoretic Approach to Subgroups of Dihedral Groups, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.

Cf. A000005 (d), A000203 (sigma), A007503, A083874, A322254 (greater members).

t[n_] := DivisorSigma[0,n] + DivisorSigma[1,n] - n; s={}; Do[n = t[m]; If[n>m && t[n]==m, AppendTo[s,m]], {m, 1, 100000}];s

f(n) = numdiv(n) + sigma(n) - n;
isok(n) = my(nn = f(n)); (nn > n) && (n == f(nn)); \\ Michel Marcus, Dec 04 2018

A322254 Greater members of dihedral amicable pairs: numbers (m, k) such that t(m) = t(k) = m + k, where t(k) = sigma(k) + d(k).

Original entry on oeis.org

274, 586, 11470, 18802, 19270, 22184, 23288, 39790, 38744, 65392, 68476, 163676, 198628, 263890, 463390, 512116, 596258, 1070492, 1100384, 1342004, 1590452, 2139722, 2122946, 2262628, 2389562, 2562844, 2344436, 2831470, 2642656, 2949628, 3464008, 5476346

Offset: 1

Amiram Eldar, Dec 01 2018

nonn

Jensen and Bussian suggested the calculation of this sequence as a part of a student research project.

The terms are ordered according to their lesser counterparts (A320457). - Amiram Eldar, Jul 03 2025

			274 is in the sequence since (144, 274) is a pair of dihedral amicable numbers: sigma(144) + d(144) = 403 + 15 = 418, sigma(274) + d(274) = 414 + 4 = 418, and 144 + 274 = 418.

Amiram Eldar, Table of n, a(n) for n = 1..420
David W. Jensen and Michael K. Keane, A Number-Theoretic Approach to Subgroups of Dihedral Groups, USAFA-TR-90-2, Air Force Academy Colorado Springs, Colorado, 1990.
David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, The College Mathematics Journal, Vol. 23, No. 2 (1992), pp. 150-152.

Cf. A000005 (d), A000203 (sigma), A007503, A083874, A320457 (lesser members).

t[n_] := DivisorSigma[0,n] + DivisorSigma[1,n]-n; s={}; Do[n=t[m]; If[n>m && t[n]==m, AppendTo[s,n]], {m,1,100000}]; s

f(n) = numdiv(n) + sigma(n) - n;
isok(n) = my(nn = f(n)); (nn < n) && (n == f(nn)); \\ Michel Marcus, Dec 04 2018

A363660 a(n) = Sum_{d|n} binomial(d+n,n).

Original entry on oeis.org

2, 9, 24, 90, 258, 1043, 3440, 13419, 48850, 187836, 705444, 2725099, 10400614, 40233015, 155133856, 601820876, 2333606238, 9079958260, 35345263820, 137876637843, 538259060526, 2104292500739, 8233430727624, 32248866496625, 126410606580284

Offset: 1

Seiichi Manyama, Jun 14 2023

nonn

Cf. A007503, A116963, A363628.

Cf. A000984, A134758, A343548, A343549, A363661.

a[n_] := DivisorSum[n, Binomial[# + n, n] &]; Array[a, 25] (* Amiram Eldar, Jul 17 2023 *)

PARI
```
a(n) = sumdiv(n, d, binomial(d+n, n));
```

a(n) = [x^n] Sum_{k>0} (1/(1 - x^k)^(n+1) - 1).

a(n) = [x^n] Sum_{k>0} binomial(k+n,n) * x^k/(1 - x^k).

cp's OEIS Frontend

A272060 Numbers k such that sigma((k-1)/2) + tau((k-1)/2) is prime.

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A272061 Primes p such that sigma((p-1)/2) + tau((p-1)/2) is prime.

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A348219 a(n) = tau(n) - omega(n) + n * Sum_{p|n, p prime} 1/p.

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A366983 a(n) = Sum_{k=1..n} (k+1) * floor(n/k).

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A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d*(m-1),m*(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1.

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A087801 Greatest common divisor of tau(n)+sigma(n) and tau(n)*sigma(n), where tau = A000005 and sigma = A000203.

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A163163 a(n) = sigma(n) + tau(n) - n.

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A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d(m-1),m(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1.