A076530 -id:A076530 - cp's OEIS frontend

A065900 Numbers n such that sigma(n) = sigma(n-1) + sigma(n-2).

3, 4, 24, 1968, 3264, 5016, 60456, 1016508, 4420164, 12055512, 14365608, 25726728, 27896424, 66562308, 72764736, 98734968, 175186656, 224868312, 253694928, 288657204, 386668344, 421575408, 504737748, 630645456, 1493548000, 1653797796, 2120325012, 2221315152

Offset: 1

Jason Earls, Dec 07 2001

nonn

No other terms less than 1000000000. - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 23 2004

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..112 (terms < 10^13, first 50 terms from Donovan Johnson)

Cf. A000203, A076530, A104149.

s1=1; s2=3; for(n=3, 10^8, s3=sigma(n); if(s3==s1+s2, print1(n ", ")); s1=s2; s2=s3) /* Donovan Johnson, Apr 06 2013 */

a(n) = A076530(n) + 1 = A104149(n) + 2. - Alex Ratushnyak, Jul 06 2013

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 23 2004

A104149 Numbers k such that sigma(k+2) = sigma(k+1) + sigma(k).

Original entry on oeis.org

1, 2, 22, 1966, 3262, 5014, 60454, 1016506, 4420162, 12055510, 14365606, 25726726, 27896422, 66562306, 72764734, 98734966, 175186654, 224868310, 253694926, 288657202, 386668342, 421575406, 504737746, 630645454, 1493547998, 1653797794, 2120325010, 2221315150

Offset: 1

Neven Juric (neven.juric(AT)apis-it.hr), Aug 16 2010

nonn

Apparently all terms > 1 are even. - Zak Seidov, Mar 23 2015

For n <= 95, no a(n) is divisible by 3; a(2), a(25) and a(57) == 2 (mod 3), the rest == 1 (mod 3). - Robert Israel, Mar 23 2015

			sigma(22) = 1+2+11+22 = 36.
sigma(23) = 1+23 = 24.
sigma(24) = 1+2+3+4+6+8+12+24 = 60.
sigma(24) = sigma(23) + sigma(22).

Giovanni Resta, Table of n, a(n) for n = 1..112 (terms < 10^13; first 50 terms from Donovan Johnson)

[n: n in [1..2*10^6] | SumOfDivisors(n+2) eq (SumOfDivisors(n+1)+SumOfDivisors(n))]; // Vincenzo Librandi, Mar 24 2015

Select[Range@ 100000, DivisorSigma[1, # + 2] == DivisorSigma[1, # + 1] + DivisorSigma[1, #] &] (* Michael De Vlieger, Mar 23 2015 *)
Position[Partition[DivisorSigma[1,Range[3*10^7]],3,1],?(#[[1]]+#[[2]]==#[[3]]&),1,Heads->False]//Flatten (* The program generates the first 13 terms *) (* _Harvey P. Dale, May 08 2018 *)

s1=1; s2=3; for(n=1, 10^8, s3=sigma(n+2); if(s3==s1+s2, print1(n ", ")); s1=s2; s2=s3) /* Donovan Johnson, Apr 08 2013 */

a(n) = A065900(n) - 2. - R. J. Mathar, Aug 19 2010

a(n) = A076530(n) - 1. - M. F. Hasler, Aug 19 2010

More terms from Zak Seidov and R. J. Mathar, Aug 19 2010

A227982 Numbers n such that sigma(n+1) - sigma(n-1) > sigma(n); where sigma(n) = A000203(n) = sum of the divisors of n.

Original entry on oeis.org

47, 59, 83, 107, 119, 143, 167, 179, 215, 239, 251, 263, 299, 335, 359, 383, 395, 419, 431, 467, 479, 503, 527, 539, 587, 599, 623, 629, 647, 659, 671, 719, 755, 767, 779, 839, 863, 887, 899, 923, 935, 959, 983, 1007, 1019, 1055, 1079, 1091, 1103, 1139, 1175

Offset: 1

Jaroslav Krizek, Aug 08 2013

nonn

See A076530 - numbers n such that sigma(n+1) - sigma(n-1) = sigma(n).

Complement of union of A076530, A227983 and number 1.

			Number 59 is in the sequence because sigma(60) - sigma(58) = 168 - 90 = 78 > sigma(59) = 60.

Jaroslav Krizek, Table of n, a(n) for n = 1..2000

Cf. A076530, A000203, A227983.

Flatten[Position[Partition[DivisorSigma[1,Range[1200]],3,1],?(Last[#]- First[#]> #[[2]]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Mar 31 2014 *)

A227983 Numbers n such that sigma(n+1) - sigma(n-1) < sigma(n); where sigma(n) = A000203(n) = sum of the divisors of n.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Jaroslav Krizek, Aug 08 2013

nonn

See A076530 - numbers n such that sigma(n+1) - sigma(n-1) = sigma(n).

Complement of union of A076530, A227982 and number 1.

			Number 60 is in the sequence because sigma(61) - sigma(59) = 62 - 60 = 2 < sigma(60) = 168.

Jaroslav Krizek, Table of n, a(n) for n = 1..2000

Cf. A076530, A000203, A227982.

Flatten[Position[Partition[DivisorSigma[1,Range[100]],3,1],?(#[[3]]- #[[1]]< #[[2]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Jul 25 2021 *)

A260420 Numbers n such that sigma(n+1) - sigma(n-1) = n+1.

Original entry on oeis.org

2, 3, 23, 14927, 31049, 69107, 246263, 5860169, 307164671, 881198663, 1489455647, 2386555631, 8225563703, 14311679063, 111494234183, 154357775303, 299004519623, 870455062823, 970388922263, 991817878343, 1677028870823, 1782783762503, 1830446935223

Offset: 1

M. F. Hasler, Jul 25 2015

nonn

Proposed by Jaroslav Krizek in A260071.

Also: numbers n such that A001065(n+1) = A000203(n-1).

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

Cf. A260071, A076530, A246852.

[n: n in [2..5*10^6] | DivisorSigma(1, n+1) - DivisorSigma(1, n-1) eq n+1]; // Vincenzo Librandi, Jul 26 2015

Select[Range@ 1000000, DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1] == # + 1 &] (* Michael De Vlieger, Jul 25 2015 *)

for(n=2,1e9,sigma(n+1)-sigma(n-1)==n+1&&print1(n","))

a(n) = A246852(n) + 1.

A260071 Primes p such that sigma(p) = sigma(p+1) - sigma(p-1).

Original entry on oeis.org

2, 3, 23, 970388922263, 991817878343, 1677028870823

Offset: 1

Jaroslav Krizek, Jul 14 2015

Primes from A076530 (numbers n such that sigma(n) = sigma(n+1) - sigma(n-1)).
Also primes from sequence A260420 (numbers n such that sigma(n+1) - sigma(n-1) = n+1).
If a number from A246852(n) + 1 is a prime p, then p is in the sequence.
If a(7) exists, it must be bigger than 10^13.

			23 is in the sequence because sigma(24) - sigma(22) = 60 - 36 = 24 = sigma(23).

Cf. A000203, A076530, A246852.

[n: n in [1..1000000] | IsPrime(n) and SumOfDivisors(n) eq ((SumOfDivisors(n+1)) - (SumOfDivisors(n-1)))];

Magma
```
[n: n in [A076530(n)] | IsPrime(n)];
```

is_ok(index)=my(p=prime(index)); p+1==sigma(p+1)-sigma(p-1);
main(size)=my(v=vector(size),index=1);for(i=1,size,while(!is_ok(index),index++);v[i]=prime(index);index++); v \\ Anders Hellström, Jul 14 2015

has(p)=p+1==sigma(p+1)-sigma(p-1)
select(has, primes(1000)) \\ Charles R Greathouse IV, Jul 22 2015

cp's OEIS Frontend

A065900 Numbers n such that sigma(n) = sigma(n-1) + sigma(n-2).

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A104149 Numbers k such that sigma(k+2) = sigma(k+1) + sigma(k).

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A227982 Numbers n such that sigma(n+1) - sigma(n-1) > sigma(n); where sigma(n) = A000203(n) = sum of the divisors of n.

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A227983 Numbers n such that sigma(n+1) - sigma(n-1) < sigma(n); where sigma(n) = A000203(n) = sum of the divisors of n.

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A260420 Numbers n such that sigma(n+1) - sigma(n-1) = n+1.

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A260071 Primes p such that sigma(p) = sigma(p+1) - sigma(p-1).

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