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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A325376 Terms k of A228058 such that gcd(k - A048250(k), A162296(k) - k) = A162296(k) - k.

Original entry on oeis.org

153, 477, 801, 1773, 2097, 2421, 3725, 4041, 4689, 4753, 5013, 5337, 6309, 6957, 7281, 7929, 8577, 8725, 9549, 9873, 11225, 11493, 13437, 14357, 14409, 14733, 15381, 17001, 17973, 18621, 19269, 19917, 21213, 21537, 23481, 24777, 25101, 25749, 26073, 26225, 26721, 27369, 28989, 29161, 29313, 29961, 31225, 32229, 32553, 33849
Offset: 1

Views

Author

Antti Karttunen, Apr 22 2019

Keywords

Comments

Also, terms of this sequence are A228058(k) for all such k that A325375(k) = A325320(k).
In range 1 .. 2^27 there are no such terms k of A228058 that gcd(k-A048250(k), A162296(k)-k) = k - A048250(k).
If any odd perfect number exists, then it must occur in this sequence, but should also satisfy the other condition given above.

Links

Crossrefs

Cf. A048250, A162296, A228058, A325313, A325314, A325320, A325375, A325822.

Programs

  • PARI
    A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n) && (gcd(n-A048250(n),A162296(n)-n) == A162296(n)-n),k++; print1(n,", ")));

A325316 a(n) = A048250(n) OR A162296(n), where OR is the bitwise-OR, A003986.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 31, 20, 26, 32, 36, 24, 60, 31, 42, 36, 56, 30, 72, 32, 63, 48, 54, 48, 79, 38, 60, 56, 90, 42, 96, 44, 52, 62, 72, 48, 124, 57, 91, 72, 58, 54, 108, 72, 120, 80, 90, 60, 104, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 191, 74, 114, 124, 124, 96, 168, 80
Offset: 1

Views

Author

Antti Karttunen, Apr 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A003986, A048250, A162296, A325317, A325318.

Programs

  • Mathematica
    Array[BitOr @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 79] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325316(n) = bitor(A048250(n),A162296(n));

Formula

a(n) = A003986(A048250(n), A162296(n)).
a(n) = A000203(n) - A325318(n) = A325317(n) + A325318(n).

A325317 a(n) = A048250(n) XOR A162296(n), where XOR is the bitwise-XOR, A003987.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 23, 20, 10, 32, 36, 24, 60, 31, 42, 32, 56, 30, 72, 32, 63, 48, 54, 48, 67, 38, 60, 56, 90, 42, 96, 44, 20, 46, 72, 48, 124, 57, 89, 72, 18, 54, 96, 72, 120, 80, 90, 60, 40, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 187, 74, 114, 124, 108, 96, 168, 80
Offset: 1

Views

Author

Antti Karttunen, Apr 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A003987, A048250, A162296, A325316, A325318.

Programs

  • Mathematica
    Array[BitXor @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 79] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325317(n) = bitxor(A048250(n),A162296(n));

Formula

a(n) = A003987(A048250(n), A162296(n)).
a(n) = A000203(n) - 2*A325318(n) = A325316(n) - A325318(n).

A325318 a(n) = A048250(n) AND A162296(n), where AND is the bitwise-AND, A004198.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 32, 16, 0, 0, 0, 0, 2, 0, 40, 0, 12, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 16, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 16, 32, 2, 0, 0, 0, 40, 0
Offset: 1

Views

Author

Antti Karttunen, Apr 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A004198, A048250, A162296, A325316, A325317.

Programs

  • Mathematica
    Array[BitAnd @@ Map[Total, {#3, Complement[#2, #3]}] & @@ {#1, #2, Select[#2, SquareFreeQ]} & @@ {#, Divisors[#]} &, 105] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325318(n) = bitand(A048250(n),A162296(n));

Formula

a(n) = A004198(A048250(n), A162296(n)).
a(n) = A000203(n) - A325316(n) = (A000203(n) - A325317(n))/2.
a(n) = A325316(n) - A325317(n).

A357495 Lesser of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

880, 10480, 20080, 24928, 42976, 69184, 110565, 252080, 267712, 489472, 566656, 569240, 603855, 626535, 631708, 687424, 705088, 741472, 786896, 904365, 1100385, 1234480, 1280790, 1425632, 1749824, 1993750, 2012224, 2401568, 2439712, 2496736, 2542496, 2573344, 2671856
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The larger counterparts are in A357496.
Both members of each pair are necessarily nonsquarefree numbers.

Examples

			880 is a term since s(880) = 1136 and s(1136) = 880.

Links

Crossrefs

Cf. A162296, A325314, A322609, A357493, A357494, A357496.
Subsequence of A013929.
Similar sequences: A002025, A002952, A126165, A126169, A259038, A292980, A322541, A324708, A348343.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, n]], {n, 2, 3*10^6}]; seq

A357496 Greater of a pair of amicable numbers k < m such that s(k) = m and s(m) = k, where s(k) = A162296(k) - k is the sum of aliquot divisors of k that have a square factor.

Original entry on oeis.org

1136, 11696, 22256, 25472, 43424, 73664, 131355, 304336, 267968, 492608, 612704, 674920, 640305, 788697, 691292, 705344, 723392, 813728, 809776, 1117395, 1258335, 1559696, 1518570, 1598368, 1821376, 2218250, 2058944, 2678752, 2744288, 2765024, 2848864, 2610656, 3134224
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2022

Keywords

Comments

Analogous to amicable numbers (A002025 and A002046) with nonsquarefree divisors.
The terms are ordered according to their lesser counterparts (A357495).
Both members of each pair are necessarily nonsquarefree numbers.

Examples

			1136 is a term since s(1136) = 880 and s(880) = 1136.

Links

Crossrefs

Cf. A162296, A325314, A322609, A357493, A357494, A357495.
Subsequence of A013929.
Similar sequences: A002046, A002953, A126166, A126170, A259039, A292981, A322542, A324709, A348344.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) - n]; seq = {}; Do[m = s[n]; If[m > n && s[m] == n, AppendTo[seq, m]], {n, 2, 3*10^6}]; seq

A357607 Odd numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

4725, 6615, 7875, 8505, 11025, 14175, 15435, 17325, 19845, 20475, 22275, 23625, 24255, 25515, 26775, 28665, 29925, 31185, 33075, 36225, 36855, 37125, 37485, 38115, 39375, 40425, 41895, 42525, 46305, 47775, 48195, 50715, 51975, 53235, 53865, 55125, 57915, 59535
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

The least term that is not divisible by 3 is a(89047132) = 134785275625.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 4, 60, 640, 6650, 66044, 660230, 6604594, 66073470, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000660... .

Examples

			4725 is a term since it is odd, and A162296(4725) = 9728 > 2*4725.

Links

Crossrefs

Cf. A162296.
Subsequence of A005231, A013929 and A357605.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

Programs

  • Mathematica
    q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[3, 60000, 2], q]

A362402 Positive numbers m such that a record number of numbers k have m as the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

1, 4, 48, 72, 216, 288, 864, 1440, 1728, 2880, 3456, 4320, 5184, 5760, 8640, 12096, 17280, 25920, 34560, 48384, 51840, 69120, 103680, 120960, 155520, 181440, 207360, 241920, 311040, 362880, 483840, 622080, 725760, 967680, 1088640, 1209600, 1451520, 2177280, 2903040
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Comments

The value 0 appears in the range of A162296 for all squarefree numbers (A005117) and therefore it is excluded from this sequence.
The corresponding record values are in A362403.
Except for 1, a subsequence of A362401.

Links

Crossrefs

Cf. A162296, A362401, A362403.
Similar sequences: A097942, A100827, A145899, A238895.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {1}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[1]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(1, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(k, ", "))); }

A362403 Number of times that the number A362402(n) occurs as a sum of divisors that have a square factor (A162296).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 10, 13, 15, 16, 20, 22, 23, 28, 34, 46, 53, 60, 62, 78, 81, 113, 115, 122, 132, 154, 184, 185, 222, 248, 254, 343, 346, 350, 354, 497, 569, 701, 711, 860, 941, 1088, 1221, 1222, 1235, 1263, 1306, 1572, 1721, 1737, 1948, 2191, 2315, 2418, 2877
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Links

Crossrefs

Cf. A162296, A362401, A362402.
Similar sequences: A131934, A101373, A206027, A238896.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {0}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[2]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(0, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(v[k], ", "))); }

A362400 Numbers k such that A162296(k) = A162296(k+1) > 0.

Original entry on oeis.org

135, 819, 1863, 9207, 10340, 41124, 75051, 95336, 278972, 305091, 465596, 544924, 570411, 711027, 903804, 977876, 1114695, 1327095, 1444779, 1520684, 1760571, 1987371, 2083491, 2303091, 2581928, 2842324, 2869011, 3062631, 3243140, 4043624, 4335848, 4469984, 4598091
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Comments

A162296(k) = A162296(k+1) = 0 if and only if k and k+1 are both squarefree (A005117), i.e., k is in A007674.

Examples

			135 is a term since A162296(135) = A162296(136) = 216.

Links

Crossrefs

Cf. A005117, A007674, A162296.
Subsequence of A013929 and A068781.
Similar sequences: A002961, A064115, A064125, A164522, A164557, A293183, A306985, A325175, A324295.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; Select[Range[2, 5*10^6], (sn = s[#]) > 0 && sn == s[# + 1] &]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(s1 = s(1), s2); for(k=2, kmax, s2 = s(k); if(s1 > 0 && s2 == s1, print1(k-1, ", ")); s1 = s2); }
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