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 1-10 of 32 results. Next

A293183 Numbers k such that bsigma(k) = bsigma(k+1), where bsigma(k) is the sum of the bi-unitary divisors of k (A188999).

Original entry on oeis.org

14, 27, 44, 459, 620, 957, 1334, 1634, 1652, 2204, 2685, 3195, 3451, 3956, 5547, 8636, 8907, 9844, 11515, 11745, 16874, 19491, 20145, 20155, 27643, 31724, 33998, 38180, 41265, 41547, 42818, 45716, 48364, 64665, 74875, 74918, 79316, 79826, 79833, 83780, 84134
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2017

Keywords

Examples

			14 is in the sequence since bsigma(14) = bsigma(15) = 24.

Links

Crossrefs

Cf. A002961, A064125, A188999.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=
DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; a = {}; b1 = 0; Do[b2 = bsigma[k]; If[b1 == b2, a = AppendTo[a, k - 1]]; b1 = b2, {k, 1, 10^6}]; a (* after Michael De Vlieger at A188999 *)

A306985 Numbers k such that isigma(k) = isigma(k+1), where isigma(k) is the sum of the infinitary divisors of k (A049417).

Original entry on oeis.org

14, 27, 44, 459, 620, 957, 1334, 1634, 1652, 2204, 2685, 3195, 3451, 3956, 4064, 4544, 5547, 8495, 8636, 8907, 9844, 11515, 15296, 19491, 20145, 20155, 27643, 31724, 33998, 38180, 41265, 41547, 42818, 45716, 48364, 61964, 64665, 74875, 74918, 79316, 79826
Offset: 1

Views

Author

Amiram Eldar, Mar 18 2019

Keywords

Comments

a(n) differs from A293183(n) starting at n = 15.

Examples

			14 is in the sequence since isigma(14) = isigma(15) = 24.

Links

Crossrefs

Cf. A002961, A049417, A064125, A293183.

Programs

  • Mathematica
    fun[p_,e_] := Module[{ b = IntegerDigits[e,2]}, m=Length[b]; Product[If[b[[j]]>0, 1+p^(2^(m-j)),1], {j,1,m}]]; isigma[1]=1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; aQ[n_] := isigma[n] == isigma[n+1]; Select[Range[1000], aQ]

A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Original entry on oeis.org

2, 3, 4, 186, 318, 434, 473, 582, 730, 978, 1024, 1035, 1245, 1357, 1397, 1506, 1661, 1902, 2085, 2116, 2224, 2329, 2453, 2505, 2506, 2770, 2954, 3144, 3345, 3377, 3624, 3641, 3765, 3790, 3882, 4037, 4172, 4438, 4898, 4938, 4975, 5221, 6126, 6285, 6312, 6356
Offset: 1

Views

Author

Amiram Eldar, Sep 03 2019

Keywords

Examples

			186 is in the sequence since A070039(186) = A070039(187) = 12.

Links

Crossrefs

Cf. A002961, A064115, A064125, A070039, A164522, A164557, A293183, A306985, A325175.

Programs

  • Mathematica
    s[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; seq={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 6500}]; seq

A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 4, 7, 20, 31, 57, 94, 98, 118, 122, 127, 201, 213, 218, 230, 242, 243, 244, 334, 384, 393, 423, 429, 481, 565, 603, 633, 694, 704, 729, 766, 844, 921, 1138, 1141, 1221, 1262, 1401, 1533, 1654, 1726, 1761, 1837, 1838, 1862, 1882, 1942, 2162, 2245, 2361, 2362
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2020

Keywords

Comments

Numbers k such that A093653(k) = A093653(k+1).
The Mersenne primes (A000668) are terms since if 2^p - 1 is a prime then A093653(2^p-1) = A093653(2^p) = p+1.

Examples

			3 is a term since A093653(3) = A093653(4) = 3.

Links

Crossrefs

Cf. A000120, A093653.
A000668 is a subsequence.
Similar sequences: A002961, A005237, A054004, A064125, A238380, A293183, A306985.

Programs

  • Mathematica
    f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; f1 = f[1]; Do[f2 = f[n]; If[f1 == f2, AppendTo[s, n - 1]]; f1 = f2, {n, 2, 240}]; s

A164522 Numbers k such that sigma_odd(k) = sigma_odd(k+1), where sigma_odd(k) is the sum of the odd divisors of k (A000593).

Original entry on oeis.org

1, 27089, 115289, 233729, 2529090, 2880989, 14059709, 17192909, 17540250, 18693990, 34902630, 54722249, 58517910, 82200689, 83087730, 92991990, 93623250, 93862230, 96578369, 111681990, 112244369, 155120129, 206450369, 269626769, 293182469, 303206310, 324764910
Offset: 1

Views

Author

Amiram Eldar, Aug 12 2019

Keywords

Examples

			27089 is in the sequence since A000593(27089) = A000593(27089 + 1) = 27456.

Links

Crossrefs

Cf. A000593, A002961, A064115, A064125, A293183, A306985.

Programs

  • Magma
    v:=[&+[d:d in Divisors(m)|IsOdd(d)] :m in [1..5000000]]; [k:k in [1..#v-1]| v[k] eq v[k+1]]; // Marius A. Burtea, Aug 12 2019
  • Mathematica
    f[p_, e_] := If[p == 2, 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); s1=0; seq={}; Do[s2 = s[n]; If[s2 == s1, AppendTo[ seq, n-1]]; s1 = s2, {n, 1, 10^6}]; seq

A164557 Numbers k such that s(k) = s(k+1), where s(k) is the sum of divisors d of k such that k/d is odd (A002131).

Original entry on oeis.org

3, 6, 7, 10, 22, 31, 46, 58, 69, 82, 106, 127, 140, 154, 160, 166, 178, 226, 262, 286, 346, 358, 382, 466, 478, 502, 562, 586, 718, 748, 781, 838, 862, 886, 982, 1001, 1018, 1066, 1186, 1282, 1299, 1306, 1318, 1366, 1438, 1486, 1522, 1614, 1618, 1672, 1704, 1822
Offset: 1

Views

Author

Amiram Eldar, Aug 12 2019

Keywords

Examples

			3 is in the sequence since A002131(3) = A002131(3 + 1) = 4.

Links

Crossrefs

Cf. A002131, A002961, A064115, A064125, A293183, A306985.

Programs

  • Magma
    v:=[&+[d:d in Divisors(m)|IsOdd(Floor(m/d))] :m in [1..2000]]; [k:k in [1..#v-1]| v[k] eq v[k+1]]; // Marius A. Burtea, Aug 12 2019
  • Mathematica
    f[p_, e_] := If[p == 2, p^e, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); s1=0; seq={}; Do[s2 = s[n]; If[s2 == s1, AppendTo[seq, n-1]]; s1 = s2, {n, 1, 2000}]; seq

A334020 Numbers k such that s(k) = s(k+1), where s(k) is the sum of unitary divisors of k that are smaller than sqrt(k) (A334019).

Original entry on oeis.org

2, 3, 4, 7, 8, 16, 31, 127, 186, 256, 318, 434, 473, 574, 582, 588, 730, 735, 819, 978, 1245, 1357, 1374, 1397, 1420, 1421, 1500, 1506, 1661, 1694, 1902, 1956, 1988, 2059, 2085, 2147, 2166, 2329, 2453, 2505, 2506, 2534, 2754, 2770, 2868, 2954, 2988, 3345, 3377
Offset: 1

Views

Author

Amiram Eldar, Apr 12 2020

Keywords

Examples

			2 is a term since A334019(2) = A334019(3) = 1.

Links

Crossrefs

Cf. A064125, A324295, A334019,

Programs

  • Mathematica
    s[n_] := DivisorSum[n, # &, #^2 < n && CoprimeQ[#, n/#] &]; Select[Range[3000], s[#] == s[# + 1] &]

A348346 Numbers k such that k and k+1 have the same positive sum of noninfinitary divisors (A348271).

Original entry on oeis.org

20150, 52767, 99296, 835515, 1241504, 2199392, 6294015, 11158496, 12770450, 17016416, 19127907, 20128544, 23686748, 24790688, 26580554, 33366015, 34385247, 39687651, 42106976, 44157087, 45466676, 59825349, 60832449, 73780244, 75268775, 81654650, 84696849, 111457213
Offset: 1

Views

Author

Amiram Eldar, Oct 13 2021

Keywords

Comments

Numbers k such that A348271(k) = A348271(k+1) > 0.
The terms are restricted to have a positive sum of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

Examples

			20150 is a term since A348271(20150) = A348271(20151) = 6720.

Links

Crossrefs

Cf. A036537, A348271, A348345.
Subsequence of A162643.
Similar sequences: A002961, A064115, A064125, A293183, A306985.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[10^5], (s1 = s[#]) > 0 && s1 == s[# + 1] &]

A333949 Numbers k such that s(k) = s(k+1), where s(k) is the sum of recursive divisors of k (A333926).

Original entry on oeis.org

14, 206, 957, 1334, 1364, 1485, 1634, 2685, 2974, 4136, 4364, 14841, 20145, 24957, 33998, 36566, 42818, 64672, 74918, 79826, 79833, 84134, 86343, 92685, 109864, 111506, 122073, 138237, 147454, 159711, 162602, 166934, 187863, 190773, 193893, 201597, 274533, 288765
Offset: 1

Views

Author

Amiram Eldar, Apr 11 2020

Keywords

Examples

			14 is a term since A333926(14) = A333926(15) = 24.

Links

Crossrefs

Cf. A333926.
Analogous sequences: A002961, A064115 (nonunitary), A064125 (unitary), A293183 (bi-unitary), A306985 (infinitary).

Programs

  • Mathematica
    recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == recDivSum[# + 1] &]

A290303 Values of usigma(n) = usigma(n+1).

Original entry on oeis.org

24, 60, 72, 180, 1440, 2160, 1872, 2640, 2400, 3000, 2880, 3024, 4320, 4320, 4320, 5280, 5280, 7400, 8640, 10080, 10200, 11520, 11880, 11520, 11088, 12960, 12096, 14400, 25920, 21600, 26640, 34560, 25200, 40320, 34560, 36000, 51840, 60480, 63360, 60480, 65280
Offset: 1

Views

Author

Amiram Eldar, Jul 26 2017

Keywords

Comments

The sum of unitary divisors of numbers n such that n and n+1 have the same sum.
The unitary version of A053215.

Links

Crossrefs

Cf. A002961, A034448, A053215, A064125.

Programs

  • Mathematica
    usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]];a={}; u1=0; For[k=0, k<10^5, k++; u2=usigma[k]; If[u1==u2, a = AppendTo[a, u1]]; u1=u2]; a

Formula

a(n) = A034448(A064125(n)).
Showing 1-10 of 32 results. Next

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

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