A232324 -id:A232324 - cp's OEIS frontend

A352908 Numbers k such that A232324(k) is prime.

3, 4, 5, 13, 21, 36, 37, 57, 61, 70, 73, 93, 100, 129, 130, 154, 157, 193, 201, 205, 217, 237, 250, 253, 277, 301, 310, 313, 322, 333, 381, 397, 406, 417, 421, 442, 453, 457, 493, 513, 517, 541, 565, 597, 603, 613, 646, 661, 673, 682, 685, 697, 733, 757, 781, 813, 826, 877, 913, 921, 925, 994

Offset: 1

J. M. Bergot and Robert Israel, Apr 14 2022

nonn

Numbers k such that the k-th triangular number mod the sum of divisors of k is prime.

			a(5) = 21 is a term because A232324(21) = 231 mod 32 = 7 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Contains A005383.

filter:= n -> isprime((n*(n+1)/2) mod numtheory:-sigma(n)):
select(filter, [$1..1000]);

A076617 Numbers k such that sum of the divisors d of k divides 1 + 2 + ... + k = k(k+1)/2.

Original entry on oeis.org

1, 2, 15, 20, 24, 95, 104, 207, 224, 287, 464, 1023, 1199, 1952, 4095, 4607, 8036, 12095, 15872, 16895, 19359, 22932, 23519, 28799, 45440, 45695, 54144, 77375, 101567, 102024, 130304, 159599, 163295, 223199, 296207, 317184, 352799, 522752, 524160, 635904

Offset: 1

Joseph L. Pe, Oct 22 2002

nonn

Alternately, numbers k such that sum of the divisors d of k divides the sum of the non-divisors d' of k, where 1 <= d, d' <= k.

Numbers k such that A232324(k) = antisigma(k) mod sigma(k) = A024816(n) mod A000203(n) = 0. - Jaroslav Krizek, Jan 24 2014

			The sum of the divisors of 15 is sigma(15) = 24; the sum of the non-divisors of 15 that are between 1 and 15 is 2 + 4 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 = 96. Since 24 divides 96, 15 is a term of the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A000203, A024816, A066860.

a = {}; Do[ s = DivisorSigma[1, i]; n = (i (i + 1) / 2) - s; If[Mod[n, s] == 0, a = Append[a, i]], {i, 1, 10^5}]; a
Select[Range[640000],Divisible[(#(#+1))/2,DivisorSigma[1,#]]&] (* Harvey P. Dale, Aug 01 2019 *)

is(n)=n*(n+1)/2%sigma(n)==0 \\ Charles R Greathouse IV, May 02 2013

a(n+2) = A066860(n) - Alex Ratushnyak, Jul 02 2013

New name from J. M. Bergot, May 02 2013

More terms from T. D. Noe, May 02 2013

A066860 The sum of the non-divisors of n (less than n) is a multiple of the sum of the divisors of n.

Original entry on oeis.org

15, 20, 24, 95, 104, 207, 224, 287, 464, 1023, 1199, 1952, 4095, 4607, 8036, 12095, 15872, 16895, 19359, 22932, 23519, 28799, 45440, 45695, 54144, 77375, 101567, 102024, 130304, 159599, 163295, 223199, 296207, 317184, 352799, 522752, 524160

Offset: 1

Joseph L. Pe, Jan 25 2002

nonn

			Divisors of 15 = {1, 3, 5, 15}, which sum to 24. Non-divisors of 15 less than 15 = {2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14}, which sum to 96, a multiple of 24. So 15 is a term of the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..300

Cf. A000203, A024816, A076617, A232324.

f[n_] := Module[{a, b, c}, a = Divisors[n]; b = Apply[Plus, Complement[Range[1, n], a]]; c = Apply[Plus, a]; Mod[b, c] == 0]; Do[If[f[n] == True, Print[n]], {n, 3, 23519}]
Select[Range[3, 10000], Mod[# (# + 1)/2, DivisorSigma[1, #]] == 0 &] (* T. D. Noe, Nov 27 2013 *)

a(n) = A076617(n+2). - Alex Ratushnyak, Jul 02 2013

A232324(a(n)) = A024816(a(n)) mod A000203(a(n)) = 0. - Jaroslav Krizek, Nov 25 2013

More terms from Lior Manor, Feb 10 2002

A232538 Numbers n such that (n(n+1)/2) modulo sigma(n) = n.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Nov 25 2013

nonn

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.

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

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

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

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

A232324(a(n)) = n.

More terms from Michel Marcus, Nov 25 2013

A232540 Numbers n such that (n(n+1)/2) modulo sigma(n) = 1.

Original entry on oeis.org

10, 22, 34, 46, 58, 82, 94, 106, 118, 142, 166, 178, 202, 214, 226, 262, 274, 298, 334, 346, 358, 382, 385, 394, 430, 454, 466, 478, 502, 514, 526, 538, 562, 586, 622, 634, 694, 706, 718, 766, 778, 802, 838, 862, 886, 898, 922, 934, 958, 982, 1006, 1018, 1042

Offset: 1

Jaroslav Krizek, Nov 25 2013

nonn

Also numbers n such that antisigma(n) modulo sigma(n) = 1. 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) = 1.
Number 5950 is only squareful number from first 1400 terms (< 50000) of this sequence.
Conjecture: supersequence of A112774 (semiprimes of the form 6n+4).

			106 is in sequence because antisigma(106) mod sigma(106) = 5509 mod 162 = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000203, A076617, A024816, A112774, A232324, A232538.

Select[Range[1100],Mod[(#(#+1))/2,DivisorSigma[1,#]]==1&] (* Harvey P. Dale, Sep 08 2017 *)

A272338 Numbers such that antisigma(n) mod sigma(n) = phi(n), where antisigma(n) is the sum of the numbers less than n that do not divide n, sigma(n) is the sum of the divisors of n and phi(n) is the Euler totient function of n.

Original entry on oeis.org

3, 9, 27, 81, 243, 319, 729, 2187, 3615, 6561, 8159, 9807, 19683, 32791, 59049, 103679, 177147, 432864, 531441, 788852, 871215, 1594323, 2779519, 2826863, 2858240, 4782969, 7213536, 10036415, 14348907, 20428863, 24423359, 29036799, 33385279, 43046721

Offset: 1

Paolo P. Lava, Apr 26 2016

A000244 is a subset of this sequence.

			27*28/2 mod sigma(27) = 378 mod 40 = 18 = phi(27).

Giovanni Resta, Table of n, a(n) for n = 1..73 (terms < 2*10^12)

Cf. A000010, A000203, A000244, A024816, A232324, A272337.

with(numtheory): P:=proc(q) local n;
for n from 1 to q do if (n*(n+1)/2) mod sigma(n)=phi(n) then print(n); fi;
od; end: P(10^6);

Select[Range[10^5], Function[n, Mod[Total@ First@ #, Total@ Last@ #] == EulerPhi@ n &@ {Complement[Range@ n, #], #} &@ Divisors@ n]] (* Michael De Vlieger, Apr 27 2016 *)

Solutions of the equation A024816(n) mod A000203(n) = A000010(n).

a(27)-a(34) from Giovanni Resta, May 01 2016

A353001 Numbers k such that the k-th triangular number mod the sum (with multiplicity) of prime factors of k, and the k-th triangular number mod the sum of divisors of k, are both prime.

Original entry on oeis.org

4, 57, 70, 93, 129, 217, 322, 381, 417, 453, 513, 565, 597, 646, 682, 781, 813, 921, 925, 1057, 1081, 1102, 1137, 1165, 1197, 1261, 1317, 1393, 1405, 1558, 1582, 1641, 1750, 1798, 1846, 1857, 1918, 1929, 2073, 2101, 2110, 2173, 2181, 2305, 2329, 2361, 2482, 2506, 2569, 2577, 2626, 2649, 2653

Offset: 1

Robert Israel, Apr 14 2022

nonn

Numbers k such that A232324(k) and A352996(k) are prime.

			a(3) = 70 is a term because 70*71/2 = 2485, A000217(70) = 144, A001414(70) = 14, and both 2485 mod 144 = 37 and 2485 mod 14 = 7 are prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A352908 and A352997.

Cf. A000217, A001414, A232324, A352996.

filter:= proc(n) local a,b,c,t;
  a:= n*(n+1)/2;
  b:= add(t[1]*t[2],t=ifactors(n)[2]);
  if not isprime(a mod b) then return false fi;
  c:= numtheory:-sigma(n);
  isprime(a mod c)
end proc:
select(filter, [$2..3000]);

Select[Range[3000], And @@ PrimeQ[Mod[#*(# + 1)/2, {DivisorSigma[1, #], Plus @@ Times @@@ FactorInteger[#]}]] &] (* Amiram Eldar, Apr 15 2022 *)

A244926 Numbers m such that there is an integer k with the property that antisigma(m) = k * sigma(m) + k.

Original entry on oeis.org

1, 2, 247, 2279, 9167, 57479, 200479, 518039, 2119207, 3685439, 9240079, 16384279, 31536647, 101601359, 140558807, 189771287, 299142967, 354032447, 384150199, 486103279, 565468637, 802008239, 853795074, 1107541759, 1328438479, 1494742004, 1580837719, 1768013279

Offset: 1

Jaroslav Krizek, Jul 08 2014

nonn

Numbers m such that A244329(m) = floor(antisigma(m) / sigma(m)) = antisigma(m) mod sigma(m) = A232324(n).
Corresponding values of integers k: 0, 0, 108, 1092, 4488, 28500, 99792, 258300, 1058148, ...
Numbers m such that sigma(m) + 1 divides antisigma(m). - Kevin P. Thompson, Nov 27 2021

			247 is in sequence because 30348 = antisigma(247) = 108 * sigma(247) + 108 = 108*280 + 108.

Kevin P. Thompson, Table of n, a(n) for n = 1..49

Cf. A024816 (antisigma), A000203 (sigma), A244329, A232324.

[n: n in [1..100000] | Floor(((n*(n+1)div 2) - (SumOfDivisors(n))) div (SumOfDivisors(n))) eq ((n*(n+1)div 2) - (SumOfDivisors(n))) mod (SumOfDivisors(n))]

isok(m) = my(s=sigma(m)); denominator((m*(m+1)/2-s)/(s+1)) == 1; \\ Michel Marcus, Jan 21 2022

a(10)-a(28) from Kevin P. Thompson, Nov 27 2021

A272337 Numbers such that antisigma(n) mod sigma(n) = d(n), where antisigma(n) is the sum of the numbers less than n that do not divide n, sigma(n) is the sum of the divisors of n and d(n) is the number of divisors of n.

Original entry on oeis.org

3, 4, 52, 164, 275, 332, 388, 556, 668, 724, 892, 1004, 1172, 1228, 1396, 1676, 1732, 1844, 2012, 2348, 2404, 2572, 2908, 3076, 3188, 3244, 3356, 3412, 3524, 3748, 4084, 4196, 4252, 4364, 4868, 4924, 5036, 5204, 5596, 5708, 5932, 6044, 6212, 6268, 6436, 6548

Offset: 1

Paolo P. Lava, Apr 26 2016

			52*53/2 mod sigma(52) = 1378 mod 98 = 6 = d(52).

Cf. A000005, A000203, A024816, A232324, A272338.

with(numtheory): P:=proc(q) local n;
for n from 1 to q do if (n*(n+1)/2) mod sigma(n)=tau(n) then print(n); fi;
od; end: P(10^6);

Select[Range@ 6600, Function[n, Mod[Total@ First@ #, Total@ Last@ #] == Length@ Last@ # &@ {Complement[Range@ n, #], #} &@ Divisors@ n]] (* faster, or *)
Select[Range@ 6600, Mod[Total[Select[Range[# - 1], Function[m, ! Divisible[#, m]]]], DivisorSigma[1, #]] == DivisorSigma[0, #] &] (* Michael De Vlieger, Apr 27 2016 *)

isok(n) = n*(n+1)/2 % sigma(n) == numdiv(n); \\ Michel Marcus, Apr 29 2016

Solutions of the equation A024816(n) mod A000203(n) = A000005(n).

A353002 Numbers k such that the k-th triangular number mod the sum (with multiplicity) of prime factors of k, and the k-th triangular number mod the sum of divisors of k, are the same prime.

Original entry on oeis.org

93, 2653, 30433, 1922113, 15421122, 28776673, 240409057, 611393953, 2713190397, 5413336381

Offset: 1

J. M. Bergot and Robert Israel, Apr 15 2022

			a(1) = 93 is a term because 93*94/2 = 4371, A000217(93) = 128, A001414(93) = 34, and 4371 mod 128 = 4371 mod 34 = 19, which is prime.

Cf. A000217, A001414, A232324, A352996, A353001.

filter:= proc(n) local a,b,c,t;
  a:= n*(n+1)/2;
  b:= add(t[1]*t[2],t=ifactors(n)[2]);
  t:= a mod b; if not isprime(t) then return false fi;
  c:= numtheory:-sigma(n);
  a mod c = t
end proc:
select(filter, [$2..2*10^7]);

Select[Range[2*10^6], (r = Mod[#*(# + 1)/2, DivisorSigma[1, #]]) == Mod[#*(# + 1)/2, Plus @@ Times @@@ FactorInteger[#]] && PrimeQ[r] &] (* Amiram Eldar, Apr 15 2022 *)

a(8) from Amiram Eldar, Apr 15 2022

a(9)-a(10) from Daniel Suteu, May 12 2022

cp's OEIS Frontend

A352908 Numbers k such that A232324(k) is prime.

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A076617 Numbers k such that sum of the divisors d of k divides 1 + 2 + ... + k = k(k+1)/2.

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A066860 The sum of the non-divisors of n (less than n) is a multiple of the sum of the divisors of n.

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A232538 Numbers n such that (n(n+1)/2) modulo sigma(n) = n.

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A232540 Numbers n such that (n(n+1)/2) modulo sigma(n) = 1.

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A272338 Numbers such that antisigma(n) mod sigma(n) = phi(n), where antisigma(n) is the sum of the numbers less than n that do not divide n, sigma(n) is the sum of the divisors of n and phi(n) is the Euler totient function of n.

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A353001 Numbers k such that the k-th triangular number mod the sum (with multiplicity) of prime factors of k, and the k-th triangular number mod the sum of divisors of k, are both prime.

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