cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A205524 Numbers n such that gcd(n, sigma(n)) is not equal to sigma(n) mod n.

Original entry on oeis.org

4, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100
Offset: 1

Views

Author

Jaroslav Krizek, Jan 28 2012

Keywords

Comments

All terms are nonprime numbers. Complement of A205523.

Examples

			Number 25 is in sequence because sigma(25)=31, gcd(25,31) = 1, and 31 mod 25 = 6.

Crossrefs

Cf. A000203, A000040, A009194, A054024, A205524, A205525.

Programs

  • Mathematica
    Select[Range[100], Mod[GCD[#, DivisorSigma[1, #]] - DivisorSigma[1, #], #] > 0 &] (* T. D. Noe, Feb 03 2012 *)

Extensions

Corrected by T. D. Noe, Feb 03 2012

A230606 Numbers n such that sigma(n) = k*(n+1) for some integer k.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 104, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263
Offset: 1

Views

Author

Jaroslav Krizek, Nov 29 2013

Keywords

Comments

Numbers n such that A108775(n) = floor(sigma(n) / n) = sigma(n) mod n = A054024(n).
Union primes (A000040) and composite numbers A045768 (k = 1 for primes p, k = 2 for composite numbers).

Examples

			20 is in sequence because sigma(20) = 42 = 2*21.

Links

Crossrefs

Cf. A000203(sigma(n)), A054024 (sigma(n) mod n), A108775.
Cf. A045768 (sigma(n) == 2 (mod n)).

Programs

  • Mathematica
    Select[Range[300],Divisible[DivisorSigma[1,#],#+1]&] (* Harvey P. Dale, May 28 2019 *)

Extensions

Example clarified by Harvey P. Dale, May 28 2019

A205525 Nonprime numbers k such that gcd(k, sigma(k)) == sigma(k) (mod k).

Original entry on oeis.org

1, 6, 12, 18, 20, 24, 28, 40, 56, 88, 104, 120, 180, 196, 224, 234, 240, 360, 368, 420, 464, 496, 540, 600, 650, 672, 780, 992, 1080, 1344, 1504, 1872, 1888, 1890, 1952, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3724, 3744, 4284, 4320, 4680, 5292, 5376, 5624
Offset: 1

Views

Author

Jaroslav Krizek, Jan 28 2012

Keywords

Comments

Complement of primes (A000040) with respect to A205523.

Examples

			24 is in the sequence because gcd(24; sigma(24)=60) = (sigma(24)=60) mod 24 = 12.

Crossrefs

Cf. A000203, A000040, A009194, A054024, A205524, A205523.

Programs

  • Mathematica
    Select[Range[10000], ! PrimeQ[#] && Mod[GCD[#, DivisorSigma[1, #]] - DivisorSigma[1, #], #] == 0 &] (* T. D. Noe, Feb 03 2012 *)
  • PARI
    isok(k) = if (!isprime(k), my(s=sigma(k)); Mod(gcd(k, s), k) == Mod(s, k)); \\ Michel Marcus, Feb 09 2021

Extensions

Corrected by T. D. Noe, Feb 03 2012

A229115 Numbers n such that sigma(n) mod n - antisigma(n) mod n = 14, where sigma(n) = A000203(n) = sum of divisor of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

Original entry on oeis.org

32, 44, 52, 68, 76, 92, 116, 124, 144, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964, 1004, 1028
Offset: 1

Views

Author

Jaroslav Krizek, Oct 24 2013

Keywords

Comments

Numbers n such that A229087(n) = A000203(n) mod n - A024816(n) mod n = A054024(n) - A229110(n) = 14.
Value 14 has in sequence A229087(n) anomalous increased frequency.
Subsequence of A229090 (numbers n such that sigma(n) mod n > antisigma(n) mod n).

Examples

			Number 32 is in sequence because sigma(32) mod 32 - antisigma(32) mod 32 = 63 mod 32 - 465 mod 32 = 31 - 17 = 14.

Links

Crossrefs

Cf. A000203 (sigma(n)), A024816 (antisigma(n)), A229110 (antisigma(n) mod n), A054024 (sigma(n) mod n), A229090.

Programs

  • PARI
    isok(n) = ((sigma(n) % n) - (n*(n+1)/2 - sigma(n)) % n) == 14; \\ Michel Marcus, Oct 31 2013

A239868 Sum of sigma(i) mod i for i from 1 to n.

Original entry on oeis.org

0, 1, 2, 5, 6, 6, 7, 14, 18, 26, 27, 31, 32, 42, 51, 66, 67, 70, 71, 73, 84, 98, 99, 111, 117, 133, 146, 146, 147, 159, 160, 191, 206, 226, 239, 258, 259, 281, 298, 308, 309, 321, 322, 362, 395, 421, 422, 450, 458, 501, 522, 568, 569, 581, 598, 606, 629, 661
Offset: 1

Views

Author

Jaroslav Krizek, Mar 28 2014

Keywords

Examples

			a(3) = 2 because sigma(3) = 4 = 1 mod 3 and a(2) + 1 = 2.
a(4) = 5 because sigma(4) = 7 = 3 mod 4 and a(3) + 3 = 5.
a(5) = 6 because sigma(5) = 6 = 1 mod 5 and a(4) + 1 = 6.

Links

Crossrefs

Cf. A000203, A054024, A239869 (values of n for which a(n)/n is an integer).

Programs

  • Magma
    [&+[SumOfDivisors (k) mod k: k in [1..n]]: n in [1..1000]]
  • Mathematica
    Table[Sum[Mod[DivisorSigma[1, i], i], {i, n}], {n, 60}] (* Alonso del Arte, Mar 30 2014 *)
Accumulate[Table[Mod[DivisorSigma[1,n],n],{n,60}]] (* Harvey P. Dale, Jun 06 2021 *)
  • PARI
    a(n) = sum(i=1, n, sigma(i) % i); \\ Michel Marcus, Jan 12 2025

Formula

a(n) = Sum_{k = 1...n} sigma(k) mod k = Sum_{k = 1...n} A054024(k).
a(n) = a(n - 1) for multiply-perfect numbers n (A007691).
a(p) = a(p - 1) + 1 for prime p.

A300657 a(n) = Sum_{d|n} sigma(d) mod d.

Original entry on oeis.org

0, 1, 1, 4, 1, 2, 1, 11, 5, 10, 1, 9, 1, 12, 11, 26, 1, 9, 1, 15, 13, 16, 1, 28, 7, 18, 18, 15, 1, 32, 1, 57, 17, 22, 15, 35, 1, 24, 19, 32, 1, 36, 1, 59, 48, 28, 1, 71, 9, 59, 23, 67, 1, 34, 19, 30, 25, 34, 1, 89, 1, 36, 58, 120, 21, 44, 1, 83, 29, 38, 1, 105
Offset: 1

Views

Author

Jaroslav Krizek, Mar 10 2018

Keywords

Comments

a(n) >= A054024(n). Conjecture: a(n) = A054024(n) only for the noncomposite numbers A008578.
a(p) = 1 for p = primes.
a(n) = n for numbers: 4, 10, 294, 8388, 612018, 1037952, 3357600, ...
n divides a(n) for numbers: 1, 4, 10, 294, 8388, 218088, 612018, 883386, 1037952, 3357600, ... Corresponding quotients: 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, ...
From Robert Israel, Mar 11 2018: (Start)
a(p*q) = 3+p+q if p < q are distinct primes and q>3.
a(p^k) = (p^(k+1)-(1+k)*p + k)/(p-1)^2 if p is prime and k >= 0. (End)

Examples

			For n = 4; a(n) = (sigma(1) mod 1 + sigma(2) mod 2 + sigma(4) mod 4) = (0 + 1 + 3) = 4.

Links

Crossrefs

Cf. A008578, A054024.

Programs

  • Magma
    [(&+[SumOfDivisors(d) mod d: d in Divisors(n)]): n in [1..100]];
  • Maple
    A300657 := n -> add(numtheory:-sigma(d) mod d, d = numtheory:-divisors(n)):
map(A300657, [$1..100]); # Robert Israel, Mar 11 2018
  • Mathematica
    Array[DivisorSum[#, Mod[DivisorSigma[1, #], #] &] &, 72] (* or *)
Fold[Function[{a, n}, Append[a, {Total@ Map[a[[#, -1]] &, Most@ Divisors@ n] + #, #} &@ Mod[DivisorSigma[1, n], n]]], {{0, 0}}, Range[2, 72]][[All, 1]] (* Michael De Vlieger, Mar 10 2018 *)
  • PARI
    a(n) = sumdiv(n, d, sigma(d) % d); \\ Michel Marcus, Mar 11 2018

Formula

a(n) = Sum_{d|n} A054024(d).

A308150 Numbers k such that sigma(k) mod k is prime, where sigma = A000203.

Original entry on oeis.org

4, 8, 18, 20, 21, 27, 32, 35, 36, 39, 50, 55, 57, 63, 65, 77, 85, 98, 100, 104, 111, 115, 125, 128, 129, 155, 161, 171, 175, 185, 187, 189, 196, 201, 203, 205, 209, 221, 235, 237, 242, 245, 265, 275, 279, 291, 299, 305, 309, 319, 323, 324, 325, 327, 335, 338, 341, 365, 371, 377, 381, 385, 391
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, May 14 2019

Keywords

Comments

Includes 1+A000668.

Examples

			a(3) = 18 is in the sequence because sigma(18) = 39, 39 == 3 (mod 18), and 3 is prime.

Links

Crossrefs

Cf. A000203, A000668, A054024.
Includes A037020 and A075081.

Programs

  • Maple
    select(n -> isprime(numtheory:-sigma(n) mod n), [$2..1000]);
  • PARI
    isok(n) = isprime(sigma(n) % n); \\ Michel Marcus, May 15 2019

A237719 Numbers n such that k(n) = (n(n+1)/2 mod n) = (antisigma(n) mod n) + (sigma(n) mod n).

Original entry on oeis.org

1, 2, 6, 12, 18, 20, 24, 28, 30, 40, 42, 54, 56, 66, 70, 78, 80, 88, 100, 102, 104, 112, 114, 120, 126, 138, 140, 150, 160, 162, 174, 176, 180, 186, 196, 198, 200, 204, 208, 220, 222, 224, 228, 234, 240, 246, 258, 260, 272, 276, 282, 294, 304, 306, 308, 318, 320
Offset: 1

Views

Author

Jaroslav Krizek, Mar 16 2014

Keywords

Comments

Numbers n such that k(n) = A142150(n) = A229110(n) + A054024(n).
Numbers n such that k(n) = (A000217(n) mod n) = (A024816(n) mod n) + (A000203(n) mod n).
k(n) = 0 for odd n, k(n) = n/2 for even n.
If there are any odd multiply-perfect numbers, they are members of this sequence.
If there is no odd multiply-perfect number, then:
(1) the only odd number in this sequence is 1,
(2) corresponding sequence of numbers k(n): {0; a(n) / 2 for n > 1}.
Supersequence of A159907, A007691 and A000396.

Examples

			12 is in the sequence because k(12) = (12*(12+1)/2) mod 12 = antisigma(12) mod 12 + sigma(12) mod 12; k(12) = 6 = 4 + 2 = n/2.

Crossrefs

Cf. A000203, A000217, A024816, A054024, A142150, A229110.

Programs

  • Magma
    [n: n in [1..320] | IsZero(n*(n+1)div 2 mod n - SumOfDivisors(n) mod n - (n*(n+1)div 2-SumOfDivisors(n)) mod n)]

A239869 Numbers k that divide A239868(k).

Original entry on oeis.org

1, 6, 7, 9, 14, 21, 37, 69, 670, 3471, 3477, 5160, 9500, 15432, 67054, 302224, 791733, 861781, 1425779, 6242689, 71511220, 187308565, 318733908, 336841318, 343858368, 499742949, 4088136488, 4172097579, 9981207398
Offset: 1

Views

Author

Jaroslav Krizek, Mar 28 2014

Keywords

Comments

A239868 = partial sums of A054024, where A054024(n) = sigma(n) mod n.
Values of k for which A239868(k) / k is an integer.
a(30) > 3*10^11. - Giovanni Resta, Mar 29 2014

Examples

			a(4) = 9 is in the sequence because A239868(9) / 9 = 18 / 9 = 2 is an integer.

Crossrefs

Cf. A000203, A054024, A239868.

Programs

  • Magma
    [n: n in [1..10000] | u eq 0 where u is ((&+[SumOfDivisors (k) mod k: k in [1..n]]) mod n)]

Extensions

a(16)-a(29) from Giovanni Resta, Mar 29 2014

A253628 Psi(n) mod n, where Psi is the Dedekind psi function (A001615).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 4, 3, 8, 1, 0, 1, 10, 9, 8, 1, 0, 1, 16, 11, 14, 1, 0, 5, 16, 9, 20, 1, 12, 1, 16, 15, 20, 13, 0, 1, 22, 17, 32, 1, 12, 1, 28, 27, 26, 1, 0, 7, 40, 21, 32, 1, 0, 17, 40, 23, 32, 1, 24, 1, 34, 33, 32, 19, 12, 1, 40, 27, 4, 1, 0, 1, 40, 45
Offset: 1

Views

Author

Tom Edgar, Jan 06 2015

Keywords

Comments

a(n) = A054024(n) when n is squarefree.
Indices of 1 appear to be given by primes A000040 (see conjecture in A068494). The (weaker) statement that a(prime(i)) = 1 is a direct consequence of the multiplicity of A001615.
a(n) = 0 if n is a member of A187778.

Examples

			A001615(12) = 24 and 24 == 0 (mod 12) so a(12) = 0.
A001615(15) = 24 and 24 == 9 (mod 15) so a(15) = 9.

Links

Crossrefs

Cf. A001615, A068494, A054024, A006093, A187778.

Programs

  • Maple
    A253628 := proc(n)
    modp(A001615(n),n) ;
end proc: # R. J. Mathar, Jan 09 2015
  • Mathematica
    a253628[n_] :=
Mod[DirichletConvolve[j, MoebiusMu[j]^2, j, #], #] & /@ Range@n; a253628[75] (* Michael De Vlieger, Jan 07 2015, after Jan Mangaldan at A001615 *)
  • Sage
    [(n*mul(1+1/p for p in prime_divisors(n)))%n for n in [1..100]]

Formula

a(n) = A001615(n) mod n.
Previous Showing 31-40 of 43 results. Next

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