A073255 -id:A073255 - cp's OEIS frontend

A074469 Least m such that Sigma-Composite-Harmonic series Sum_{k=1..m} 1/A000203(A002808(k)) >= n.

32, 301, 2123, 13172, 76105, 420007, 2245009, 11719362, 60071831, 303487314, 1515211979

Offset: 1

Labos Elemer, Sep 05 2002

Cf. A004080, A002387, A002808, A000203, A073255, A046024, A074631, A074633, A074468, A074470.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] {s=0, s1=0}; Do[s=s+(1/DivisorSigma[1, c[n]]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

a(n)=my(m,s=0.);for(c=4,(2*n+2)^(n+2),if(isprime(c),next,m++);s+=1/sigma(c);if(s>=n,return(m))) \\ Charles R Greathouse IV, Feb 19 2013

a(6)-a(11) from Donovan Johnson, Aug 22 2011

A074470 Least m such that Phi-Composite-Harmonic series Sum_{k=1..m} 1/A000010(A002808(k)) >= n.

Original entry on oeis.org

2, 7, 16, 31, 60, 113, 205, 371, 663, 1176, 2069, 3631, 6341, 11039, 19159, 33164, 57287, 98763, 169967, 292061, 501165, 858892, 1470334, 2514423, 4295912, 7333264, 12508213, 21319360, 36312685, 61811287, 105152840, 178787270, 303829041, 516074615, 876190239

Offset: 1

Labos Elemer, Sep 05 2002

Cf. A004080, A002387, A002808, A000203, A073255, A046024, A074631, A074633, A074468, A074469.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] {s=0, s1=0}; Do[s=s+(1/EulerPhi[c[n]]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

More terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 23 2005

a(30)-a(35) from Donovan Johnson, Aug 21 2011

A269065 Irregular triangle read by rows: row n lists divisors of n-th composite number.

Original entry on oeis.org

1, 2, 4, 1, 2, 3, 6, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 2, 3, 4, 6, 12, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 2, 3, 6, 9, 18, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34

Offset: 1

Ilya Gutkovskiy, Feb 21 2016

Subsequence of A027750.
Row sums give A073255.
Right border gives A002808.

			Triangle begins:
1,  2,  4;
1,  2,  3,  6;
1,  2,  4,  8;
1,  3,  9;
1,  2,  5,  10;
1,  2,  3,  4,  6,  12;
1,  2,  7,  14;
1,  3,  5,  15
1,  2,  4,  8,  16;
1,  2,  3,  6,  9,  18;
1,  2,  4,  5,  10, 20;
1,  3,  7,  21;
1,  2,  11, 22;
1,  2,  3,  4,  6,  8,  12, 24;
1,  5,  25;
1,  2,  13, 26;
1,  3,  9,  27;
1,  2,  4,  7,  14, 28;
1,  2,  3,  5,  6,  10, 15, 30;
1,  2,  4,  8,  16, 32;
1,  3,  11, 33;
1,  2,  17, 34;
...

Ilya Gutkovskiy, Extended example
Ilya Gutkovskiy, Graphic additions
Eric Weisstein's World of Mathematics, Composite Number
Eric Weisstein's World of Mathematics, Divisor
Index entries for sequences related to divisors of numbers

Cf. A002808, A027750, A035004 (row length), A133021, A133031, A138881.

Flatten[Table[Divisors[Composite[n]], {n, 22}]]

tabf(nn) =  forcomposite(c=1, nn, print(divisors(c), ", ")); \\ Michel Marcus, Feb 21 2016

A347155 Sum of divisors of nontriangular numbers.

Original entry on oeis.org

3, 7, 6, 8, 15, 13, 12, 28, 14, 24, 31, 18, 39, 20, 42, 36, 24, 60, 31, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 84, 72, 48, 124, 57, 93, 72, 98, 54, 120, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140

Offset: 1

Omar E. Pol, Aug 20 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys.
So knowing this characteristic shape we can know if a number is a nontriangular number (or not) just by looking at the diagram, even ignoring the concept of nontriangular number.
Therefore we can see a geometric pattern of the distribution of the nontriangular numbers in the stepped pyramid described in A245092.
If both Dyck paths have peaks on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317303.
If both Dyck paths have valleys on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317304.

			a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13.
On the other we can see that in the main diagonal of the diagrams both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys as shown below.
Illustration of initial terms:
m(n) = A014132(n).
.
   n   m(n) a(n)   Diagram
.                    _   _ _   _ _ _   _ _ _ _   _ _ _ _ _   _ _ _ _ _ _
                   _| | | | | | | | | | | | | | | | | | | | | | | | | | |
   1    2    3    |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | |
                   _ _|  _|_| | | | | | | | | | | | | | | | | | | | | | |
   2    4    7    |_ _ _|    _|_| | | | | | | | | | | | | | | | | | | | |
   3    5    6    |_ _ _|  _|  _ _|_| | | | | | | | | | | | | | | | | | |
                   _ _ _ _|  _| |  _ _|_| | | | | | | | | | | | | | | | |
   4    7    8    |_ _ _ _| |_ _|_|    _ _|_| | | | | | | | | | | | | | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | | | | | | | | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | | | | | | | | |
                   _ _ _ _ _ _|  _ _|    _| |    _ _ _|_| | | | | | | | |
   7   11   12    |_ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|_| | | | | | |
   8   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| | |  _ _ _ _|_| | | | |
   9   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _|_| | |
  10   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|_|
                   _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       | | |
  11   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|_| |
  12   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|      _| |  _ _|
  13   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _|_|
  14   19   20    |_ _ _ _ _ _ _ _ _ _| | |       |_ _|
  15   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|
                   _ _ _ _ _ _ _ _ _ _ _| | |  _ _| |
  16   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _|
  17   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  18   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  19   25   31    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  20   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
  21   27   40    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column m gives the nontriangular numbers.
Also the diagrams have on the main diagonal the following property: diagram [1] has peaks, diagrams [2, 3] have valleys, diagrams [4, 5, 6] have peaks, diagrams [7, 8, 9, 10] have valleys, and so on.
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 3 + 3 = 6, so a(3) = 6.
For more information see A237593.

Cf. A000203, A000217, A014132, A237591, A237593, A245092, A262626, A317303, A317304, A346873.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).

Array[DivisorSigma[1,#+Round@Sqrt[2#]]&,100] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

a(n) = sigma(n + round(sqrt(2*n))); \\ Michel Marcus, Aug 21 2021

a(n) = A000203(A014132(n)).

A072553 Sigma of n-th composite number equals a(n)-th composite number if it is also a composite or equals zero if sigma[c] is prime.

Original entry on oeis.org

0, 6, 8, 0, 10, 18, 14, 14, 0, 26, 28, 20, 24, 42, 0, 28, 27, 39, 51, 44, 32, 37, 32, 66, 42, 39, 65, 71, 60, 56, 51, 93, 40, 68, 51, 72, 89, 51, 89, 57, 65, 128, 71, 76, 0, 60, 109, 95, 71, 109, 150, 83, 93, 105, 71, 128, 143, 90, 95, 175, 79, 99, 89, 138, 182, 82, 128, 96

Offset: 1

Labos Elemer, Aug 06 2002

nonn

			n=1: c[1]=4, sigma[4]=1+2+4=7 prime, a(1)=0; n=10: c[10]=18, sigma[18]=1+2+3+6+9+18=39 composite and 39 is the 26th composite number, so a(10)=26.

Cf. A065855, A000203, A002808, A073255.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] G[x_] := x-PrimePi[x]-1 Do[s=c[n]; s1=DivisorSigma[1, s]; s2=G[s1]; If[PrimeQ[s1], Print[0]]; If[ !PrimeQ[s1], Print[s2]], {n, 1, 128}]

a(n)=G[sigma[c[n]]]=A065855[A000203[A002808(n)]]]= A065855[A073255[n]] if sigma[c]=A000203[A002808(n)]] is composite and a(n)=0 if A073255[n]=A000203[A002808(n)]] is prime.

A073261 Length of FixedPointList approximating (2^n)-th composite number. See program link below.

Original entry on oeis.org

4, 4, 3, 3, 3, 4, 3, 5, 4, 4, 5, 4, 5, 6, 6, 6, 6, 5, 6, 6, 7, 6, 6, 6, 7, 7, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 9, 10

Offset: 0

Labos Elemer, Jul 22 2002

nonn

Number of iterations needed to reach the composite number using the formula in the program.

			n=30: {1073741824, 1128141853, 1130754984, 1130880243, 1130886219, 1130886489, 1130886503, 1130886504}, so a(30)=8.

Cf. A002808, A073255-A073264, A065856.

Table[ Length[ FixedPointList[ 2^n+PrimePi[ # ]+1 &, 2^n]]-1, {n, 0, 45}]

Extended by Robert G. Wilson v, Jul 24 2002

cp's OEIS Frontend

A074469 Least m such that Sigma-Composite-Harmonic series Sum_{k=1..m} 1/A000203(A002808(k)) >= n.

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A074470 Least m such that Phi-Composite-Harmonic series Sum_{k=1..m} 1/A000010(A002808(k)) >= n.

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A269065 Irregular triangle read by rows: row n lists divisors of n-th composite number.

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A347155 Sum of divisors of nontriangular numbers.

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A072553 Sigma of n-th composite number equals a(n)-th composite number if it is also a composite or equals zero if sigma[c] is prime.

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A073261 Length of FixedPointList approximating (2^n)-th composite number. See program link below.

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