A152770 -id:A152770 - cp's OEIS frontend

A152992 a(n) = sigma(n) - d(n) - pi(n).

0, 0, 0, 2, 1, 5, 2, 7, 6, 10, 5, 17, 6, 14, 14, 20, 9, 26, 10, 28, 20, 24, 13, 43, 19, 29, 27, 41, 18, 54, 19, 46, 33, 39, 33, 71, 24, 44, 40, 70, 27, 75, 28, 64, 58, 54, 31, 99, 39, 72, 53, 77, 36, 96, 52, 96, 60, 70, 41, 139, 42, 74, 80, 102, 62, 118, 47, 101, 73, 117, 50

Offset: 1

Omar E. Pol, Dec 19 2008, Dec 31 2008

			a(15) = 24 - 4 - 6 = 14 because the sum of divisors of 15 is 1 + 3 + 5 + 15 = 24, the number of divisors of 15 is 4 (1,3,5,15) and the number of primes not exceeding 15 is 6 (2,3,5,7,11,13). - _Emeric Deutsch_, Dec 30 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A000720, A065608, A152770, A152991.

with(numtheory): seq(sigma(n)-tau(n)-pi(n), n = 1 .. 75); # Emeric Deutsch, Dec 30 2008

Table[DivisorSigma[1,n]-DivisorSigma[0,n]-PrimePi[n],{n,75}] (* Harvey P. Dale, Sep 19 2011 *)

a(n) = A000203(n) - A000005(n) - A000720(n) = A065608(n) - A000720(n) = A152991(n) - A000005(n).

Corrected and extended by Emeric Deutsch, Dec 30 2008

A153011 Sum of proper divisors, minus the number of proper divisors, minus cototient of n, plus 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 5, 0, 6, 0, 0, 0, 14, 0, 0, 2, 8, 0, 14, 0, 11, 0, 0, 0, 24, 0, 0, 0, 20, 0, 18, 0, 12, 8, 0, 0, 36, 0, 9, 0, 14, 0, 24, 0, 26, 0, 0, 0, 54, 0, 0, 10, 26, 0, 26, 0, 18, 0, 22, 0, 65, 0, 0, 10, 20, 0, 30, 0, 50, 10, 0, 0, 70, 0, 0, 0, 38, 0, 68, 0, 24, 0

Offset: 1

Omar E. Pol, Dec 23 2008

Antti Karttunen, Table of n, a(n) for n = 1..11880
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000005, A000010, A000203, A001065, A032741, A051953, A152770.

A000203 := proc(n) numtheory[sigma](n) ; end: A000005 := proc(n) numtheory[tau](n) ; end: A152770 := proc(n) A000203(n)-A000005(n)-n+ 1; end: A051953 := proc(n) n - numtheory[phi](n) ; end: A153011 := proc(n) A152770(n)-A051953(n)+1 ; end: seq(A153011(n),n=1..120) ; # R. J. Mathar, Feb 19 2009

A153011(n) = (2+eulerphi(n)+sigma(n)-numdiv(n)-(2*n)); \\ Antti Karttunen, Oct 08 2018

a(n) = A152770(n) - A051953(n) + 1.

More terms from R. J. Mathar, Feb 19 2009

A153825 Sum of proper divisors minus the number of proper divisors of n!.

Original entry on oeis.org

0, 0, 0, 3, 29, 225, 1669, 14245, 118705, 1118001, 11705019, 144091717, 1738439017, 24817157329, 355309322689, 5378578597729, 86081749391905, 1570394279028289, 28281459220178401, 572031558109560385, 11458497230555053681

Offset: 0

Omar E. Pol, Jan 02 2009

nonn

a(n) is the sum of proper divisors minus the number of proper divisors of factorial number A000142(n).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000005, A000142, A000203, A001065, A032741, A062569, A152770, A153823, A153824.

[DivisorSigma(1,Factorial(n)) - Factorial(n) - (DivisorSigma(0,Factorial(n))-1): n in [0..20]]; // Vincenzo Librandi, Aug 31 2016

with(numtheory): seq(sigma(factorial(n))-factorial(n)-tau(factorial(n))+1, n = 0 .. 22); # Emeric Deutsch, Jan 07 2009

Table[DivisorSigma[1,n!]-n!-(DivisorSigma[0,n!]-1),{n,0,20}] (* Harvey P. Dale, Jan 14 2012 *)

a(n) = (sigma(n!) - n!) - (numdiv(n!) - 1); \\ Michel Marcus, Aug 31 2016

a(n) = A153824(n) - A153823(n) = A152770(A000142(n)).

Extended by Emeric Deutsch, Jan 07 2009

A162196 Sum of proper divisors minus the number of proper divisors of nonprime number A018252(n).

Original entry on oeis.org

0, 1, 3, 4, 2, 5, 11, 7, 6, 11, 16, 17, 8, 11, 29, 4, 13, 10, 23, 35, 26, 12, 17, 10, 47, 19, 14, 43, 47, 35, 28, 23, 67, 6, 38, 18, 41, 59, 14, 57, 20, 29, 97, 31, 36, 57, 16, 71, 53, 24, 67, 112, 37, 44, 59, 16, 83, 97, 36, 41, 129, 20, 43, 30

Offset: 1

Omar E. Pol, Jul 04 2009

Also, zero together with the positive integers of A152770.

Note that the k-th positive integer of this sequence is equal to the sum of proper divisors minus the number of proper divisors of the composite number A002808(k).

Cf. A001065, A018252, A002808, A032741, A152770, A152988, A152989, A152990, A162194, A162195.

A162196 := proc(n)
        A152770(A018252(n)) ;
end proc: # R. J. Mathar, Sep 28 2011

a(n) = A152770(A018252(n)).

a(n) = A001065(A018252(n)) - A032741(A018252(n)).

A152758 Numbers k such that the deficiency of k plus the number of proper divisors of k is not a prime number (see A152864).

Original entry on oeis.org

1, 8, 12, 15, 24, 25, 30, 32, 33, 35, 36, 39, 40, 42, 44, 48, 50, 51, 54, 56, 60, 63, 65, 66, 68, 69, 72, 78, 80, 81, 84, 85, 87, 90, 92, 96, 98, 100, 102, 105, 108, 112, 114, 116, 117, 120, 121, 123, 126, 128, 129, 130, 132, 136, 138, 140, 141, 143, 144, 148, 150, 153

Offset: 1

Omar E. Pol, Dec 14 2008

Cf. A000005, A000040, A000203, A032741, A033879, A152757, A152770, A152864.

with(numtheory): a := proc (n) if isprime(2*n-sigma(n)+tau(n)-1) = false then n else end if end proc: seq(a(n), n = 1 .. 200); # Emeric Deutsch, Jan 08 2009

Extended by Emeric Deutsch, Jan 08 2009

Name edited by Jon E. Schoenfield, Jan 06 2019

A152985 Sum of proper divisors minus the number of proper divisors of the square A000290(n).

Original entry on oeis.org

0, 1, 2, 11, 4, 47, 6, 57, 36, 109, 10, 245, 12, 195, 170, 247, 16, 509, 18, 547, 292, 439, 22, 1055, 152, 597, 358, 969, 28, 1895, 30, 1013, 632, 985, 534, 2431, 36, 1215, 850, 2317, 40, 3397, 42, 2173, 1712, 1747, 46, 4313, 396, 2953, 1382, 2955, 52, 4715, 1090, 4083

Offset: 1

Omar E. Pol, Dec 21 2008

B. D. Swan, Table of n, a(n) for n=1,...,50000

Cf. A000005, A000203, A000290, A001065, A032741, A152770.

a(n) = sigma(n^2) - n^2 - (numdiv(n^2) - 1); \\ Michel Marcus, Jan 28 2014

a(n) = A001065(A000290(n)) - A032741(A000290(n)) = A152770(A000290(n)).

A152986 Sum of proper divisors minus the number of proper divisors of pentagonal number A000326(n).

Original entry on oeis.org

0, 0, 11, 11, 10, 18, 67, 71, 60, 32, 187, 351, 30, 46, 519, 337, 128, 220, 577, 483, 366, 286, 507, 1153, 248, 336, 2489, 847, 70, 818, 871, 2181, 1108, 116, 2861, 2275, 694, 130, 2763, 3645, 100, 2352, 2823, 1863, 2278, 158, 3607, 6617, 636, 920, 6181, 4019

Offset: 1

Omar E. Pol, Dec 22 2008

B. D. Swan, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A000326, A001065, A032741, A152770.

A000326 := proc(n) n*(3*n-1)/2 ; end: A000203 := proc(n) numtheory[sigma](n) ; end: A000005 := proc(n) numtheory[tau](n) ; end: A152770 := proc(n) A000203(n)-A000005(n)-n+1 ; end: A152986 := proc(n) A152770(A000326(n)) ; end: for n from 1 to 80 do printf("%a,",A152986(n)) ; od: # R. J. Mathar, Jan 03 2009

DivisorSigma[1,#]-#-DivisorSigma[0,#]+1&/@PolygonalNumber[5,Range[60]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 02 2021 *)

a(n)=A001065(A000326(n))-A032741(A000326(n)) = A152770(A000326(n)).

Extended by R. J. Mathar, Jan 03 2009

A152987 Sum of proper divisors minus the number of proper divisors of the number of partitions of n, A000041(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 11, 35, 47, 57, 16, 0, 98, 187, 146, 176, 184, 525, 326, 1525, 1007, 254, 1632, 1275, 4261, 3311, 2859, 1476, 7489, 4383, 4408, 7624, 9859, 7450, 0, 5428, 9086, 38472, 50191, 29898, 33867, 41264, 22030, 47947, 109323, 107783, 77168

Offset: 1

Omar E. Pol, Dec 21 2008

Note that if a(n) != 0 then the number of partitions of n (A000041(n)) is a composite number (A002808), otherwise A000041(n) is a noncomposite number (A008578). See A152770.

Cf. A000005, A000041, A000203, A001065, A002808, A008578, A032741, A152770.

A000041 := proc(n) combinat[numbpart](n) ; end: A001065 := proc(n) numtheory[sigma](n)-n ; end: A032741 := proc(n) if n = 0 then 0; else numtheory[tau](n)-1 ; fi; end: A152987 := proc(n) local np ; np := A000041(n) ; A001065(np)-A032741(np) ; end: for n from 1 to 80 do printf("%d,",A152987(n)) ; end: # R. J. Mathar, Jan 22 2009

a(n) = A001065(A000041(n)) - A032741(A000041(n)) = A152770(A000041(n)).

More terms from R. J. Mathar, Jan 22 2009

A152993 a(n) = n - d(n) - pi(n) + 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 2, 6, 5, 6, 6, 9, 6, 10, 7, 10, 11, 13, 8, 14, 14, 15, 14, 18, 13, 19, 16, 19, 20, 21, 17, 24, 23, 24, 21, 27, 22, 28, 25, 26, 29, 31, 24, 32, 30, 33, 32, 36, 31, 36, 33, 38, 39, 41, 32, 42, 41, 40, 40, 44, 41, 47, 44, 47, 44

Offset: 1

Omar E. Pol, Dec 19 2008

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000720, A152770, A152991, A152992.

Table[n-DivisorSigma[0,n]-PrimePi[n]+1,{n,70}] (* Harvey P. Dale, Sep 16 2020 *)

a(n) = {n - numdiv(n) - primepi(n) + 1} \\ Andrew Howroyd, Jan 03 2020

a(n) = n - A000005(n) - A000720(n) + 1.

Terms a(17) and beyond from Andrew Howroyd, Jan 03 2020

A154349 Sum of proper divisors minus the number of proper divisors of Motzkin number A001006(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 18, 0, 34, 170, 1643, 3603, 0, 25118, 139063, 474559, 284490, 984006, 6536387, 24265729, 18678366, 96214018, 277799290, 1282283434, 2077807072, 1899874612, 19252363859, 44221482383, 1967547352, 29743945396, 1265868622

Offset: 0

Omar E. Pol, Jan 07 2009

nonn

Note that, if a(n) != 0 then Motzkin number A001006(n) is a composite number (A002808), otherwise A001006(n) is a noncomposite number (A008578). See A152770.

Amiram Eldar, Table of n, a(n) for n = 0..201

Cf. A001006, A001065, A002808, A032741, A008578, A152770, A152981, A152982, A152983, A152988, A152990.

with(numtheory): M := proc (n) options operator, arrow: (sum((-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n), j = 0 .. floor((1/3)*n)))/(n+1) end proc: seq(sigma(M(n))-M(n)-tau(M(n))+1, n = 0 .. 30); # Emeric Deutsch, Jan 12 2009

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; diff[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Table[diff[mot[n]], {n, 0, 30}] (* Amiram Eldar, Nov 26 2019 *)

a(n) = A001065(A001006(n)) - A032741(A001006(n)) = A152770(A001006(n)).

Extended by Emeric Deutsch, Jan 12 2009

cp's OEIS Frontend

A152992 a(n) = sigma(n) - d(n) - pi(n).

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A153011 Sum of proper divisors, minus the number of proper divisors, minus cototient of n, plus 1.

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A153825 Sum of proper divisors minus the number of proper divisors of n!.

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A162196 Sum of proper divisors minus the number of proper divisors of nonprime number A018252(n).

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A152758 Numbers k such that the deficiency of k plus the number of proper divisors of k is not a prime number (see A152864).

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A152985 Sum of proper divisors minus the number of proper divisors of the square A000290(n).

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A152986 Sum of proper divisors minus the number of proper divisors of pentagonal number A000326(n).

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A152987 Sum of proper divisors minus the number of proper divisors of the number of partitions of n, A000041(n).

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