A067816 -id:A067816 - cp's OEIS frontend

A076629 Duplicate of A067816.

1, 5, 8585, 16119, 29886159

Offset: 1

dead

A246852 Numbers n such that sigma(n+2) - sigma(n) = n + 2.

1, 2, 22, 14926, 31048, 69106, 246262, 5860168, 307164670, 881198662, 1489455646, 2386555630, 8225563702, 14311679062, 111494234182, 154357775302, 299004519622, 870455062822, 970388922262, 991817878342, 1677028870822, 1782783762502, 1830446935222

Offset: 1

Jaroslav Krizek, Sep 05 2014

nonn

Also numbers n such that A001065(n+2) = A000203(n). - Michel Marcus, Sep 06 2014

			Number 22 is in sequence because sigma(22+2) - sigma(22) = 60 - 36 = 24 = 22 + 2.

Giovanni Resta, Table of n, a(n) for n = 1..33 (terms < 10^13)

Cf. A000203, A067816, A246851, A246853, A246854, A246855.

[n:n in[1..10^7] | SumOfDivisors(n+2)-SumOfDivisors(n) eq n+2]

Select[Range[6*10^6], DivisorSigma[1, # + 2] - DivisorSigma[1, #] == # + 2 &] (* Jake L Lande, Jun 30 2024 *)

for(n=1,10^7,if(sigma(n+2)-sigma(n)==n+2,print1(n,", "))) \\ Derek Orr, Sep 05 2014

a(9)-a(14) from Hiroaki Yamanouchi, Sep 10 2014

a(15)-a(23) from Giovanni Resta, Jul 13 2015

A231366 Number of numbers whose sum of non-divisors (A024816) is equal to n.

Original entry on oeis.org

2, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Jaroslav Krizek, Nov 09 2013

nonn

a(n) = frequency of values n in A024816(m), where A024816(m) = sum of non-divisors of m = antisigma(m).
From Charles R Greathouse IV, Nov 11 2013: (Start)
So far all n such that a(n) > 1 correspond to members of A067816:
a(0) = 2 from 1, 2;
a(9) = 2 from 5, 6;
a(36844389) = 2 from 8585, 8586;
a(129894940) = 2 from 16119, 16120;
a(446591224981504) = 2 from 29886159, 29886160.
I checked this, and thus Krizek's conjecture below, up to 4*10^19.
(End)

			a(9) = 2 because there are two numbers m (5, 6) with antisigma(m) = 9.

Antti Karttunen, Table of n, a(n) for n = 0..32001

Cf. A054973 (number of numbers whose divisors sum to n), A231365, A231368, A231367, A231369, A067816.

up_to = 105;
A024816(n) = (n*(n+1)/2-sigma(n));
A231366list(up_to) = { my(v=vector(1+up_to), u); for(n=1, 2+up_to, if((u = A024816(n))<=up_to, v[1+u]++)); (v); };
v231366 = A231366list(up_to);
A231366(n) = v231366[1+n]; \\ Antti Karttunen, Jan 19 2025

Conjecture: max a(n) = 2.
a(A231368(n)) >= 1, a(A231369(n)) = 0.
a(n) = 0 for such n that A231367(n) = 0, a(n) = 0 if A024816(m) = n has no solution.
a(n) >= 1 for such n that A231367(n) = 1, a(n) >= 1 if A024816(m) = n for any m.
Conjecture: a(n) = 2 iff n is number from A225775 (0, 9, 36844389, 129894940, 446591224981504, …)

Data section extended to a(105) by Antti Karttunen, Jan 19 2025

A246851 a(n) = smallest number k such that sigma(k+n) - sigma(k) = k + n, or -1 if no solution exists.

Original entry on oeis.org

1, 1, 7, 1, 3577, 1, 25, 8, 13, 1, 403668223, 1, 833, 262, 19, 1, 27, 1, 793, 5, 45, 1, 1795, 66, 8, 9, 31, 1, 2005, 1, 309, 32, 261, 4238, 22490141, 1, 21, 40, 43, 1, 399, 1, 1897, 262, 193, 1, 27, 1252907952, 711, 49, 1158765, 1, 271259, 27, 129, 20518072

Offset: 1

Jaroslav Krizek, Sep 05 2014

sign

a(p-1) = 1 for any prime p.
a(11) > 11*10^8. - Derek Orr, Sep 05 2014
a(35) > 88*10^7. - Derek Orr, Sep 05 2014
a(185), a(385) and a(869) > 10^11. - Hiroaki Yamanouchi, Sep 11 2014

			Sequence of numbers k < 10^7  such that sigma(k+n) - sigma(k) = k + n for 1 <= n <= 10:
n = 1: 1, 5, 8585, 16119, ... (A067816).
n = 2: 1, 2, 22, 14926, 31048, 69106, 246262, 5860168, ... (A246852).
n = 3: 7, 6285, 4693485, ... (A246853).
n = 4: 1, 4, 26, 122, 146, 458, 746, 3746, 47612, ... (A246854).
n = 5: 3577, 14773, 2843579, ... (A246855).
n = 6: 1, 3, 114, 116058, 340014, ...
n = 7: 25, 65017, ...
n = 8: 8, 34, 76, 13474, 19042, ...
n = 9: 13, 1743, 1773, 4323, 53175, 109035, 138535, ...
n = 10: 1, 20, 1958, 35150, 49010, 246686, 1030046, 1240694, ...

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..184

Cf. A000203, A067816, A246852, A246853, A246854, A246855.

A246851:=func; [A246851(n): n in[1..100]]

a(11)-a(56) from Hiroaki Yamanouchi, Sep 11 2014

A246853 Numbers n such that sigma(n+3) - sigma(n) = n + 3.

Original entry on oeis.org

7, 6285, 4693485, 54028959, 75898473, 724416741, 2359059709, 4901493769, 321212249593, 511578306649, 534245763769, 6158645822473

Offset: 1

Jaroslav Krizek, Sep 05 2014

Also numbers n such that A001065(n+3) = A000203(n). - Michel Marcus, Sep 06 2014

a(13) > 10^13. - Giovanni Resta, Jul 13 2015

			Number 7 is in sequence because sigma(7+3) - sigma(7) = 18 - 8 = 10 = 7 + 3.

Cf. A000203, A067816, A246851, A246852, A246854, A246855.

[n:n in[1..10^7] | SumOfDivisors(n+3)-SumOfDivisors(n) eq n+3]

for(n=1,10^7,if(sigma(n+3)-sigma(n)==n+3,print1(n,", "))) \\ Derek Orr, Sep 05 2014

a(4)-a(8) from Hiroaki Yamanouchi, Sep 10 2014

a(9)-a(12) from Giovanni Resta, Jul 13 2015

A246854 Numbers n such that sigma(n+4) - sigma(n) = n + 4.

Original entry on oeis.org

1, 4, 26, 122, 146, 458, 746, 3746, 47612, 16065500, 388978292, 5313509288, 64278616556

Offset: 1

Jaroslav Krizek, Sep 05 2014

Also numbers n such that A001065(n+4) = A000203(n). - Michel Marcus, Sep 06 2014
a(14) (if it exists) > 10^11. - Hiroaki Yamanouchi, Sep 10 2014
a(14) (if it exists) > 10^13. - Giovanni Resta, Jul 13 2015

			Number 26 is in sequence because sigma(26+4) - sigma(26) = 72 - 42 = 30 = 26 + 4.

Cf. A000203, A067816, A246851, A246852, A246853, A246855.

[n:n in[1..10^7] | SumOfDivisors(n+4)-SumOfDivisors(n) eq n+4]

for(n=1,10^7,if(sigma(n+4)-sigma(n)==n+4,print1(n,", "))) \\ Derek Orr, Sep 05 2014

a(10)-a(13) from Hiroaki Yamanouchi, Sep 10 2014

A246855 Numbers k such that sigma(k+5) - sigma(k) = k + 5.

Original entry on oeis.org

3577, 14773, 2843579

Offset: 1

Jaroslav Krizek, Sep 05 2014

Also numbers k such that A001065(k+5) = A000203(k). - Michel Marcus, Sep 06 2014
a(4) (if it exists) > 10^11. - Hiroaki Yamanouchi, Sep 10 2014
a(4) (if it exists) > 10^13. - Giovanni Resta, Jul 13 2015
No other terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			Number 3577 is in sequence because sigma(3577+5) - sigma(3577) = 7800 - 4218 = 3582 = 3577 + 5.

Cf. A000203, A067816, A246851, A246852, A246853, A246854.

[n:n in[1..10^7] | SumOfDivisors(n+5)-SumOfDivisors(n) eq n+5];

Select[Range[285*10^4],DivisorSigma[1,#+5]-DivisorSigma[1,#]==#+5&] (* Harvey P. Dale, Jun 21 2024 *)

for(n=1,10^7,if(sigma(n+5)-sigma(n)==n+5,print1(n,", "))) \\ Derek Orr, Sep 05 2014

A231545 Numbers n such that n = sigma(n) - sigma(n-1).

Original entry on oeis.org

2, 6, 8586, 16120, 29886160

Offset: 1

Jaroslav Krizek, Nov 11 2013

nonn

Also, numbers n such that antisigma(n) = antisigma(n-1), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.

Numbers n such that A163553(n-1) = 0.

			6 is in sequence because antisigma(6) = antisigma(5) = 9.

Cf. A067816, A024816 (antisigma(n)), A231546 (numbers n such that sigma(n) = sigma(n-1)).

a(n) = A067816(n) + 1.

A225775 Numbers k such that antisigma(x) = antisigma(x+1) = k has solution.

Original entry on oeis.org

0, 9, 36844389, 129894940, 446591224981504

Offset: 1

Jaroslav Krizek, Jul 26 2013

nonn

Antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n.

Conjecture: also numbers k such that antisigma(n) = antisigma(m) = k has solution for distinct numbers n and m. - Jaroslav Krizek, Nov 10 2013

			36844389 is in the sequence because 36844389 = antisigma(8585) = antisigma(8586).

Cf. A067816 (numbers n such that antisigma(n) = antisigma(n+1)).

A227305 Numbers n such that sigma(n) - sigma(n-1) divides n.

Original entry on oeis.org

2, 3, 5, 6, 10, 19, 52, 118, 1054, 3201, 8586, 9802, 16120, 60556, 140698, 145216, 11273536, 29886160, 44868748, 8748377956, 325377469696, 2368739714188

Offset: 1

Alex Ratushnyak, Jul 05 2013

a(23) > 2.5*10^12. - Giovanni Resta, Jul 13 2013

Cf. A053222, A067816, A223138, A225211.

a(21)-a(22) from Giovanni Resta, Jul 13 2013

cp's OEIS Frontend

A076629 Duplicate of A067816.

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A246852 Numbers n such that sigma(n+2) - sigma(n) = n + 2.

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A231366 Number of numbers whose sum of non-divisors (A024816) is equal to n.

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A246851 a(n) = smallest number k such that sigma(k+n) - sigma(k) = k + n, or -1 if no solution exists.

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A246853 Numbers n such that sigma(n+3) - sigma(n) = n + 3.

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A246854 Numbers n such that sigma(n+4) - sigma(n) = n + 4.

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A246855 Numbers k such that sigma(k+5) - sigma(k) = k + 5.

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A231545 Numbers n such that n = sigma(n) - sigma(n-1).

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