A001015 -id:A001015 - cp's OEIS frontend

A075668 Sum of next n 7th powers.

1, 2315, 374445, 17703664, 394340375, 5265954441, 48574262275, 338837482880, 1900477947429, 8950536157375, 36536761179281, 132397570996560, 433806511149115, 1303971065324669, 3637715990646375, 9507513902672896, 23461050872397545, 55013865421504275

Offset: 1

Zak Seidov, Sep 24 2002

			a(1) = 1^7 = 1; a(2) = 2^7 + 3^7 = 2315; a(3) = 4^7 + 5^7 + 6^7 = 374445; a(4) = 7^7 + 8^7 + 9^7 + 10^7 = 17703664.

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A001015 (7th powers).

Cf. A006003 (for natural numbers), A072474 (for squares), A075664 - A075671 (for 3rd to 10th powers), A069876 (n-th powers).

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=7; Table[Sum[i^s, {i, i1, i2}], {n, 20}]

a(n) = Sum_{i=n*(n-1)/2+1..n*(n-1)/2+n} i^7.
a(n) = (3*n^15 + 42*n^13 + 168*n^11 + 206*n^9 - 11*n^7 - 56*n^5 + 32*n^3)/384. - Charles R Greathouse IV, Sep 17 2009
G.f.: x*(x^14 +2299*x^13 +337525*x^12 +11989784*x^11 +154720571*x^10 +875467853*x^9 +2397170367*x^8 +3336829200*x^7 +2397170367*x^6 +875467853*x^5 +154720571*x^4 +11989784*x^3 +337525*x^2 +2299*x +1)/(x-1)^16. - Colin Barker, Jul 22 2012

A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

Original entry on oeis.org

32318, 45807, 110443, 123932, 188568, 202057, 266693, 280182, 344818, 358307, 422943, 436432, 501068, 514557, 579193, 592682, 657318, 670807, 735443, 748932, 813568, 827057, 891693, 905182, 969818, 983307

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			32318 is in the sequence because 32318^2 + 1 =  5 ^ 7 * 29 * 461.
6826318 is in the sequence because 6826318^2 + 1 = 5 ^ 3 * 13 ^ 8 * 457.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562, A218563, A218564, A218565.

Cf. A001015 (7th powers).

Select[Range[2,1500000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>6&]

A257659 Numbers that are not seventh powers, but can be written as the sum of the seventh powers of two or more of their prime factors.

Original entry on oeis.org

275223438741, 4561072096211306682, 9306119954843409393442022085025276

Offset: 1

Felix Fröhlich, Jul 26 2015

From Robert Israel, Nov 02 2016: (Start)
Each term is the sum of the seventh powers of three or more of its prime factors (since the sum of seventh powers of two distinct primes would not be divisible by those primes).
It is possible that the three terms shown are just the smallest examples presently known - there may be smaller ones.
Other terms include the following (and these too may not be the next terms):
48174957112005843444270083236899591347874 = 2^7 + 1259^7 + 648383^7.
343628633008268493930426179988576850614546787655 = 5^7 + 97^7 + 6178313^7.
1556588247952374145751498792380776025975963817566087335 = 5^7 + 941^7 + 55174589^7.
6777869034345885139001456808449377853222864558972446987604 = 2^7 + 337^7 + 182635307^7.
8652931112104420195217156139788964690213217995925746635175635 = 5^7 + 29^7 + 507351601^7.
33684756195335243623428442147352712728560450053586233129585039130540009686445977 = 3^7 + 2731^7 + 229647602339^7.
4218418507660286246537768294375414778864666339784229288571328866079146694717894140 = 5^7 + 7^7 + 2677^7 + 457863123059^7.
(End)

			275223438741 is not a seventh power, i.e., not a term of A001015, but is equal to the product of prime numbers 3 * 23 * 43 * 92761523, and 3^7 + 23^7 + 43^7 = 275223438741, so 275223438741 is a term of the sequence.

J. M. De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, page 362, ISBN 978-0-8218-4807-4.

Jean-Marie De Koninck & Florian Luca, Partial Sums of Powers of Prime Factors, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.6 (see p. 7).

Cf. A001015, A092759.

Edited by Robert Israel, Nov 02 2016

A291828 Numbers k such that k^2 is sum of two positive 7th powers.

Original entry on oeis.org

16, 2048, 34992, 262144, 1250000, 4478976, 13176688, 33554432, 76527504, 160000000, 276922881, 311794736, 573308928, 1003976272, 1686616064, 2733750000, 4294967296, 6565418768, 9795520512, 14301947824, 20480000000, 28817416656, 35446128768, 39909726208

Offset: 1

XU Pingya, Sep 03 2017

nonn

When a^7 + b^7 = m, (ma)^7 + (mb)^7 = m^8 is square.

When k in this sequence, k*(n^7) (n = 2, 3, ... ) is also in this sequence.

			16^2 = 2^8 = 2^7 + 2^7, so 16 is in the sequence.
276922881^2 = 129^7 + 358^7, so 276922881 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..447

Cf. A000290, A001015, A004431, A050801.

lst={};Do[If[IntegerQ[(n^2-a^7)^(1/7)],AppendTo[lst,n]],{n,4*10^10},{a,(n^2/2)^(1/7)}]; lst

A343617 Decimal expansion of P_{3,2}(7) = Sum 1/p^7 over primes == 2 (mod 3).

Original entry on oeis.org

0, 0, 7, 8, 2, 5, 3, 5, 4, 1, 1, 3, 0, 5, 0, 4, 9, 2, 8, 7, 4, 2, 5, 1, 7, 0, 1, 6, 7, 0, 7, 5, 5, 9, 2, 0, 6, 0, 3, 3, 0, 7, 9, 3, 0, 9, 7, 5, 1, 3, 2, 4, 4, 3, 3, 1, 4, 6, 8, 0, 4, 8, 8, 3, 3, 9, 4, 0, 3, 5, 4, 3, 7, 0, 6, 3, 8, 0, 9, 2, 1, 8, 4, 3, 5, 7, 0, 1, 1, 0, 5, 8, 6, 5, 3, 8, 3, 8, 6, 4, 5, 6, 2, 9, 5

Offset: 0

M. F. Hasler, Apr 25 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			0.0078253541130504928742517016707559206033079309751324433146804883394...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, value P(m=3, n=2, s=7), p. 21.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A001015 (n^7), A085967 (PrimeZeta(7)).
Cf. A343612 - A343619 (P_{3,2}(s): analog for 1/p^s, s = 2 .. 9).
Cf. A343627 (for primes 3k+1), A086037 (for primes 4k+1), A085996 (for primes 4k+3).

A343617_upto(N=100)={localprec(N+5); digits((PrimeZeta32(7)+1)\.1^N)[^1]} \\ see A343612 for the function PrimeZeta32

P_{3,2}(7) = Sum_{p in A003627} 1/p^7 = P(7) - 1/3^7 - P_{3,1}(7).

A017575 a(n) = (12n+4)^7.

Original entry on oeis.org

16384, 268435456, 13492928512, 163840000000, 1028071702528, 4398046511104, 14645194571776, 40867559636992, 100000000000000, 221068140740608, 450766669594624, 860542568759296, 1555363874947072, 2684354560000000, 4453476124377088, 7140436495826944, 11112006825558016

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001015, A013665, A016783, A017569.

a[n_] := (12*n + 4)^7; Array[a, 20, 0] (* Amiram Eldar, Jul 14 2024 *)

From Amiram Eldar, Jul 14 2024: (Start)
a(n) = A001015(A017569(n)) = A017569(n)^7.
a(n) = 16384 * A016783(n).
Sum_{n>=0} 1/a(n) = 7*Pi^7/(403107840*sqrt(3)) + 1093*zeta(7)/35831808. (End)

More terms from Amiram Eldar, Jul 14 2024

A070726 a(n) = n^7 mod 46.

Original entry on oeis.org

0, 1, 36, 25, 8, 17, 26, 5, 12, 27, 14, 7, 16, 9, 42, 11, 18, 43, 6, 15, 44, 33, 22, 23, 24, 13, 2, 31, 40, 3, 28, 35, 4, 37, 30, 39, 32, 19, 34, 41, 20, 29, 38, 21, 10, 45, 0, 1, 36, 25, 8, 17, 26, 5, 12, 27, 14, 7, 16, 9, 42, 11, 18, 43, 6, 15, 44, 33, 22, 23, 24, 13, 2, 31, 40

Offset: 0

N. J. A. Sloane, May 13 2002

nonn

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

CF. A001015 (n^7).

PowerMod[Range[0,80],7,46] (* Harvey P. Dale, Mar 14 2023 *)

a(n)=(n^7)%46 \\ Edward Jiang, Sep 06 2014

a(n) = lift(Mod(n, 46)^7); \\ Michel Marcus, Jun 15 2018

[power_mod(n,7,46)for n in range(0, 75)] # Zerinvary Lajos, Oct 29 2009

A094075 Denominator of I(n)=integral_{x=0 to 1/n}(x^2-1)^3 dx.

Original entry on oeis.org

105, 13440, 229635, 1720320, 8203125, 29393280, 86472015, 220200960, 502211745, 1050000000, 2046152955, 3762339840, 6588594285, 11068417920, 17940234375, 28185722880, 43085560665, 64283103360, 93856532595, 134400000000

Offset: 1

Al Hakanson (hawkuu(AT)excite.com), Apr 29 2004

The numerator is b(n) = -105*n^6+105*n4-63*n^2+15. E.g., b(3) = -68392.

			I(3) = -68592/229635.

Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

f[n_] := Integrate[(x^2 - 1)^3, {x, 0, 1/n}]; Table[(-105n^6 + 105n^4 - 63n^2 + 15)/f[n], {n, 20}] (* Robert G. Wilson v, May 03 2004 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{105,13440,229635,1720320,8203125,29393280,86472015,220200960},20] (* Harvey P. Dale, Aug 04 2023 *)

a(n) = 105*n^7 = 105*A001015(n).

G.f.: 105*x*(1+120*x+1191*x^2+2416*x^3+1191*x^4+120*x^5+x^6)/(x-1)^8. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

More terms from Robert G. Wilson v, May 03 2004

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009

A138586 a(1) = 1; a(n) = a(n-1) + (n!)^7.

Original entry on oeis.org

1, 129, 280065, 4586751489, 358322666751489, 100306488365546751489, 82606511560391889386751489, 173238283180457843219993066751489, 828593116199250458889895450218986751489

Offset: 1

Jonathan Vos Post, May 18 2008

After a(1) = 1 these are all divisible by 3. a(n)/3 is prime (i.e. a(n) is semiprime) for n = 2, 4 (i.e. (1!)^7 + (2!)^7 + (3!)^7 + (4!)^7 = 4586751489 = 3 * 1528917163) and then when next?

Cf. A000142, A001015, A104344, A100288.

a(n) = Sum_{k=1..n} (k!)^7 = Sum_{k=1..n} A001015(A000142(n)).

A167166 a(n) = n^7 mod 16.

Original entry on oeis.org

0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0, 5, 0, 15, 0, 1, 0, 11, 0, 13, 0, 7, 0, 9, 0, 3, 0

Offset: 0

Zerinvary Lajos, Oct 29 2009

Equivalently: n^(4*m+7) mod 16. - G. C. Greubel, Jun 04 2016

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Table[Mod[n^7, 16], {n, 0, 10}] (* G. C. Greubel, Jun 04 2016 *)
PowerMod[Range[0,100],7,16] (* or *) PadRight[{},100,{0,1,0,11,0,13,0,7,0,9,0,3,0,5,0,15}] (* Harvey P. Dale, Jul 29 2018 *)

a(n)=n^7%16 \\ Charles R Greathouse IV, Apr 06 2016

[power_mod(n,7,16)for n in range(0, 93)] #

From R. J. Mathar, Sep 30 2013: (Start)
a(n) = a(n-16).
G.f. -x*(1 +11*x^2 +13*x^4 +7*x^6 +9*x^8 +3*x^10 +5*x^12 +15*x^14) / ( (x-1)*(1+x)*(1+x^2)*(1+x^4)*(1+x^8) ). (End)
a(n) = A130909(A001015(n)). - Michel Marcus, Jun 04 2016

cp's OEIS Frontend

A075668 Sum of next n 7th powers.

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A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

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A257659 Numbers that are not seventh powers, but can be written as the sum of the seventh powers of two or more of their prime factors.

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A291828 Numbers k such that k^2 is sum of two positive 7th powers.

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A343617 Decimal expansion of P_{3,2}(7) = Sum 1/p^7 over primes == 2 (mod 3).

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A017575 a(n) = (12n+4)^7.

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A070726 a(n) = n^7 mod 46.

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A094075 Denominator of I(n)=integral_{x=0 to 1/n}(x^2-1)^3 dx.

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