A175522 -id:A175522 - cp's OEIS frontend

A175807 A007318-perfect numbers.

2, 3, 4, 5, 12, 22, 26, 154

Offset: 1

Vladimir Shevelev, Dec 05 2010

See definition in comment to A175522. The definition is applied to the flattened view of the binomial coefficients with a single index, without regard to fact that A007318 is a triangle.

No more terms up to 10^6. - Michel Marcus, Feb 07 2016

			Since A007318(1)+ A007318(2)+ A007318(3)+ A007318(4)+ A007318(6)=6= A007318(12), then 12 is in the sequence.

Cf. A175811, A175522, A000396.

A007318 := proc(n) option remember; local t,r; t := 0 ; for r from 0 do if t+r+1 > n then return binomial(r,n-t) ; end if; t := t+r+1 ; end do: end proc:
isA175807 := proc(n) m := 0 ; for d in numtheory[divisors](n) minus {n} do m := m+A007318(d) ; end do; m = A007318(n) ; end proc:
for n from 1 do if isA175807(n) then printf("%d,\n",n); end if; end do: # R. J. Mathar, Dec 05 2010

b(n) = {my(m = 1); while (m*(m+1)/2 < n, m++); if (! ispolygonal(n, 3), m--); binomial(m, n - m*(m+1)/2);}
isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 07 2016

A175811 A007318-deficient numbers.

Original entry on oeis.org

1, 7, 11, 13, 17, 18, 19, 23, 24, 25, 29, 30, 31, 32, 33, 37, 38, 39, 40, 41, 42, 43, 47, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 93, 94, 95, 96, 97, 98, 99, 100, 101, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117

Offset: 1

Vladimir Shevelev, Dec 05 2010

Definition see in comment to A175522. The same criticism on index-selection as in A175807 applies. All primes greater than 5 are in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007318, A175522, A175807 (perfect version), A005100, A005101.

A007318 := proc(n) option remember; local t, r; t := 0 ; for r from 0 do if t+r+1 > n then return binomial(r, n-t) ; end if; t := t+r+1 ; end do: end proc:
isA175811 := proc(n) m := 0 ; for d in numtheory[divisors](n) minus {n} do m := m+A007318(d) ; end do; m < A007318(n) ; end proc:
for n from 1 to 120 do if isA175811(n) then printf("%d,", n); end if; end do: # R. J. Mathar, Dec 06 2010

b(n) = {my(m = 1); while (m*(m+1)/2 < n, m++); if (! ispolygonal(n, 3), m--); binomial(m, n - m*(m+1)/2);}
isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 07 2016

{n: sum_{d|n, dA007318(d) < A007318(n)}.

Terms >25 from R. J. Mathar, Dec 06 2010

A175853 (2*n-1)-perfect numbers.

Original entry on oeis.org

20, 128768, 33501184

Offset: 1

Vladimir Shevelev, Dec 05 2010

For the definition see A175837.

Considering only terms of the form 2^k * p with p prime results in a linear equation for p for a given k. Solving quickly generates other terms of the sequence: 137433972736, 2199001235456, 649037107316852462774394019643392, and a 157-digit term associated with k=260. - D. S. McNeil, Dec 08 2010

			Proper divisors of 20 are: 1,2,4,5,10. Since 2*20-1=(2*1-1)+(2*2-1)+(2*4-1)+(2*5-1)+(2*10-1)=39, then 20 is in the sequence.

Cf. A175837 (abundant version), A175522, A033880, A000005.

A033880(a(n))=A000005(a(n))/2-1.

a(3) from D. S. McNeil, Dec 08 2010

A177050 Ceiling(n/2)-perfect numbers.

Original entry on oeis.org

2, 4, 8, 10, 16, 32, 64, 110, 128, 136, 256, 512, 884, 1024, 2048, 4096, 8192, 16384, 18632, 32768, 32896, 65536, 70564, 100804, 116624, 131072, 262144, 391612, 449295, 524288, 1048576, 2097152, 4194304, 8388608, 15370304, 16777216, 33554432, 67108864, 73995392

Offset: 1

Vladimir Shevelev, Dec 09 2010

nonn

All powers of 2 except for 1 are terms of the sequence. All numbers of the form 2^(2^k-1)*p, where p=2^(2^k)+1 is a Fermat prime (k >= 1) are in the sequence. Thus numbers 136, 32896, 2147516416 are in the sequence. It is interesting that in this construction Fermat primes play the same role that Mersenne primes in construction of usual even perfect numbers. Unfortunately, the conversion for even ceiling(n/2)-perfect numbers is false: the first counterexample, found by D. S. McNeil, is 110 = 2*5*11. Besides, the first odd term, found by D. S. McNeil, is 449295 = 3*5*7*11*389.

Amiram Eldar, Table of n, a(n) for n = 1..47

Cf. A000396, A175522, A175807, A175853.

isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 08 2016

is_A177050 = lambda n: sum(ceil(d/2) for d in divisors(n)) == 2*ceil(n/2) # D. S. McNeil, Dec 10 2010

a(31)-a(39) from Michel Marcus, Feb 08 2016

A324101 Numbers whose "unary-binary encoded prime factorization" (A156552) is not A000120-deficient.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105

Offset: 1

Antti Karttunen, Feb 18 2019

nonn

Numbers n for which A192895(A156552(n)) >= 0.

Numbers n such that A156552(n) is either in A175522 or in A175526.

Antti Karttunen, Table of n, a(n) for n = 1..4133

Cf. A324102 (complement, apart from 1 which is in neither sequence).

Cf. A175522, A175526, A192895.

A175821 A007318-abundant numbers.

Original entry on oeis.org

6, 8, 9, 10, 14, 15, 16, 20, 21, 27, 28, 34, 35, 36, 44, 45, 46, 54, 55, 56, 64, 65, 66, 76, 77, 78, 80, 90, 91, 92, 102, 104, 105, 118, 119, 120, 122, 135, 136, 138, 150, 152, 153, 168, 170, 171, 172, 188, 189, 190, 192, 207, 208, 209, 210, 228, 230, 231, 232, 250, 252, 253, 254, 255, 256, 275, 276, 278, 296, 297, 298, 299, 300, 320, 322, 324, 325, 326, 327, 328, 348, 350, 351, 352, 354, 375, 376, 377, 378, 380, 381, 400, 402, 404, 405, 406, 408

Offset: 1

Vladimir Shevelev, Dec 05 2010

The comment in A175522 contains a definition.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A175807 (perfect version), A175811 (deficient version), A007318, A005100, A005101.

A000027 \ { A175807 U A175811}. [R. J. Mathar, Dec 06 2010]

Terms beyond 27 from R. J. Mathar, Dec 06 2010

A175837 (2n-1)-abundant numbers.

Original entry on oeis.org

12, 18, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252

Offset: 1

Vladimir Shevelev, Dec 05 2010

nonn

A number k is (2n-1)-abundant if sum_{d|k, d 2*k-1, a specialization of the definition in A175522.
Adding 2k-1 on both sides of the condition yields the equivalent condition A129246(k) > 2*(2k-1).
Adding 2k-1 on both sides also yields sum_{d|k} (2*d-1) > 2*(2k-1), equivalent to 2*sum_{d|k}d - tau(k) > 2*(2k-1) or sigma(k) > 2k-1+tau(k)/2, equivalent to A033880(k) > tau(k)/2-1.

Cf. A175522, A033880, A000203, A005100, A005101.

aQ[n_] := DivisorSum[n, 2#-1&, # 2n-1; Select[Range[252], aQ] (* Amiram Eldar, Feb 18 2019 *)

A175837 = { n | A033880(n) > A000005(n)/2-1 }.

More terms from Amiram Eldar, Feb 18 2019

A176234 Floor(sqrt(n))-perfect numbers.

Original entry on oeis.org

2, 3, 4, 21, 26, 27, 33, 35, 38, 46, 58, 62, 74, 475, 605, 1083, 1719, 2007, 2151, 2169, 2259, 2313, 2421, 2431, 2439, 2493, 2529, 2547, 2637, 2737, 2763, 2799, 2979, 3123, 3303, 3357, 3367, 3451, 3619, 3681, 3698, 4255, 4465, 4625, 5035, 5125, 5185, 5695, 6205

Offset: 1

Vladimir Shevelev, Dec 07 2010

nonn

See definition in comment to A175522.

The even terms begin: 2, 4, 26, 38, 46, 58, 62, 74, 3698, 34226, 34726, ... - Michel Marcus, Feb 08 2016

			floor(sqrt(35))=5; floor(sqrt(1))+floor(sqrt(5))+floor(sqrt(7))=5. Therefore, 35 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A175522, A000396, A175807.

f[n_] := Sum[Floor[Sqrt[Divisors[n][[i]]]], {i, 1, Length[Divisors[n]] - 1}]; Select[Range[3000], f[#] == Floor[Sqrt[#]] &]

isok(n) = sumdiv(n, d, (dMichel Marcus, Feb 08 2016

More terms from Michel Marcus, Feb 08 2016

A177052 Ceiling(n/2)-abundant numbers.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228

Offset: 1

Vladimir Shevelev, Dec 09 2010

nonn

For definition, see A175522.

All positive numbers == 0 (mod 6) are in the sequence (basically A008588). In addition, note that all odd primes are ceiling(n/2)-deficient numbers. The first odd term of the sequence is 315.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A177050 (perfect version), A005100, A005101, A175524, A175526, A175811, A175821, A175853.

isok(n) = sumdiv(n, d, (d ceil(n/2); \\ Michel Marcus, Feb 08 2016

is_A177052 = lambda n: sum(ceil(d/2) for d in divisors(n)) > 2*ceil(n/2) # D. S. McNeil, Dec 10 2010

{n : Sum_{d|n, dA004526(1+d) > A004526(1+n)}. [R. J. Mathar, Dec 11 2010]

A177084 Ceiling(n/3)-perfect numbers.

Original entry on oeis.org

2, 3, 4, 10, 14, 50, 52, 130, 184, 315, 688, 988, 2528, 6490, 35456, 396916, 537088, 538112, 801376, 1297312, 8452096, 8456192, 35221184, 53996590, 134520832, 222469702

Offset: 1

Vladimir Shevelev, Dec 09 2010

For definition, see comment of A175522.

Cf. A000396, A002264, A175522, A175807, A175853, A177050.

aQ[n_] := DivisorSum[n, Ceiling[#/3] &, # < n &] == Ceiling[n/3]; Select[Range[10^6], aQ] (* Amiram Eldar, Jul 20 2019 *)

is_A177084 = lambda n: sum(ceil(d/3) for d in divisors(n)) == 2*ceil(n/3) # D. S. McNeil, Dec 10 2010

{n: Sum_{d|n, dA002264(2+d) = A002264(2+n)}. - R. J. Mathar, Dec 11 2010

a(21)-a(26) from Amiram Eldar, Jul 20 2019

cp's OEIS Frontend

A175807 A007318-perfect numbers.

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A175811 A007318-deficient numbers.

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A175853 (2*n-1)-perfect numbers.

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A177050 Ceiling(n/2)-perfect numbers.

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A324101 Numbers whose "unary-binary encoded prime factorization" (A156552) is not A000120-deficient.

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A175821 A007318-abundant numbers.

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A175837 (2n-1)-abundant numbers.

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A176234 Floor(sqrt(n))-perfect numbers.

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