A007407 -id:A007407 - cp's OEIS frontend

A242241 Least prime p such that H(2,n) = sum_{k=1..n}1/k^2 == 0 (mod p) but there is no 0 < k < n with H(2,k) == 0 (mod p), or 1 if such a prime p does not exist.

1, 5, 7, 41, 11, 13, 266681, 17, 19, 178939, 23, 18500393, 40799043101, 29, 31, 619, 601, 8821, 86364397717734821, 421950627598601, 2621, 295831, 47, 2237, 157, 53, 307, 7741, 6823, 61, 205883, 487, 67, 21767149, 71, 73, 149, 2004383, 79, 34033

Offset: 1

Zhi-Wei Sun, May 09 2014

nonn

Conjecture: (i) a(n) is prime for any n > 1.

(ii) For any prime p > 5, there exists a prime q < p/2 such that H(2,q-1) = sum_{0

See also A242222 and A242223 for similar conjectures involving harmonic numbers H(n) = sum_{k=1..n}1/k (n > 0).

			a(4) = 41 since H(2,4) = 5*41/(2^4*3^2) but none of H(2,1) = 1, H(2,2) = 5/2^2 and H(2,3) = 7^2/(2^2*3^2) is congruent to 0 modulo 41.

Zhi-Wei Sun, Table of n, a(n) for n = 1..108
Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Cf. A000040, A007406, A007407, A242210, A242213, A242222, A242223.

h[n_]:=Numerator[HarmonicNumber[n,2]]
f[n_]:=FactorInteger[h[n]]
p[n_]:=p[n]=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[h[n]<2, Goto[cc]]; Do[Do[If[Mod[h[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n,1,40}]

A266581 Numerators of expansion of PolyLog(-2, x)/PolyLog(2, x), where PolyLog(m, x) is the polylogarithm function.

Original entry on oeis.org

1, 15, 1145, 7795, 10605889, 59526571, 139954552433, 34217723087, 806539298609929, 3932874930141827, 4100492004734957581, 96658551584623754987, 838219558485468722155050481, 142916593419748754034403361, 158366688967470905539833679601, 102317913027622943383626250477

Offset: 0

Ilya Gutkovskiy, May 07 2016

Numerators of expansion of (Sum_{k>=1} x^k*k^2)/(Sum_{k>=1} x^k/k^2).

Numerators of numbers for which convolution with Sum_{k=1..n} 1/k^2 = A007406(n)/A007407(n) gives Sum_{k=1..n} k^2 = A000330(n).

			1, 15/4, 1145/144, 7795/576, 10605889/518400, 59526571/2073600, 139954552433/3657830400, 34217723087/696729600, 806539298609929/13168189440000, …

Eric Weisstein's World of Mathematics, Dilogarithm, Polylogarithm, and Wolstenholme Number

Cf. A232193 (numerators of expansion of PolyLog(-1, x)/PolyLog(1, x)), A232248 (denominators of expansion of PolyLog(-1, x)/PolyLog(1, x)).

Cf. A000330, A007406, A007407, A273698 (denominators).

Table[Numerator[SeriesCoefficient[PolyLog[-2, x]/PolyLog[2, x], {x, 0, n}]], {n, 0, 15}]

A273698 Denominators of expansion of PolyLog(-2, x)/PolyLog(2, x), where PolyLog(m, x) is the polylogarithm function.

Original entry on oeis.org

1, 4, 144, 576, 518400, 2073600, 3657830400, 696729600, 13168189440000, 52672757760000, 45888506560512000, 917770131210240000, 6840049010896797696000000, 1013340594206932992000000, 984967057569138868224000000, 562838318610936496128000000

Offset: 0

Ilya Gutkovskiy, May 28 2016

Denominators of expansion of (Sum_{k>=1} x^k*k^2)/(Sum_{k>=1} x^k/k^2).

Denominators of numbers for which convolution with Sum_{k=1..n} 1/k^2 = A007406(n)/A007407(n) gives Sum_{k=1..n} k^2 = A000330(n).

			1, 15/4, 1145/144, 7795/576, 10605889/518400, 59526571/2073600, 139954552433/3657830400, 34217723087/696729600, 806539298609929/13168189440000, ...

Eric Weisstein's World of Mathematics, Dilogarithm, Polylogarithm, and Wolstenholme Number

Cf. A232193 (numerators of expansion of PolyLog(-1, x)/PolyLog(1, x)), A232248 (denominators of expansion of PolyLog(-1, x)/PolyLog(1, x)).

Cf. A000330, A007406, A007407, A266581 (numerators).

Table[Denominator[SeriesCoefficient[PolyLog[-2, x]/PolyLog[2, x], {x, 0, n}]], {n, 0, 15}]

A322266 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{j=1..n} 1/j^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 8, 36, 12, 1, 1, 16, 216, 144, 60, 1, 1, 32, 1296, 1728, 3600, 20, 1, 1, 64, 7776, 20736, 216000, 3600, 140, 1, 1, 128, 46656, 248832, 12960000, 24000, 176400, 280, 1, 1, 256, 279936, 2985984, 777600000, 12960000, 8232000, 705600, 2520, 1

Offset: 1

Ilya Gutkovskiy, Dec 01 2018

			Square array begins:
  1,       1,          1,              1,                  1,  ...
  2,     3/2,        5/4,            9/8,              17/16,  ...
  3,    11/6,      49/36,        251/216,          1393/1296,  ...
  4,   25/12,    205/144,      2035/1728,        22369/20736,  ...
  5,  137/60,  5269/3600,  256103/216000,  14001361/12960000,  ...

Eric Weisstein's World of Mathematics, Harmonic Number

Columns k=0..10 give A000012, A002805, A007407, A007409, A007480, A069052, A103346, A103348, A103350, A103352, A103717.
Numerators are in A322265.
Cf. A103438, A276487.

Table[Function[k, Denominator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
Table[Function[k, Denominator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

G.f. of column k: PolyLog(k,x)/(1 - x), where PolyLog() is the polylogarithm function (for rationals Sum_{j=1..n} 1/j^k).

A322557 Smallest k such that floor(Nsqrt(Sum_{m=1..k} 6/m^2)) = floor(NPi), where N = 10^n.

Original entry on oeis.org

7, 23, 600, 1611, 10307, 359863, 1461054, 17819245, 266012440, 1619092245, 10634761313, 97509078554, 1203836807622, 10241799698090, 294871290395291, 4004525174270251, 24827457879988026, 112840588371964574, 2064072875704476882, 15243903003939891921

Offset: 0

Zachary Russ, Aug 28 2019

6*A007406(k)/A007407(k) = Sum_{m=1..k} 6/m^2.

It seems nearly certain that, for all n >= 0, a(n) = ceiling(z - 1/2 - 1/(12*z)) where z = 6/(Pi^2 - (floor(Pi*10^n)/10^n)^2). - Jon E. Schoenfield, Aug 31 2019

			floor((10^0)*sqrt(Sum_{m=1..7} 6/m^2)) = 3.
floor((10^1)*sqrt(Sum_{m=1..23} 6/m^2)) = 31.
floor((10^2)*sqrt(Sum_{m=1..600} 6/m^2)) = 314.
floor((10^3)*sqrt(Sum_{m=1..1611} 6/m^2)) = 3141.
floor((10^4)*sqrt(Sum_{m=1..10307} 6/m^2)) = 31415.
floor((10^5)*sqrt(Sum_{m=1..359863} 6/m^2)) = 314159.

Zachary Russ, Klarice Sequence
Jon E. Schoenfield, Magma program

Cf. A000796, A013661, A013679, A002388, A274982, A007406, A007407.

Cf. A011545 (floor(Pi*10^n)).

a(n) = {my(k = 1); t = floor(10^(n)*Pi); while(floor(10^(n)*sqrt(sum(m = 1, k, 6/m^2))) != t, k++); k; } \\ Jinyuan Wang, Aug 30 2019

a(6)-a(19) from Jon E. Schoenfield, Aug 31 2019

A384818 Denominator of the sum of the reciprocals of all square divisors of all positive integers <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 2, 18, 18, 18, 36, 36, 36, 36, 144, 144, 144, 144, 144, 144, 144, 144, 144, 3600, 3600, 1200, 1200, 1200, 1200, 1200, 600, 600, 600, 600, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 600, 600, 600, 1200, 58800, 58800, 58800, 58800, 58800

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

			1, 2, 3, 17/4, 21/4, 25/4, 29/4, 17/2, 173/18, 191/18, 209/18, 463/36, ...

Cf. A007407, A017668, A284650, A309125, A373440, A384817 (numerators).

nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 53}] // Denominator

a(n) = denominator(sum(k=1, n, sumdiv(k, d, if (issquare(d), 1/d)))); \\ Michel Marcus, Jun 10 2025

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^(k^2) / (k^2*(1 - x^(k^2))).
a(n) is the denominator of Sum_{k=1..floor(sqrt(n))} floor(n/k^2) / k^2.
A384817(n) / a(n) ~ Pi^4 * n / 90.

A119785 Numerator of the product of the n-th square pyramidal number and the n-th generalized harmonic number in power 2.

Original entry on oeis.org

1, 25, 343, 1025, 57959, 488579, 266681, 18321733, 185784679, 21651619, 5507071447, 15632832085, 40799043101, 1187015026009, 6362282386111, 13990468150733, 238357395880861, 167890966963712483, 86364397717734821

Offset: 1

Alexander Adamchuk, Jun 25 2006

p^2 divides a(p-1) for prime p>3. p^2 divides a((p-1)/2) for prime p>3.

Cf. A000330, A007406, A007407.

Numerator[Table[n(n+1)(2n+1)/6*Sum[1/k^2,{k,1,n}],{n,1,30}]]
Numerator[Table[Sum[Sum[i^2/j^2, {i, 1, n}], {j, 1, n}],{n,1,30}]]

a(n) = numerator[Sum[i^2,{i,1,n}] * Sum[1/j^2,{j,1,n}]] = numerator[n(n+1)(2n+1)/6 * Sum[1/j^2,{j,1,n}]] = numerator[A000330(n) * ( A007406(n)/A007407(n) )]. Also a(n) = numerator[Sum[Sum[i^2/j^2, {i, 1, n}], {j, 1, n}]].

A120287 Numerator of 1/n^3 + 2/(n-1)^3 + 3/(n-2)^3 +...+ (n-1)/2^3 + n.

Original entry on oeis.org

1, 17, 355, 7715, 203413, 492527, 49601051, 1823359051, 16684019407, 186004308017, 22757931053507, 298630937704541, 50872538998767329, 51223731720255509, 103063783892301061, 7045407930432340853

Offset: 1

Alexander Adamchuk, Jul 07 2006

p divides a(p-1) and a(p-2) for prime p>3.

Numerators of the Eulerian numbers T(-3,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - Michael J. Collins, Oct 10 2024

Cf. A001008, A007406, A007407, A007408, A007410.

Numerator[Table[Sum[Sum[1/i^3,{i,1,k}],{k,1,n}],{n,1,25}]]

a(n) = numerator[Sum[Sum[1/i^3,{i,1,k}],{k,1,n}]].

A120288 Numerator of 1/n^4 + 2/(n-1)^4 + 3/(n-2)^4 +...+ (n-1)/2^4 + n.

Original entry on oeis.org

1, 33, 2033, 87425, 11440331, 82653347, 58026230977, 472474237481, 38806560342253, 431701520479427, 579954645879123307, 7598398013722878661, 16804804720323979155637, 16901141697896969645401

Offset: 1

Alexander Adamchuk, Jul 07 2006

p^2 divides a(p-1) for prime p>3. p divides a(p-2) for prime p>5.

Cf. A001008, A007406, A007407, A007408, A007410.

Numerator[Table[Sum[Sum[1/i^4,{i,1,k}],{k,1,n}],{n,1,20}]]

a(n) = numerator[Sum[Sum[1/i^4,{i,1,k}],{k,1,n}]].

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cp's OEIS Frontend

A073527 Numbers n such that denominator of Sum_{k=1..n} 1/k^2 is not a square.

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A242241 Least prime p such that H(2,n) = sum_{k=1..n}1/k^2 == 0 (mod p) but there is no 0 < k < n with H(2,k) == 0 (mod p), or 1 if such a prime p does not exist.

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A266581 Numerators of expansion of PolyLog(-2, x)/PolyLog(2, x), where PolyLog(m, x) is the polylogarithm function.

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A273698 Denominators of expansion of PolyLog(-2, x)/PolyLog(2, x), where PolyLog(m, x) is the polylogarithm function.

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A322266 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{j=1..n} 1/j^k.

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A322557 Smallest k such that floor(N*sqrt(Sum_{m=1..k} 6/m^2)) = floor(N*Pi), where N = 10^n.

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A384818 Denominator of the sum of the reciprocals of all square divisors of all positive integers <= n.

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PARI

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A119785 Numerator of the product of the n-th square pyramidal number and the n-th generalized harmonic number in power 2.

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A120287 Numerator of 1/n^3 + 2/(n-1)^3 + 3/(n-2)^3 +...+ (n-1)/2^3 + n.

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A322557 Smallest k such that floor(Nsqrt(Sum_{m=1..k} 6/m^2)) = floor(NPi), where N = 10^n.