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.

Previous Showing 11-20 of 20 results.

A229088 Numbers k such that sigma(k) mod k = antisigma(k) mod k, where sigma(k) = A000203(k) = sum of divisors of k, antisigma(k) = A024816(k) = sum of non-divisors of k.

Original entry on oeis.org

1, 4, 40, 224, 360, 2016, 47616, 174592, 293760, 524160, 1571328, 1782144, 3485664, 134209536, 282977280, 492101632, 746444160, 1459956960, 1684126080, 1716728832, 4428914688, 27298252800, 41233360896, 376591138560, 719045268480, 1622308746240
Offset: 1

Views

Author

Jaroslav Krizek, Oct 24 2013

Keywords

Comments

Numbers k such that A229087(k) = A000203(k) mod k - A024816(k) mod k = A054024(k) - A229110(k) = 0.
Complement of union A229089 and A229090 with respect to A000027; where A229089 = numbers k such that sigma(k) mod k < antisigma(k) mod k, A229090 = numbers k such that sigma(k) mod k > antisigma(k) mod k.
719045268480 and 1622308746240 are also terms. - Donovan Johnson, Oct 25 2013
If a number m is in this sequence and k(m) = A054024(m)/m = A229110(m)/m then k(m) = 0 for odd m (for number 1 and eventually odd multiply-perfect numbers m > 1). Conjecture: k(m) = 1/4 or 3/4 for all even m. Sequence of values k(m): 0, 3/4, 1/4, 1/4, 1/4, 1/4, 3/4, 1/4, 3/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4, 1/4, 3/4, 3/4, 3/4, 3/4, 1/4, 3/4, 3/4, ... . Value k(m) = 3/4 also for m = 719045268480 and 1622308746240. - Jaroslav Krizek, Jun 19 2014
Also, the denominator of sigma(k)/k (reduced to lowest terms) of the currently known terms, except 1, are all 4: 1, 7/4, 9/4, 9/4, 13/4, 13/4, 11/4, 9/4, 15/4, 17/4, 13/4, 15/4, 15/4, 11/4, 15/4, 9/4, 19/4, 19/4, 19/4, 15/4, 13/4, 19/4, 15/4. - Michel Marcus, Jun 21 2014
Conjecture: For k>1, numbers k such that GCD(sigma(k), k) = n/4. - Jaroslav Krizek, Sep 23 2014

Examples

			40 is in sequence because sigma(40) mod 40 = 90 mod 40 = antisigma(40) mod 40 = 730 mod 40 = 10.

Crossrefs

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

Programs

  • PARI
    for(n=1, 10^9, s=sigma(n); t=n*(n+1)/2; if(s%n==(t-s)%n, print1(n ", "))) /* Donovan Johnson, Oct 24 2013 */

Extensions

a(8)-a(23) from Donovan Johnson, Oct 24 2013
a(24)-a(26) from Jud McCranie, Oct 10 2023

A229090 Numbers n such that sigma(n) mod n > antisigma(n) mod n, where sigma(n) = A000203(n) = sum of divisors of n, antisigma(n) = A024816(n) = sum of non-divisors of n.

Original entry on oeis.org

2, 8, 10, 12, 15, 16, 21, 24, 30, 32, 42, 44, 45, 50, 52, 60, 63, 64, 68, 75, 76, 80, 92, 99, 105, 110, 116, 117, 124, 126, 128, 130, 135, 136, 140, 144, 147, 148, 150, 152, 153, 154, 160, 164, 165, 168, 170, 171, 172, 182, 184, 188, 189, 190, 195, 198, 200
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) > 0.
Complement of union A229088 and A229089 with respect to A000027, where A229088 = numbers n such that sigma(n) mod n = antisigma(n) mod n, A229089 = numbers n such that sigma(n) mod n < antisigma(n) mod n.

Examples

			Number 12 is in sequence because sigma(12) mod 12 = 28 mod 12 = 4 > antisigma(12) mod 12 = 50 mod 12 = 2.

Links

Crossrefs

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

Programs

  • Mathematica
    smQ[n_]:=Module[{sig=DivisorSigma[1,n]},Mod[sig,n]>Mod[(n(n+1))/2-sig,n]]; Select[Range[200],smQ] (* Harvey P. Dale, Dec 23 2013 *)

A229089 Numbers n such that sigma(n) mod n < antisigma(n) mod n, 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

3, 5, 6, 7, 9, 11, 13, 14, 17, 18, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93
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) < 0.
Complement of union A229088 and A229090 with respect to A000027, where
A229088 = numbers n such that sigma(n) mod n = antisigma(n) mod n,
A229090 = numbers n such that sigma(n) mod n > antisigma(n) mod n.

Examples

			Number 11 is in sequence because sigma(11) mod 11 = 12 mod 11 = 1 < antisigma(11) mod 11 = 54 mod 11 = 10.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[100],Mod[Total[Complement[Range[#],Divisors[#]]],#]> Mod[ DivisorSigma[ 1,#],#]&] (* Harvey P. Dale, Jan 24 2022 *)

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

A244324 Numbers n such that floor(antisigma(n) / n) = antisigma(n) mod n.

Original entry on oeis.org

1, 2, 15, 20, 104, 207, 464, 650, 1023, 1952, 2975, 19359, 130304, 147455, 522752, 1207359, 5017599, 8382464
Offset: 1

Views

Author

Jaroslav Krizek, Jun 25 2014

Keywords

Comments

Antisigma(n) = A024816(n) = sum of numbers less than n which do not divide n.
Also numbers n such that there is some number k > 0 with property: antisigma(n) = k*(n+1). Corresponding values of numbers k: 0, 0, 6, 8, 50, 102, 230, 323, 510, 974, 1486, 9678, …
Numbers n such that A244325(n) = A229110(n).

Examples

			Antisigma(19359) = 187366080 = 9678*19359 + 9678.

Crossrefs

Cf. A024816, A244325, A229110.

Programs

  • Magma
    [n: n in [1..1000000] | u eq 0 where u is (Floor((((n*(n+1)) div 2  - SumOfDivisors(n)) div n))) - (((((n*(n+1)) div 2)-SumOfDivisors(n)) mod (n)))]
  • PARI
    isok(n) = my(as = n*(n+1)/2 - sigma(n)); (as\n == as % n); \\ Michel Marcus, Jun 26 2014

Extensions

a(16)-a(18) from Michel Marcus, Jun 26 2014

A227231 Numbers k such that antisigma(k) mod k = k - 1.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 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, Sep 26 2013

Keywords

Comments

Antisigma(k) = A024816(k) = sum of numbers less than k which do not divide k.
Union of A065091 (odd primes) and sequence nonprimes 1, 4, 36, ... (all terms < 10^5).
No more composite terms to 10^10. - Charles R Greathouse IV, Nov 02 2014

Examples

			antisigma(36) mod 36 => 575 mod 36 = 35.

Links

Crossrefs

Cf. A024816 (antisigma(n)), A065091, A229110 (antisigma(n) mod n).

Programs

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)]

A239877 Numbers k that divide A239876(k).

Original entry on oeis.org

1, 2, 6, 14, 23, 93, 95, 343, 1924, 13358, 27385, 54709, 150554, 445242, 581211, 589819, 14733535, 18859421, 19861702, 371619757, 775908129, 1076759948, 1083679128, 7402437933, 42679464436
Offset: 1

Views

Author

Jaroslav Krizek, Mar 29 2014

Keywords

Comments

Values of k for which A239876(k) / k is an integer.
A239876 = partial sums of A229110 where A229110(n) = antisigma(n) mod n = A024816(n) mod n.
a(26) > 3*10^11. - Giovanni Resta, Mar 29 2014

Examples

			a(4) = 14 is in the sequence because A239876(14) / 14 = 70 / 14 = 5 is an integer.

Crossrefs

Cf. A024816, A229110, A239876.

Programs

  • Magma
    [n: n in [1..1000] | u eq 0 where u is ((&+[(k*(k+1)div 2 - SumOfDivisors (k)) mod k: k in [1..n]]) mod n)]

Extensions

a(13)-a(25) from Giovanni Resta, Mar 29 2014

A341162 Computing a(n) modulo (digit sum of a(n)) for the successive terms of the sequence results in a sequence that has the same succession of digits as this one.

Original entry on oeis.org

11, 13, 17, 15, 41, 29, 43, 38, 14, 56, 16, 19, 25, 23, 31, 28, 91, 44, 53, 34, 97, 46, 101, 57, 22, 55, 26, 33, 35, 68, 74, 69, 106, 64, 89, 61, 39, 47, 92, 93, 37, 104, 58, 49, 79, 71, 32, 67, 124, 125, 82, 62, 51, 52, 59, 137, 66, 88, 77, 116, 83, 99, 65, 96, 134, 151, 118, 109, 136, 98, 119, 154, 85, 129, 86
Offset: 1

Views

Author

Eric Angelini and Carole Dubois, Feb 06 2021

Keywords

Comments

This is the lexicographically earliest sequence of distinct terms > 0 with this property.

Examples

			............n:  1   2   3   4   5   6   7   8 ...
.........a(n): 11, 13, 17, 15, 41, 29, 43, 38
Digitsum (DS):  2   4   8   6   5  11   7  11
..a(n) mod DS:  1   1   1   3   1   7   1   5 <- same dig. succ. as the sequence.

Links

Crossrefs

Cf. A007953, A070635, A229110.

A229114 Numbers k such that antisigma(k) mod k = antisigma(k+1) mod (k+1).

Original entry on oeis.org

1, 8, 27, 3115, 3451, 4725, 10611, 15951, 20155, 27643, 74875, 2767675, 18390564, 27923284, 50293331, 425018875, 897002491, 10561657872, 15193530235, 20939306635, 40882585915, 80585844499
Offset: 1

Views

Author

Jaroslav Krizek, Sep 26 2013

Keywords

Comments

Antisigma(k) = A024816(k) = sum of numbers less than k which do not divide k.
Numbers k such that A229110(k) = A229110(k+1).
For k < 10^8, 2 is the only number such that sigma(k) mod k = sigma(k+1) mod (k+1).
a(23) > 10^11. - Donovan Johnson, Sep 27 2013

Examples

			a(3) = 27 because antisigma(27) mod 27 = 338 mod 27 = antisigma(28) mod 28 = 350 mod 28 = 14.

Crossrefs

Cf. A024816 (antisigma(n)), A229110 (antisigma(n) mod n).

Programs

  • PARI
    s=1; r=0; for(n=1, 10^9, n1=n+1; s=s+n1; r1=(s-sigma(n1))%n1; if(r==r1, print(n)); r=r1) /* Donovan Johnson, Sep 27 2013 */

Extensions

a(12)-a(22) from Donovan Johnson, Sep 27 2013
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