cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A227759 Numbers n such that A227758(n) = sigma(sigma(n)) - sigma(n) - n < 0, where sigma(n) = A000203(n) = sum of the divisors of n.

Original entry on oeis.org

1, 2, 4, 9, 13, 16, 18, 25, 36, 37, 43, 49, 50, 61, 64, 67, 73, 81, 97, 98, 100, 109, 121, 144, 151, 157, 163, 169, 181, 193, 211, 225, 229, 241, 242, 256, 277, 283, 289, 313, 324, 331, 337, 338, 361, 373, 397, 400, 409, 421, 433, 441, 457, 484, 487, 523, 529
Offset: 1

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Author

Jaroslav Krizek, Jul 29 2013

Keywords

Comments

Numbers n such that A051027(n) - A000203(n) - n < 0, where A000203(n) = sum of the divisors of n , A051027(n) = A000203(A000203(n)) = sigma(sigma(n)) = sum of the divisors of the sum of the divisors of n.
Conjecture: a(n) = complement of union A000668 and A227760, where A000668 = Mersenne primes, A227760 = numbers n such that sigma(sigma(n)) - sigma(n) - n > 0.

Examples

			Number 16 is in sequence because sigma(sigma(16)) - sigma(16) - 16 = 32 - 31 - 16 = -15 < 0.

Crossrefs

Cf. A000203, A051027, A000668, A227760, A227758.

Formula

A227758(a(n)) < 0.

A227760 Numbers n such that A227758(n) = sigma(sigma(n)) - sigma(n) - n > 0, where sigma(n) = A000203(n) = sum of the divisors of n.

Original entry on oeis.org

5, 6, 8, 10, 11, 12, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87
Offset: 1

Views

Author

Jaroslav Krizek, Jul 29 2013

Keywords

Comments

Numbers n such that A051027(n) - A000203(n) - n < 0, where A000203(n) = sum of the divisors of n , A051027(n) = A000203(A000203(n)) = sigma(sigma(n)) = sum of the divisors of the sum of the divisors of n.
Conjecture: a(n) = complement of union A000668 and A227759, where A000668 = Mersenne primes, A227759 = numbers n such that sigma(sigma(n)) - sigma(n) - n < 0.

Examples

			Number 15 is in sequence because sigma(sigma(15)) - sigma(15) - 15 = 60 - 24 - 15 = 21 > 0.

Crossrefs

Cf. A000203, A051027, A000668, A227759, A227758.

Programs

  • Mathematica
    sgmaQ[n_]:=Module[{s=DivisorSigma[1,n]},Positive[DivisorSigma[1,s]-s-n]]; Select[Range[100],sgmaQ] (* Harvey P. Dale, Aug 08 2013 *)

Formula

A227758(a(n)) > 0.

A227756 Primes p such that antisigma(p) = antisigma(p+1) + 12, where antisigma = A024816.

Original entry on oeis.org

23, 29, 41, 53, 101, 113, 137, 173, 257, 281, 317, 353, 401, 617, 641, 653, 677, 761, 821, 941, 977, 1181, 1193, 1361, 1373, 1433, 1613, 1697, 1877, 1901, 2081, 2153, 2237, 2273, 2297, 2333, 2381, 2633, 2657, 2693, 2741, 2777, 2801, 3137, 3413, 3461, 3557
Offset: 1

Views

Author

Jaroslav Krizek, Jul 26 2013

Keywords

Comments

Primes p such that sigma(p + 1) = 2*p + 14.
This is the subsequence of primes in A227757.
Also primes p such that sigma(sigma(p)) - sigma(p) - p = 13 (see A227758). The composite numbers with this property are 333, 37377, 972691, 1089871,...

Examples

			The prime 41 is in sequence because antisigma(41) = 819 = antisigma(42) + 12 = 807 + 12.

Links

Crossrefs

Cf. A024816, A227757, A051027.

Programs

  • Mathematica
    Select[Prime[Range[500]],DivisorSigma[1,# + 1] == 2*# + 14 &] (* Stefano Spezia, Apr 18 2025 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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