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.

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A252234 Numbers n such that there exists an m so that squarefree kernel of n = squarefree kernel of m, and n is the sum of the proper divisors of m (m may equal n).

Original entry on oeis.org

6, 28, 36, 50, 240, 312, 384, 450, 496, 810, 1008, 1344, 4256, 4536, 8128, 10800, 11700, 14112, 15288, 19656, 23040, 49686, 90720, 95040, 98280, 98553, 124848, 129024, 153760, 249018, 256932, 260100, 378225, 404586, 454860, 532224, 561834, 700245, 714240
Offset: 1

Views

Author

Naohiro Nomoto, Dec 15 2014

Keywords

Comments

Since m=n is allowed, perfect numbers (A000396) are terms of this sequence. - Michel Marcus, Jan 02 2015
m: 6, 24, 28, 40, 120, 216, 234, 270, 360, 496, 588, 672, 2016, ..., . - Robert G. Wilson v, Feb 28 2015
Odd members are 98553, 378225, 700245, ..., . - Robert G. Wilson v, Feb 28 2015

Examples

			For n = 36, m = 24, 36 is the sum of the proper divisors of 24, and rad(36) = rad(24) = 6.

Links

Crossrefs

Cf. A001065 (sum of proper divisors of n), A007947 (the squarefree kernel of n).
Cf. A048138, A152454, A252997.

Programs

  • Mathematica
    rad[n_] := Times @@ (First@ # & /@ FactorInteger@ n); f[n_] := Block[{sd = DivisorSigma[1, n] - n}, If[ rad[n] == rad[sd], sd, 0]]; k = 1; lst = {}; While[k < 1000001, a = f@ k; If[a > 0, AppendTo[lst, a]]; k++]; Sort@ lst (* Robert G. Wilson v, Feb 28 2015 *)

A253302 Smallest integer m such that gcd{x | sum of proper divisors of x is m} is equal to n, when there are at least two such x's.

Original entry on oeis.org

1, 16, 300, 136, 50316, 186, 97748, 1352, 72318, 1340, 1652, 1356
Offset: 1

Views

Author

Michel Marcus, Dec 30 2014

Keywords

Comments

Only integers m that satisfy A048138(m) > 1 are considered here.
Other known terms with odd indices: a(25)=39926, a(49)=46850.
For terms with even indices see A253303.

Examples

			For every prime p the sum of proper divisors of p is 1, and the GCD of the primes is 1, so a(1) = 1.

Crossrefs

Cf. A001065 (sum of proper divisors), A048138, A152454, A253303.

Programs

  • PARI
    build(nb) = {vs = vector(nb); nc = nb^2; forcomposite(n=2, nc, val = sigma(n)-n; if (val <= nb, if (vs[val] == 0, vs[val] = -n, vs[val] = gcd(vs[val], n)););); vs[1] = 1; vs;}
lista() = {vd = build(100000); vmax = 20; vr = vector(vmax); for (ig=1, vmax, for (i=1, #vd, w = vd[i]; if (w ==ig, vr[ig] = i; break;););); for (i=1, #vr, if (vr[i] == 0, break, print1(vr[i], ", ")));}

Extensions

a(1)=1 suggested by Robert Israel, Jan 01 2015

A363461 Least n-untouchable number.

Original entry on oeis.org

2, 208, 388, 298, 838
Offset: 1

Views

Author

Jinyuan Wang, Jun 03 2023

Keywords

Comments

Let s^m(k) denote the m-th iterate of s(k) = sigma(k) - k. n-untouchable numbers are the numbers that lie in the image of s^(n-1)(k), but not in the image of s^n(k).

Links

Crossrefs

Cf. A005114, A152454, A283152, A284147, A284156, A284187.

A365619 a(n) is the least integer k such that A366110(k) = n, or 0 if there is no such k.

Original entry on oeis.org

0, 0, 0, 0, 0, 454, 0, 13, 0, 0, 0, 19, 0, 16, 0, 17, 15, 22, 6, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 396, 0, 0, 0, 46, 0, 148, 40, 0, 8, 0, 0, 652, 0, 15980, 0, 0, 0, 25, 0, 0, 0, 0, 0, 50, 0, 0, 0, 0, 0, 136, 0, 0, 0, 27, 0, 64, 0, 0, 0, 100, 0, 29, 21, 0, 0, 0, 0, 15574, 0, 0, 0, 0, 0, 346
Offset: 1

Views

Author

Michel Marcus, Nov 02 2023

Keywords

Comments

The nonzero values are sure, but the zeros are conjectured.
Search was conducted up to 10^5.

Links

Crossrefs

Cf. A152454, A366110.

Programs

  • PARI
    lista(ns) = my(nn=100); my(vs = vector(ns, k, [])); forcomposite (i=1, (ns/2)^2, my(x=sigma(i)-i); if (x <= ns, if (#vs[x] <= 1, vs[x] = concat(vs[x], i), vs[x][2] = i));); my(v = vector(nn)); for (i=1, #vs, if (#vs[i] > 1, my(d = vecmax(vs[i])-vecmin(vs[i])); if ((d<=nn) && (!v[d] || (v[d]>i)), v[d] = i););); v;
lista(2*10^4)

A252056 a(n) is the least m such that m = A001065(j) = A001065(k) where j != k, A000005(j) = A000005(k) = n; or 0 if no such m exists.

Original entry on oeis.org

0, 1, 0, 13, 0, 73, 0, 106, 9064940, 4001, 0, 396, 0
Offset: 1

Views

Author

Naohiro Nomoto, Dec 13 2014

Keywords

Comments

When n>2 and A001055(n)=1, then a(n)=0; because in that case, only a prime^n has n divisors, and then it is not possible to get twice the same value for sigma(x)-x. This happens for n=3, 5, 7, 11, 13, 17, 19, 23, 29, ... - Michel Marcus, Dec 16 2014
Note that for n=8, j and k do not have the same prime signature. - Michel Marcus, Dec 17 2014

Examples

			For n=2, all primes have 2 divisors and satisfy sigma(x)-x=1, so a(2) = 1.
For n=4, 27 and 35 have 4 divisors and the sum of their proper divisors is 13 for both (1+3+9 and 1+5+7).
For n=6, 98 and 175 have 6 divisors and the sum of their proper divisors is 73 for both (1+2+7+14+49 and 1+5+7+25+35).
For n=8, 104 and 110 have 8 divisors and the sum of their proper divisors is 106 for both (1+2+4+8+13+26+52 and 1+2+5+10+11+22+55).
For n=9, 163^2*167^2 and 61^2*353^2 have 9 divisors and the sum of their proper divisors is 9064940 for both.
For n=10, 7203 and 7857 have 10 divisors and the sum of their proper divisors is 4001 for both.
For n=12, 276 and 306 have 12 divisors and the sum of their proper divisors is 396 for both.

Crossrefs

Cf. A000005 (number of divisors of n), A001065 (sum of proper divisors of n).
Cf. A001055, A048138, A152454.

Extensions

a(9)-a(13) from Michel Marcus, Dec 16 2014
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