A050507 -id:A050507 - cp's OEIS frontend

A054799 Integers n such that sigma(n+2) = sigma(n) + 2, where sigma = A000203, the sum of divisors of n.

3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 434, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487

Offset: 1

Labos Elemer, May 22 2000

nonn

Only 3 composite numbers are known: 434, 8575, 8825. This sequence is the union of A050507 and A001359.

The terms are also the solutions of A001065(x) = A001065(x+2), where A001065(n) is the sum of proper divisors of n. - Michel Marcus, Nov 14 2014

			n = 434, divisors = {1, 2, 7, 14, 31, 62, 217, 434}, sigma(434) = 768, sigma(436) = 770; n = 8575, divisors = {1, 5, 7, 25, 35, 49, 175, 245, 343, 1225, 1715, 8575}, sigma(8575) = 12400, sigma(8577) = 12402; n = 8825, divisors = {1, 5, 25, 353, 1765, 8825}, sigma(8525) = 10974, sigma(8527) = 10976.

Sivaramakrishnan, R. (1989): Classical Theory of Arithmetical Functions., M.Dekker Inc., New York, Problem 12 in Chapter V., p. 81.

Cf. A000203, A001359, A050507.

Select[Range[1500],DivisorSigma[1,#+2]==DivisorSigma[1,#]+2&] (* Jayanta Basu, May 01 2013 *)

is(n)=sigma(n+2)==sigma(n)+2 \\ Charles R Greathouse IV, Feb 13 2013

A056774 Composite n such that phi(n+2) = phi(n)+2.

Original entry on oeis.org

6, 12, 14, 18, 20, 44, 62, 92, 116, 164, 212, 254, 332, 356, 452, 524, 692, 716, 764, 932, 956, 1004, 1124, 1172, 1436, 1676, 1724, 1772, 1964, 2036, 2372, 2564, 2612, 2636, 2732, 2876, 2972, 3044, 3236, 3644, 3812, 4052, 4076, 4124, 4196, 4412, 4892

Offset: 1

Labos Elemer, Aug 17 2000

nonn

Below 100000 no common composite solutions with sigma(n+2)=sigma(n)+2, while prime solutions are common.

phi(x)+2=phi(x+2) is equivalent to cototient(x+2)=cototient(x), so also defines closest numbers with identical value of cototients (A051953), either primes or composites.

			n=254, phi(254+2) = phi(256) = 128 = phi(254)+2 = 126+2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A001838, A055458, A015913-A015917, A050507, A054799.

Select[Range[5000],CompositeQ[#]&&EulerPhi[#]+2==EulerPhi[#+2]&] (* Harvey P. Dale, Jul 10 2017 *)

isok(n) = !isprime(n) && (eulerphi(n+2) == eulerphi(n)+2); \\ Michel Marcus, Aug 30 2019

A084293 a(n) = 2n + A054905(n).

Original entry on oeis.org

436, 305635361, 110, 35, 195566, 77, 26, 55, 38, 76, 938, 104, 212308, 85, 74, 106677, 86, 161

Offset: 1

Labos Elemer, May 26 2003

The sequence begins 436, 305635361, 110, 35, 195566, 77, 26, 55, 38, 76, 938, 104, 212308, 85, 74, 106677, 86, 161, ?, 91, 87, 92, 122, 111, 1585396, 145, 94, 76627, 10283, 159, 772, 133, 122, 412, 194, 142, 964, 205, 374, 925, 6725, 209, ?, 1015, 178, ?, ?, 206, 146, ?, ..., where the other missing terms (designated by "?") are unknown, if they exist (see also A206768).

			To terms of A054905, where sigma(x+2n)=sigma(x)+2n replacing x+2n=y,x=y-2n, we get sigma(y)-2n=sigma(y-2n);
For several analogous sequences, the corresponding "mirror-solutions" can be easily constructed. See: e.g. A015913-A015918; A050507, A054799, A054903-A054906; A054982-A054987; A059118; A055009, A055458, A063500, etc.

Cf. A054905.

Composite x satisfying sigma(x-2n) = sigma(x) - 2n.

A322668 Numbers k such that (k, k+2) are not twin primes yet sigma(k+2)/d(k+2) - sigma(k)/d(k) = 1.

Original entry on oeis.org

1, 350, 6497, 12317, 133787, 181427, 404471, 439097, 485237, 501182, 549378, 1410119, 2696807, 6220607, 6827369, 6954767, 9770027, 10302419, 10449347, 10887977, 11014007, 16745387, 18959111, 25883519, 27334469, 39508037, 40311149, 40551617, 42561437, 44592209

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

A variation of A050507 with average of the divisors instead of their sum.

Amiram Eldar, Table of n, a(n) for n = 1..241 (terms below 10^10)

Cf. A000005 (number of divisors), A000203 (sum of divisors).

Cf. A050507, A056774, A057020, A057021, A238380.

f[n_] := DivisorSigma[1, n]/DivisorSigma[0, n]; aQ[n_] := f[n + 2] - f[n] ==  1 && !(PrimeQ[n] && PrimeQ[n + 2]); Select[Range[1000000], aQ]

isok(k) = !(isprime(k) && isprime(k+2)) && (sigma(k+2)/numdiv(k+2) - sigma(k)/numdiv(k) == 1); \\ Michel Marcus, Jan 22 2019

A084292 a(n) = 6n + A054904(n).

Original entry on oeis.org

110, 77, 38, 104, 74, 161, 87, 111, 94, 159, 122, 142, 374, 209, 178, 206, 206, 253, 326, 302, 206, 302, 471, 249, 519, 341, 346, 303, 354, 481, 542, 377, 2057, 533, 386, 411, 5138, 662, 846, 527, 386, 437, 1034, 519, 794, 689, 626, 493, 566, 629, 873, 527, 638

Offset: 1

Labos Elemer, May 26 2003

nonn

Composite solutions y to sigma(y-6n) = sigma(y) - 6n. For terms x of A054904, where sigma(x+6n) = sigma(x) + 6n, replacing x+6n = y, x = y-6n, we get sigma(y) - 6n = sigma(y-6n).

Amiram Eldar, Table of n, a(n) for n = 1..724

Cf. A000203 (sigma), A054904, A084293.

For several analogous sequences such corresponding "mirror-solutions" can be easily constructed. See, e.g., A015913-A015918, A050507, A054799, A054903-A054906, A054982-A054987, A059118, A055009, A055458, A063500, etc.

A217259 Numbers n such that sigma(n+1) - sigma(n-1) = 2; sigma(n) = A000203(n) = sum of divisors of n.

Original entry on oeis.org

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 435, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488

Offset: 1

Jaroslav Krizek, Mar 17 2013

nonn

Also numbers n such that antisigma(n+1) - antisigma(n-1) = 2*n - 1.
Antisigma(n) = A024816(n) = sum of nondivisors of n.
Union of A014574 (average of twin prime pairs) and sequence 435, 8576, 8826, … (= all terms < 100000).
If n = average of twin prime pairs (q < p) then antisigma(p) - antisigma(q) = 2*n - 1 = p + q - 1.
No term found below 2*10^9 to continue sequence 435, 8576, 8826, ... - Michel Marcus, Mar 19 2013

			Number 435 is in sequence because antisigma(436) - antisigma(434) = 94496 - 93627 = 869 = 2*435 - 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..1227 (all terms < 100000)

Cf. A014574, A015886, A024816, A050507, A206768.

Equals A054799 + 1. - Michel Marcus, May 21 2018

Flatten[Position[Partition[DivisorSigma[1,Range[1500]],3,1],?(#[[3]]- #[[1]] == 2&),1,Heads->False]]+1 (* _Harvey P. Dale, May 03 2018 *)

isok(n) = (sigma(n+1) - sigma(n-1)) == 2; \\ Michel Marcus, May 20 2018

cp's OEIS Frontend

A054799 Integers n such that sigma(n+2) = sigma(n) + 2, where sigma = A000203, the sum of divisors of n.

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A056774 Composite n such that phi(n+2) = phi(n)+2.

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A084293 a(n) = 2n + A054905(n).

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A322668 Numbers k such that (k, k+2) are not twin primes yet sigma(k+2)/d(k+2) - sigma(k)/d(k) = 1.

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A084292 a(n) = 6n + A054904(n).

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A217259 Numbers n such that sigma(n+1) - sigma(n-1) = 2; sigma(n) = A000203(n) = sum of divisors of n.

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