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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A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

Original entry on oeis.org

1, 2, 10, 84, 1301, 15693, 376762, 6168552, 176787631, 3770427352, 142364319626, 3152758480715, 154718778284149, 4340093860950619, 210971170836848270, 7281694486114555088, 435659030617933827137, 14181121059071691716406, 1052864393300587929716722, 41673907052879908244100770
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 18 2020

Keywords

Crossrefs

Cf. A031971, A057661, A226561, A332517, A332621, A332652, A332653, A332654.

Programs

  • Magma
    [&+[(k div Gcd(n,k))^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 18 2020
  • Mathematica
    Table[Sum[(k/GCD[n, k])^n, {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^n, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

Formula

a(n) = Sum_{k=1..n} (lcm(n, k)/n)^n.
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} k^n.

A341316 Row sums in A341315.

Original entry on oeis.org

0, 3, 6, 12, 18, 33, 33, 66, 66, 93, 96, 168, 117, 237, 195, 222, 258, 411, 276, 516, 348, 453, 501, 762, 453, 783, 708, 822, 711, 1221, 663, 1398, 1026, 1167, 1230, 1356, 1008, 2001, 1545, 1650, 1356, 2463, 1356, 2712, 1833, 1923, 2283, 3246, 1797, 3153, 2346, 2868, 2592, 4137
Offset: 0

Views

Author

N. J. A. Sloane, Feb 17 2021

Keywords

Comments

This is three times A057661. See that entry for much more information.

Crossrefs

Cf. A057660, A057661, A341314, A341315.

Formula

a(n) = 3*(A057660(n)+1)/2 for n>=1. - Hugo Pfoertner, Feb 17 2021

A343513 a(n) = Sum_{k=1..n} (k/gcd(n, k))^3.

Original entry on oeis.org

1, 2, 10, 30, 101, 137, 442, 526, 1063, 1202, 3026, 1965, 6085, 4853, 7310, 8654, 18497, 10100, 29242, 17630, 29557, 30857, 64010, 30397, 77601, 60842, 89272, 71913, 164837, 60737, 216226, 139470, 188165, 180338, 265142, 152544, 443557, 282665, 371134, 275726, 672401, 251066, 815410, 461645
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 17 2021

Keywords

Comments

a(n) = 1+n^2*(n-1)^2/4 if n is prime. - Robert Israel, Apr 19 2021

Links

Crossrefs

Cf. A053819, A057661, A332654, A343497, A343514.

Programs

  • Maple
    f:= proc(n) local k;
  add((k/igcd(n,k))^3,k=1..n)
end proc:
map(f, [$1..100]); # Robert Israel, Apr 19 2021
  • Mathematica
    Table[Sum[(k/GCD[n, k])^3, {k, 1, n}], {n, 1, 44}]
  • PARI
    a(n) = sum(k=1, n, (k/gcd(n, k))^3); \\ Michel Marcus, Apr 17 2021

Formula

a(n) = Sum_{d|n} A053819(d).

A232533 a(n) = Sum_{i=1...n} Sum_{j=1..i} lcm(i,j)/i.

Original entry on oeis.org

1, 3, 7, 13, 24, 35, 57, 79, 110, 142, 198, 237, 316, 381, 455, 541, 678, 770, 942, 1058, 1209, 1376, 1630, 1781, 2042, 2278, 2552, 2789, 3196, 3417, 3883, 4225, 4614, 5024, 5476, 5812, 6479, 6994, 7544, 7996, 8817, 9269, 10173, 10784, 11425, 12186, 13268
Offset: 1

Views

Author

Zhining Yang, Nov 25 2013

Keywords

Comments

Partial sums of A057661.

Examples

			a(3) = lcm(1,1)/1 + (lcm(2,1) + lcm(2,2))/2 + (lcm(3,1) + lcm(3,2) + lcm(3,3))/3 = 7.

Links

Crossrefs

Cf. A057661.

Programs

  • Mathematica
    Table[Sum[Sum[LCM[m, k], {k, m}]/m, {m, n}], {n, 50}] (* T. D. Noe, Nov 25 2013 *)
  • PARI
    a(n)=sum(k=1,n,sum(b=1,k,lcm(b,k)/k));
  • PARI
    a(n)=sum(k=1,n,sum(m=1,k,m/gcd(m,k))); \\ Charles R Greathouse IV, Nov 25 2013

A280131 Partial sums of A029940 (Product_{d|n} phi(d)).

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 28, 40, 56, 66, 98, 110, 146, 210, 274, 290, 434, 452, 708, 852, 952, 974, 1998, 2078, 2222, 2438, 3302, 3330, 7426, 7456, 8480, 8880, 9136, 9712, 23536, 23572, 23896, 24472, 40856, 40896, 61632, 61674, 65674, 74890, 75374, 75420
Offset: 1

Views

Author

Jaroslav Krizek, Dec 27 2016

Keywords

Comments

phi(n) is the number of totatives of n (A000010).

Links

Crossrefs

Cf. A000010, A029940, A232533 (partial sums of A057661), A280132 (partial products of A029940).

Programs

  • Magma
    [&+[&*[EulerPhi(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]]
  • Maple
    A029940:= [seq(mul(numtheory:-phi(d),d=numtheory:-divisors(n)),n=1..100)]:
ListTools:-PartialSums(A029940); # Robert Israel, Jan 11 2017
  • Mathematica
    Accumulate@ Array[Product[EulerPhi@ d, {d, Divisors@ #}] &, 47] (* Michael De Vlieger, Dec 27 2016 *)

Formula

a(n) = Sum_{i=1..n} A029940(i).

A280132 Partial products of A029940 (Product_{d|n} phi(d)).

Original entry on oeis.org

1, 1, 2, 4, 16, 64, 384, 3072, 36864, 589824, 5898240, 188743680, 2264924160, 81537269760, 5218385264640, 333976656936960, 5343626510991360, 769482217582755840, 13850679916489605120, 3545774058621338910720, 510591464441472803143680, 51059146444147280314368000
Offset: 1

Views

Author

Jaroslav Krizek, Dec 27 2016

Keywords

Comments

phi(n) is the number of totatives of n (A000010).

Crossrefs

Cf. A000010, A029940, A280131 (partial sums of A029940), A280133 (partial products of A057661).

Programs

  • Magma
    [&*[&*[EulerPhi(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]];
  • Mathematica
    FoldList[Times[#1, #2] &, Array[Product[EulerPhi@ d, {d, Divisors@ #}] &, 22]] (* Michael De Vlieger, Dec 27 2016 *)

Formula

a(n) = Product_{i=1..n} A029940(i).

A280246 a(n) = Product_{d|n} psi(d), where psi(m) is the sum of totatives of m (A023896).

Original entry on oeis.org

1, 1, 3, 4, 10, 18, 21, 64, 81, 200, 55, 1728, 78, 882, 1800, 4096, 136, 26244, 171, 64000, 7938, 6050, 253, 2654208, 2500, 12168, 19683, 592704, 406, 25920000, 465, 1048576, 54450, 36992, 88200, 544195584, 666, 58482, 109512, 327680000, 820, 504094752, 903
Offset: 1

Views

Author

Jaroslav Krizek, Dec 30 2016

Keywords

Comments

a(n) = n only for n = 1, 3 and 4.
n divides a(n) for all n except 2.
Conjecture: a(n) is odd iff the sum of totatives of n (A023896) is odd.

Examples

			For n=6; sets of totatives of divisors of 6: {1}, {1}, {1, 2}, {1, 5}; a(6) = 1*1*(1+2)*(1+5) = 18.

Crossrefs

Cf. A023896, A057661, A280247, A280248.

Programs

  • Magma
    [&*[&+[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(n)]: n in [1..100]]
  • Mathematica
    Table[Product[Total@ Select[Range@ d, CoprimeQ[d, #] &], {d, Divisors@ n}], {n, 43}] (* Michael De Vlieger, Dec 30 2016 *)

Formula

a(n) = Product_{d|n} A023896(d).

A332652 a(n) = Sum_{k=1..n} n^(k/gcd(n, k)).

Original entry on oeis.org

1, 4, 15, 76, 785, 7836, 137263, 2130976, 47895489, 1010012140, 28531167071, 743044702104, 25239592216033, 797785008119932, 31147773583464735, 1157442765678719056, 51702516367896047777, 2185932446984222457444, 109912203092239643840239, 5255987282125826560192520
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 18 2020

Keywords

Crossrefs

Cf. A031972, A056665, A057661, A226561, A228640, A332620, A332621, A332653, A332655.

Programs

  • Magma
    [&+[n^(k div Gcd(n,k)):k in [1..n]]:n in [1..21]]; // Marius A. Burtea, Feb 18 2020
  • Mathematica
    Table[Sum[n^(k/GCD[n, k]), {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, n^k, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

Formula

a(n) = Sum_{k=1..n} n^(lcm(n, k)/n).
a(n) = Sum_{d|n} Sum_{k=1..d, gcd(k, d) = 1} n^k.
a(n) = n * A332653(n).

A333613 a(1) = 1; thereafter a(n) = Sum_{k = 1..n} a(k/gcd(n,k)).

Original entry on oeis.org

1, 2, 4, 7, 15, 21, 51, 78, 158, 230, 568, 661, 1797, 2595, 5117, 7789, 19095, 21702, 59892, 81801, 171329, 258028, 630942, 713093, 1887828, 2776798, 5727675, 8335692, 20702970, 21420664, 62826604, 92041835, 189376593, 281410640, 656577018, 742729123, 2087788417
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 28 2020

Keywords

Links

Crossrefs

Cf. A006874, A045545, A057661, A332791.

Programs

  • Maple
    A333613:= proc(n) option remember;
if n<3 then n;
else add( A333613(lcm(n,j)/n), j = 1..n);
end if; end proc;
seq(A333613(n), n=1..40); # G. C. Greubel, Mar 08 2021
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[a[k/GCD[n, k]], {k, n}]; Table[a[n], {n, 37}]
a[1] = 1; a[n_] := a[n] = Sum[Sum[If[GCD[k, d] == 1, a[k], 0], {k, d}], {d, Divisors[n]}]; Table[a[n], {n, 37}]
  • Sage
    @CachedFunction
def A333613(n): return 1 if n==1 else sum( A333613(lcm(n, j)/n) for j in (1..n) )
[A333613(n) for n in (1..40)] # G. C. Greubel, Mar 08 2021

Formula

a(1) = 1; a(n) = Sum_{k = 1..n} a(lcm(n, k)/n).
a(1) = 1; a(n) = Sum_{d|n} Sum_{k = 1..d, gcd(d, k) = 1} a(k).

A343514 a(n) = Sum_{k=1..n} (k/gcd(n, k))^4.

Original entry on oeis.org

1, 2, 18, 84, 355, 645, 2276, 3192, 7413, 9400, 25334, 18395, 60711, 52747, 88760, 106688, 243849, 137790, 432346, 275570, 499867, 522513, 1151404, 561415, 1542125, 1214436, 1907502, 1569673, 3756719, 1344999, 5274000, 3451216, 4970577, 4690778, 7499154, 4217504, 12948595, 8207261, 11565572
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 17 2021

Keywords

Crossrefs

Cf. A053820, A057661, A332654, A343498, A343513.

Programs

  • Mathematica
    Table[Sum[(k/GCD[n, k])^4, {k, 1, n}], {n, 1, 39}]
  • PARI
    a(n) = sum(k=1, n, (k/gcd(n, k))^4); \\ Michel Marcus, Apr 17 2021

Formula

a(n) = Sum_{d|n} A053820(d).
Previous Showing 11-20 of 32 results. Next

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

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