A048050 -id:A048050 - cp's OEIS frontend

A064439 Numbers n such that sigma(n) - n - 1 = pi(n).

4, 55, 65, 95, 125, 145, 155, 185, 205, 2779, 2863, 55297, 174691, 174779, 487903, 1301989, 1302457, 5254751, 6383483, 23140961, 48267437, 59651051, 70111213, 70111247, 92514491, 199445641, 212210443, 514269523, 514269599, 21881358361, 1602278990111

Offset: 1

Jason Earls, Oct 01 2001

nonn

a(32) > 3*10^12. - Giovanni Resta, Mar 31 2017

Cf. A000720, A048050.

Select[Range[10^5], DivisorSigma[1, #] - # - 1 == PrimePi[#] &] (* Giovanni Resta, Mar 31 2017 *)

sig(n) = sigma(n)-n-1; pi(x, c=0) = forprime(p=2,x,c++); c for(n=1,10^8, if(sig(n)==pi(n),print(n)))

{ n=0; for (m=1, 10^9, if ((sigma(m) - m - 1)==primepi(m), write("b064439.txt", n++, " ", m); if (n==100, break)) ) } \\ Harry J. Smith, Sep 14 2009

More terms from Klaus Brockhaus, Oct 05 2001. No further term < 800000.
a(16)-a(17) from Harry J. Smith, Sep 14 2009
a(18)-a(29) from Donovan Johnson, Feb 09 2013
a(30)-a(31) from Giovanni Resta, Mar 31 2017

A202239 n such that the sum of the factorials of the digits of n equals the sum of d|n, 1

Original entry on oeis.org

620, 13407, 66061, 266533, 282401, 416641, 3507607, 7036153, 7622243, 10327663, 17735167, 34802143, 57653483, 86357113, 86546363, 91203611, 112777747, 121825337, 124283381, 127316869, 176080309, 216687451, 218172511, 231811037, 243238447, 263364883, 272368301

Offset: 1

Michel Lagneau, Dec 16 2011

			620 is in the sequence because 6! + 2! + 0!  = 720 + 2 + 1 = 723, and sum of the divisors 1< d< n = sigma(620) - n - 1 = 1344 - 620 - 1 = 723.

Cf. A000142, A061602, A048050.

isA202239 := proc(n)
        A061602(n) = A048050(n) ;
end proc:
for n from 1 do
        if isA202239(n) then
        print(n) ;
        end if;
end do; # R. J. Mathar, Dec 18 2011

Q[n_]:=Module[{a=Total[Rest[Most[Divisors[n]]]]}, a == Total[IntegerDigits[n]!]]; Select[Range[2, 10^8], Q]

{n: A061602(n)= A048050(n)}. - R. J. Mathar, Dec 18 2011

a(17)-a(27) from Donovan Johnson, Jan 14 2012

A202242 n such that the sum of digit !! of n equals the sum of d|n, 1

Original entry on oeis.org

14, 56, 28361, 119507, 191557, 287039, 691259, 750889

Offset: 1

Michel Lagneau, Dec 16 2011

No further terms less than 10^8.

The double factorial n!! (A006882) of a positive integer n is the product of the positive integers <= n that have the same parity as n.

			56 is in the sequence because 5!! + 6!!  = 15 + 48 = 63, and sum of the divisors 1< d< 56 =  sigma(56) - 56 - 1 = 120 - 56 - 1 = 63.

Cf. A006882, A048050.

Q[n_]:=Module[{a=Total[Rest[Most[Divisors[n]]]]}, a == Total[IntegerDigits[n]!!]]; Select[Range[2, 10^5], Q]

A293992 Numbers k such that sigma(k) - k - 1 is a perfect number.

Original entry on oeis.org

8, 115, 187, 1375, 2455, 8143, 13543, 18261, 21103, 23479, 40615, 41623, 43279, 49183, 49441, 51703, 56743, 61063, 61279, 61423, 89287, 95551, 137887, 214303, 331567, 379807, 476071, 715471, 1422871, 1515967, 1793527, 1977127, 2431087, 3098527, 3663871

Offset: 1

Zdenek Cervenka, Oct 21 2017

nonn

			sigma(8) - 8 - 1 = 6, a perfect number, so 8 is a term;
sigma(115) - 115 - 1 = 28, a perfect number, so 115 is a term.

Cf. A000396, A048050.

With[{pn=PerfectNumber[Range[10]]},Select[Range[37*10^5],MemberQ[pn, DivisorSigma[ 1,#]-#-1]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 04 2019 *)

A380231 Alternating row sums of triangle A237591.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 3, 4, 5, 4, 3, 6, 5, 4, 7, 8, 7, 8, 7, 10, 9, 8, 7, 10, 11, 10, 9, 12, 11, 14, 13, 14, 13, 12, 15, 16, 15, 14, 13, 16, 15, 18, 17, 16, 19, 18, 17, 20, 21, 22, 21, 20, 19, 22, 21, 24, 23, 22, 21, 24, 23, 22, 25, 26, 25, 28, 27, 26, 25, 28, 27, 32, 31, 30, 29, 28, 31, 30, 29

Offset: 1

Omar E. Pol, Jan 17 2025

Consider the symmetric Dyck path in the first quadrant of the square grid described in the n-th row of A237593. Let C = (A240542(n), A240542(n)) be the middle point of the Dyck path.
a(n) is also the coordinate on the x axis of the point (a(n),n) and also the coordinate on the y axis of the point (n,a(n)) such that the middle point of the line segment [(a(n),n),(n,a(n))] coincides with the middle point C of the symmetric Dyck path.
The three line segments [(a(n),n),C], [(n,a(n)),C] and [(n,n),C] have the same length.
For n > 2 the points (n,n), C and (a(n),n) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (n,n), C and (n,a(n)) are the vertices of a virtual isosceles right triangle.
For n > 2 the points (a(n),n), (n,n) and (n,a(n)) are the vertices of a virtual isosceles right triangle.

			For n = 14 the 14th row of A237591 is [8, 3, 1, 2] hence the alternating row sum is 8 - 3 + 1 - 2 = 4, so a(14) = 4.
On the other hand the 14th row of A237593 is the 14th row of A237591 together with the 14 th row of A237591 in reverse order as follows: [8, 3, 1, 2, 2, 1, 3, 8].
Then with the terms of the 14th row of A237593 we can draw a Dyck path in the first quadrant of the square grid as shown below:
.
         (y axis)
          .
          .
          .    (4,14)              (14,14)
          ._ _ _ . _ _ _ _            .
          .               |
          .               |
          .               |_
          .                 |
          .                 |_ _
          .                C    |_ _ _
          .                           |
          .                           |
          .                           |
          .                           |
          .                           . (14,4)
          .                           |
          .                           |
          . . . . . . . . . . . . . . | . . . (x axis)
        (0,0)
.
In the example the point C is the point (9,9).
The three line segments [(4,14),(9,9)], [(14,4),(9,9)] and [(14,14),(9,9)] have the same length.
The points (14,14), (9,9) and (4,14) are the vertices of a virtual isosceles right triangle.
The points (14,14), (9,9) and (14,4) are the vertices of a virtual isosceles right triangle.
The points (4,14), (14,14) and (14,4) are the vertices of a virtual isosceles right triangle.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Other alternating row sums (ARS) related to the Dyck paths of A237593 and the stepped pyramid described in A245092 are as follows:
ARS of A237593 give A000004.
ARS of A196020 give A000203.
ARS of A252117 give A000203.
ARS of A271343 give A000593.
ARS of A231347 give A001065.
ARS of A236112 give A004125.
ARS of A236104 give A024916.
ARS of A249120 give A024916.
ARS of A271344 give A033879.
ARS of A231345 give A033880.
ARS of A239313 give A048050.
ARS of A237048 give A067742.
ARS of A236106 give A074400.
ARS of A235794 give A120444.
ARS of A266537 give A146076.
ARS of A236540 give A153485.
ARS of A262612 give A175254.
ARS of A353690 give A175254.
ARS of A239446 give A235796.
ARS of A239662 give A239050.
ARS of A235791 give A240542.
ARS of A272026 give A272027.
ARS of A211343 give A336305.
Cf. A221529, A237270, A237271, A237591, A262626, A322141.

A380231[n_] := 2*Sum[(-1)^(k + 1)*Ceiling[(n + 1)/k - (k + 1)/2], {k,  Quotient[Sqrt[8*n + 1] - 1, 2]}] - n;
Array[A380231 , 100] (* Paolo Xausa, Sep 06 2025 *)

row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
a(n) = my(orow = concat(row235791(n), 0)); vecsum(vector(#orow-1, i, (-1)^(i+1)*(orow[i] - orow[i+1]))); \\ Michel Marcus, Apr 13 2025

a(n) = 2*A240542(n) - n.
a(n) = n - 2*A322141(n).
a(n) = A240542(n) - A322141(n).

cp's OEIS Frontend

A064439 Numbers n such that sigma(n) - n - 1 = pi(n).

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A202239 n such that the sum of the factorials of the digits of n equals the sum of d|n, 1

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A202242 n such that the sum of digit !! of n equals the sum of d|n, 1

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A293992 Numbers k such that sigma(k) - k - 1 is a perfect number.

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A380231 Alternating row sums of triangle A237591.

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