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 1-6 of 6 results.

A093892 Index of first occurrence of n in A093890, or 0 if no such number exists.

Original entry on oeis.org

1, 3, 2, 51, 4, 6, 8, 10, 242, 12, 363, 16, 18, 20, 92, 50, 28, 24, 32, 2514, 30, 1002, 1434, 894, 36, 474, 678, 1338, 354, 402, 48, 64, 174, 138, 66, 105, 114, 255, 102, 60, 627, 88, 80, 470, 72, 130, 104, 100, 84, 170, 190, 90, 152, 112, 96, 136, 406, 595, 385, 108
Offset: 0

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

Conjecture: there are infinitely many zeros in this sequence.

Crossrefs

Formula

A093890[a(n)]=n. - R. J. Mathar, Feb 05 2007

Extensions

Corrected by R. J. Mathar, Feb 05 2007
More terms from David Wasserman, Apr 23 2007
Obsolete comment removed by R. J. Mathar, Jul 22 2009

A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

Original entry on oeis.org

1, 3, 3, 7, 3, 12, 3, 15, 7, 15, 3, 28, 3, 15, 15, 31, 3, 39, 3, 42, 15, 15, 3, 60, 7, 15, 15, 56, 3, 72, 3, 63, 15, 15, 15, 91, 3, 15, 15, 90, 3, 96, 3, 63, 55, 15, 3, 124, 7, 63, 15, 63, 3, 120, 15, 120, 15, 15, 3, 168, 3, 15, 59, 127, 15, 144, 3, 63, 15, 142, 3, 195, 3, 15, 63, 63
Offset: 1

Views

Author

Emeric Deutsch, May 15 2006

Keywords

Comments

If a(n)=sigma(n) (=sum of the divisors of n =A000203(n); i.e. all numbers from 1 to sigma(n) are sums of distinct divisors of n), then n is called a practical number (A005153). The actual sums obtained from the divisors of n are given in row n of the triangle A119348.
The records appear to occur at the highly abundant numbers, A002093, excluding 3 and 10. For n in A174533, a(n) = sigma(n)-2. - T. D. Noe, Mar 29 2010
The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020
Zumkeller numbers A083207 give the positions of even terms in this sequence (likewise, the positions of odd terms in A308605). - Antti Karttunen and Ilya Gutkovskiy, Nov 29 2024

Examples

			a(5)=3 because the divisors of 5 are 1 and 5 and all the possible sums: are 1,5 and 6; a(6)=12 because we can form all sums 1,2,...,12 by adding up the terms of a nonempty subset of the divisors 1,2,3,6 of 6.
		

Crossrefs

One less than A308605.
Cf. A083207 (positions of even terms).

Programs

  • Haskell
    import Data.List (subsequences, nub)
    a119347 = length . nub . map sum . tail . subsequences . a027750_row'
    -- Reinhard Zumkeller, Jun 27 2015
    
  • Maple
    with(numtheory): with(linalg): a:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)}) end: seq(a(n),n=1..90);
  • Mathematica
    a[n_] := Total /@ Rest[Subsets[Divisors[n]]] // Union // Length;
    Array[a, 100] (* Jean-François Alcover, Jan 27 2018 *)
  • PARI
    A119347(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); sum(i=1,poldegree(p),(0Antti Karttunen, Nov 28 2024
    
  • PARI
    A119347(n) = { my(c=[0]); fordiv(n, d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024
  • Python
    from sympy import divisors
    def A119347(n):
        c = {0}
        for d in divisors(n,generator=True):
            c |=  {a+d for a in c}
        return len(c)-1 # Chai Wah Wu, Jul 05 2023
    

Formula

For n > 1, 3 <= a(n) <= sigma(n). - Charles R Greathouse IV, Feb 11 2019
For p prime, a(p) = 3. For k >= 0, a(2^k) = 2^(k + 1) - 1. - Ctibor O. Zizka, Oct 19 2023
From Antti Karttunen, Nov 29 2024: (Start)
a(n) = A308605(n)-1.
a(n) = 2*(A237290(n)/A000203(n)) - 1. [Found by Sequence Machine. See A237290.]
a(n) <= A100587(n).
(End)

Extensions

Definition clarified by Antti Karttunen, Nov 29 2024

A093891 Numbers k such that every prime up to sigma(k) is a sum of divisors of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

Sequence is infinite as sigma (2^n) = 2^(n+1)-1 and a(2^n) = pi(2^(n+1)-1).
Does this sequence include any non-members of A005153 other than 10, 70 and 836? - Franklin T. Adams-Watters, Apr 28 2006
The answer to the previous comment is yes, this sequence has many terms that are not in A005153. See A174434. - T. D. Noe, Mar 19 2010

Examples

			4 is a member as sigma(4) = 7 and all the primes up to 7 are a partial sum of divisors of 4, since divisors of 4 are 1, 2 and 4 and because primes arising are 2, 3 = 1+2, 5 = 1+4 and 7 = 1+2+4.
		

Crossrefs

Programs

  • Mathematica
    Select[Range[240], SubsetQ[Total /@ Rest@ Subsets@ Divisors[#], Prime@ Range@ PrimePi@ DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 19 2021 *)
  • PARI
    isok(m) = {my(d=divisors(m), vp = primes(primepi(sigma(m)))); for (i=1, 2^#d - 1, my(b = Vecrev(binary(i)), x = sum(k=1, #b, b[k]*d[k])); if (vecsearch(vp, x), vp = setminus(vp, Set(x))); if (#vp == 0, return (1)););} \\ Michel Marcus, Mar 19 2021

Extensions

More terms from Franklin T. Adams-Watters, Apr 28 2006

A093893 Numbers n such that every sum of two or more divisors is composite.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 183, 191, 193, 197, 199, 211, 213, 217, 223, 227, 229, 233
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

All terms are odd and very few are composite. Every odd prime is a trivial member.
Very few terms have more than four divisors. The smallest such term is 4753, which has six divisors: 1,7,49,97,679,4753. - Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

Crossrefs

Programs

  • Mathematica
    For[a:=3, a<=500, s =Divisors[a];n := 1;d := False; While[(n<=2^Length[s])\[And]( ["not" character]d), If[Length[NthSubset[n, s]]>=2, If[ !PrimeQ[Plus@@NthSubset[n, s]], n++, d:= True], n++ ]]; If[ ["not" character]d, Print[a]];a+=2]; (Kalman)
    fQ[n_] := Union@ PrimeQ[Plus @@@ Subsets[ Divisors@n, {2, Infinity}]] == {False}; Select[ Range[3, 235, 2], fQ@# &] (* Robert G. Wilson v, May 25 2009 *)

Extensions

More terms from Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004
a(1)=1 prepended by Max Alekseyev, Mar 31 2015

A093894 Composite members of A093893.

Original entry on oeis.org

49, 87, 91, 121, 133, 169, 183, 213, 217, 247, 249, 259, 287, 301, 339, 343, 361, 403, 411, 427, 445, 469, 473, 481, 501, 511, 527, 529, 553, 559, 581, 589, 591, 633, 679, 699, 703, 713, 717, 721, 763, 789, 793, 817, 841, 843, 871, 889, 895, 949, 951, 961
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

Comment: Most terms of this sequence have four divisors. Some terms (the squares of primes) have three divisors; very few terms have more than four divisors (the first such term is 4753, with six). Conjecture: This sequence is infinite. - Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

Examples

			133 is a term, the divisors are 1,7,19,133 and no sum of two or more gives a prime.
		

Crossrefs

Programs

  • Mathematica
    For[a:=4, a<=2000, s =Divisors[a];n := 1;d := False; While[(n<=2^Length[s])\[And]( ["not" character]d), If[Length[NthSubset[n, s]]>=2, If[ !PrimeQ[Plus@@NthSubset[n, s]], n++, d:= True], n++ ]]; If[ ["not" character]d, Print[a]];a++;While[PrimeQ[a], a+=2]]; (* Adam M. Kalman, Nov 11 2004 *)

Extensions

Corrected and extended by Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

A331665 Numbers k with a record number of divisors d < sqrt(k) such that d + k/d is prime.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 3570, 4830, 11550, 30030, 43890, 111930, 131670, 510510, 690690, 870870, 1021020, 2459730, 9699690, 13123110, 17160990, 40750710, 146006070, 223092870, 340510170, 358888530, 688677990, 1462190730, 2445553110, 2911018110, 6469693230
Offset: 1

Views

Author

Amiram Eldar, Jan 23 2020

Keywords

Comments

The corresponding record values are 0, 1, 2, 4, 8, 12, 13, 14, 15, 21, 24, 25, 29, 40, 41, 46, 49, 51, 70, 77, 88, 89, 90, 117, 120, 147, 153, 154, 155, 161, 263, ...
Apparently all the primorial numbers (A002110) are terms. The record values of terms that are primorial numbers are terms of A103787.

Examples

			2 has one divisor below sqrt(2), 1, such that 1 + 2/1 = 3 is prime.
6 has 2 divisors below sqrt(6), 1 and 2, such that 1 + 6/1 = 7 and 2 + 6/2 = 5 are primes.
30 has 4 divisors below sqrt(30), 1, 2, 3, and 5 such that 1 + 30/1 = 31, 2 + 30/2 = 17, 3 + 30/3 = 13 and 5 + 30/5 = 11 are primes.
		

Crossrefs

Programs

  • Mathematica
    divCount[n_] := DivisorSum[n, Boole @ PrimeQ[# + n/#] &, #^2 < n &]; seq = {}; dm = -1; Do[d1 = divCount[n]; If[d1 > dm, dm = d1; AppendTo[seq, n]], {n, 1,10^6}]; seq
Showing 1-6 of 6 results.