A232540 -id:A232540

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A232324

n(n+1)/2 modulo sigma(n).

+10
11

0, 0, 2, 3, 3, 9, 4, 6, 6, 1, 6, 22, 7, 9, 0, 12, 9, 15, 10, 0, 7, 1, 12, 0, 15, 15, 18, 14, 15, 33, 16, 24, 33, 1, 6, 29, 19, 21, 52, 10, 21, 39, 22, 66, 21, 1, 24, 60, 28, 66, 30, 6, 27, 45, 28, 36, 53, 1, 30, 150, 31, 33, 40, 48, 45, 51, 34, 78, 15, 37, 36

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Also antisigma(n) modulo sigma(n). Antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n, sigma(n) = A000203(n) = the sum of the divisors of n.

a(n) = 0 for numbers from A076617, a(n) = 1 for numbers from A232540, a(n) = n for numbers from A232538.

LINKS

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = n(n+1)/2 mod A000203(n).

EXAMPLE

For n=10, a(10) = antisigma(10) mod sigma(10) = 37 mod 18 = 1.

MATHEMATICA

Table[Mod[n (n + 1)/2, DivisorSigma[1, n]], {n, 100}] (* T. D. Noe, Nov 27 2013 *)

CROSSREFS

Cf. A000203, A076617, A024816, A232538, A232540.

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Nov 25 2013

STATUS

approved

A232538

Numbers n such that (n(n+1)/2) modulo sigma(n) = n.

+10
2

33, 136, 145, 261, 897, 1441, 2016, 2241, 2353, 3808, 4320, 7201, 17101, 26937, 30721, 32896, 46593, 70561, 148960, 151633, 169345, 174592, 208801, 400401, 578593, 712801, 803800, 1040401, 1103233, 1596673, 2265121, 2377089, 3330001, 4357153, 5953024, 5962321

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Also numbers n such that antisigma(n) modulo sigma(n) = n. Antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n, sigma(n) = A000203(n) = the sum of the divisors of n.

Numbers n such that A232324(n) = n.

a(19) > 10^5.

LINKS

Table of n, a(n) for n=1..36.

FORMULA

A232324(a(n)) = n.

EXAMPLE

136 is in sequence because antisigma(136) mod sigma(136) = 9046 mod 270 = 136.

MATHEMATICA

Select[Range[6*10^6], Mod[(#(#+1))/2, DivisorSigma[1, #]]==#&] (* Harvey P. Dale, Sep 12 2019 *)

PROG

(PARI) isok(n) = (n*(n+1)/2 - sigma(n)) % sigma(n) == n; \\ Michel Marcus, Nov 25 2013

CROSSREFS

Cf. A000203, A076617, A024816, A232324, A232540.