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 31 results. Next

A064605 Numbers k such that A064602(k) is divisible by k.

Original entry on oeis.org

1, 2, 8, 74, 146, 150, 158, 307, 526, 541, 16157, 20289, 271343, 953614, 1002122, 2233204, 3015123, 15988923, 48033767, 85110518238
Offset: 1

Views

Author

Labos Elemer, Sep 24 2001

Keywords

Comments

Analogous sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A064610, A064611, A048290, A062982, A045345.
a(20) > 3*10^10. - Donovan Johnson, Aug 31 2012
a(21) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

Examples

			Summing divisor-square sums for j = 1..8 gives 1+5+10+21+26+50+50+85 = 248, which is divisible by 8, so 8 is a term and the integer quotient is 31.
		

Crossrefs

Programs

  • Mathematica
    k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[2, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Apr 25 2011 *)

Formula

(Sum_{j=1..k} sigma_2(j)) mod k = A064602(k) mod k = 0.

Extensions

a(15)-a(19) from Donovan Johnson, Jun 21 2010
a(20) from Amiram Eldar, Jan 18 2024

A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

Original entry on oeis.org

1, 35, 37, 1015, 27417, 27421, 27449, 27453, 19774739, 530743781, 530743799, 530743807, 530743813
Offset: 1

Views

Author

Labos Elemer, Sep 24 2001

Keywords

Comments

The corresponding quotients are 1, 3, 3, 5, 7, 7, 7, 7, 11, 13, 13, 13, 13, ...
a(14) > 7.5*10^10, if it exists. - Amiram Eldar, Jun 04 2021

Examples

			For n = 37, the sum A064608(37) = 1+2+2+2+2+4+2+...+4+4+4+2 = 111 = 3*37, so 37 is in the sequence.
		

Crossrefs

Cf. A064608.
Analogous "integer-mean" sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A048290, A063986, A063971, A064911, A062982, A045345.

Programs

  • Mathematica
    s[1] = 1; s[n_] := s[n] = s[n - 1] + 2^PrimeNu[n]; Select[Range[30000], Divisible[s[#], #] &] (* Amiram Eldar, Jun 04 2021 *)

Formula

{n: A064608(n) == 0 (mod n)}.

Extensions

a(10)-a(13) from Donovan Johnson, Jul 20 2012

A064611 Partial sum of usigma is divisible by n, where usigma(n) = A034448(n) and summatory-usigma(n) = A064609(n).

Original entry on oeis.org

1, 2, 8, 11, 12, 174, 212, 524, 1567, 14096, 19795, 38466, 42114, 55575, 338809, 498001, 1175281, 2424880, 3994532, 7908519, 48453784, 696840720, 5497869355, 7479239685
Offset: 1

Views

Author

Labos Elemer, Sep 24 2001

Keywords

Comments

Analogous sequences for various arithmetical functions are A050226, A056550, A064605-A064607, A064610, A064612, A048290, A062982, A045345.

Examples

			udivisor sums[=usigma(j) values] from 1 to 8 are added: 1+3+4+5+6+12+8+9=48; it is divisible by 8, thus 8 is here.
		

Crossrefs

Programs

  • Mathematica
    s = Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &], {n, 10^6}]; Module[{a = First@ s, b = {First@ s}}, Do[a += s[[i]]; If[Divisible[a, i], AppendTo[b, i]], {i, 2, Length@ s}]; b] (* Michael De Vlieger, Mar 18 2017 *)

Formula

A064609(n) mod n = 0.

Extensions

a(17)-a(22) from Donovan Johnson, Jul 20 2012
a(23)-a(24) from Amiram Eldar, Mar 17 2019

A085829 a(n) = least k such that the average number of divisors of {1..k} is >= n.

Original entry on oeis.org

1, 4, 15, 42, 120, 336, 930, 2548, 6930, 18870, 51300, 139440, 379080, 1030484, 2801202, 7614530, 20698132, 56264040, 152941824, 415739030, 1130096128, 3071920000, 8350344420, 22698590508, 61701166395, 167721158286, 455913379324, 1239301050624, 3368769533514
Offset: 1

Views

Author

Robert G. Wilson v, Jul 07 2003

Keywords

Comments

Does a(n+1)/a(n) converge to e?
Since the total number of divisors of {1..k} (see A006218) is k * (log(k) + 2*gamma - 1) + O(sqrt(k)), the average number of divisors of {1..k} approaches (log(k) + 2*gamma - 1). Since log(a(n)) + 2*gamma - 1 approaches n, a(n+1)/a(n) approaches e. - Jon E. Schoenfield, Aug 13 2007

Examples

			a(20) = 415739030 because the average number of divisors of {1..415739030} is >= 20.
		

References

  • Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.

Crossrefs

Programs

  • Mathematica
    s = 0; k = 1; Do[ While[s = s + DivisorSigma[0, k]; s < k*n, k++ ]; Print[k]; k++, {n, 1, 20}]
  • PARI
    A085829(n) = {local(s,k);s=1;k=1;while(sMichael B. Porter, Oct 23 2009

Extensions

Edited by Don Reble, Nov 06 2005
More terms from Jon E. Schoenfield, Aug 13 2007

A064607 Numbers k such that A064604(k) is divisible by k.

Original entry on oeis.org

1, 2, 7, 151, 257, 1823, 3048, 5588, 6875, 7201, 8973, 24099, 5249801, 9177919, 18926164, 70079434, 78647747, 705686794, 2530414370, 3557744074, 25364328389, 32487653727, 66843959963
Offset: 1

Views

Author

Labos Elemer, Sep 24 2001

Keywords

Comments

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(19) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(24) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

Examples

			Adding 4th-power divisor-sums for j = 1..7 gives 1+17+82+273+626+1394+2402 = 4795 which is divisible by 7, so 7 is a term and the integer quotient is 655.
		

Crossrefs

Programs

  • Mathematica
    k = 1; lst = {}; s = 0; While[k < 1000000001, s = s + DivisorSigma[4, k]; If[ Mod[s, k] == 0, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G.Wilson v, Aug 25 2011 *)

Formula

(Sum_{j=1..k} sigma_4(j)) mod k = A064604(k) mod k = 0.

Extensions

a(13)-a(18) from Donovan Johnson, Jun 21 2010
a(19)-a(23) from Amiram Eldar, Jan 18 2024

A064612 Partial sum of bigomega is divisible by n, where bigomega(n)=A001222(n) and summatory-bigomega(n)=A022559(n).

Original entry on oeis.org

1, 4, 5, 2178, 416417176, 416417184, 416417185, 416417186, 416417194, 416417204, 416417206, 416417208, 416417213, 416417214, 416417231, 416417271, 416417318, 416417319, 416417326, 416417335, 416417336, 416417338, 416417339, 416417374
Offset: 1

Views

Author

Labos Elemer, Sep 24 2001

Keywords

Comments

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
Partial sums of A001222, similarly to summatory A001221 increases like loglog(n), explaining small quotients.
a(25) > 10^13. - Giovanni Resta, Apr 25 2017

Examples

			Sum of bigomega values from 1 to 5 is: 0+0+1+1+2+1=5, which is divisible by n=5, so 5 is here, with quotient=1. For the last value,2178,below 1000000 the quotient is only 3.
		

Crossrefs

Formula

Mod[A022559(n), n]=0

Extensions

a(5)-a(24) from Donovan Johnson, Nov 15 2009

A057494 a(n) = Sum_{k = 1..10^n} d(k) where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 27, 482, 7069, 93668, 1166750, 13970034, 162725364, 1857511568, 20877697634, 231802823220, 2548286736297, 27785452449086, 300880375389757, 3239062263181054, 34693207724724246, 369957928177109416, 3929837791070240368, 41600963003695964400, 439035480966899467508
Offset: 0

Views

Author

Robert G. Wilson v, Sep 21 2000

Keywords

Comments

The Polymath project describes an algorithm for computing a(n) in time O(2.154...^n), see Tao, Croot, and Helfgott link. - Charles R Greathouse IV, Apr 16 2012

Crossrefs

Programs

  • Mathematica
    k = s = 0; Do[ While[ k < 10^n, k++; s = s + DivisorSigma[ 0, k ] ]; Print[s], {n, 0, 8} ]
  • PARI
    a(n) = sum(k=1, 10^n, numdiv(k)); \\ Michel Marcus, Feb 19 2017
    
  • Python
    from math import isqrt
    def A057494(n): return -(s:=isqrt(m:=10**n))**2+(sum(m//k for k in range(1,s+1))<<1) # Chai Wah Wu, Oct 23 2023

Formula

a(n) = A006218(10^n). - Max Alekseyev, May 10 2009

Extensions

a(10)-a(16) from Max Alekseyev, Jan 25 2010
a(17)-a(19) from Donovan Johnson, Dec 26 2012
a(20)-a(27) from Hiroaki Yamanouchi, Sep 22 2015
a(28)-a(36) from Henri Lifchitz, Feb 19 2017

A064606 Numbers k such that A064603(k) is divisible by k.

Original entry on oeis.org

1, 2, 7, 45, 184, 210, 267, 732, 1282, 3487, 98374, 137620, 159597, 645174, 3949726, 7867343, 13215333, 14153570, 14262845, 317186286, 337222295, 2788845412, 10937683400, 72836157215, 95250594634
Offset: 1

Views

Author

Labos Elemer, Sep 24 2001

Keywords

Comments

Analogous sequences for various arithmetical functions are A050226, A056650, A064605-A064607, A064610, A064611, A048290, A062982, A045345.
a(22) > 2*10^9. - Donovan Johnson, Jun 21 2010
a(26) > 10^11, if it exists. - Amiram Eldar, Jan 18 2024

Examples

			Adding divisor-cube sums for j = 1..7 gives 1+9+28+73+126+252+344 = 833 = 7*119, which is divisible by 7, so 7 is a term and the integer quotient is 119.
		

Crossrefs

Programs

Formula

(Sum_{j=1..k} sigma_3(j)) mod k = A064603(k) mod k = 0.

Extensions

a(15)-a(21) from Donovan Johnson, Jun 21 2010
a(22)-a(25) from Amiram Eldar, Jan 18 2024

A063986 Numbers k that divide Sum_{j=1..k} A051953(j) where A051953(j) = j - Phi(j). Arithmetic mean of first k cototient values is an integer.

Original entry on oeis.org

1, 4, 5, 24, 25, 249, 600, 617, 12272, 13763, 21332, 25228, 783665, 15748051, 41846733, 195853251, 2488541984, 14399065016, 21119309213, 22430204140, 43787603128, 157825075944, 206651865067, 271605149320, 374049315076, 650288309748
Offset: 1

Views

Author

Labos Elemer, Sep 06 2001

Keywords

Comments

The odd terms of A048290 and A063986 are the same. - Jud McCranie, Jun 26 2005
a(27) > 10^12. - Donovan Johnson, Dec 09 2011

Examples

			k=5: (1 + 1 + 2 + 2 + 4)/5 = 2.
		

Crossrefs

Programs

  • Mathematica
    s = 0; Do[s = s + n - EulerPhi[n]; If[ IntegerQ[s/n], Print[n]], {n, 1, 10^7} ]

Extensions

More terms from Dean Hickerson, Sep 07 2001
One more term from Robert G. Wilson v, Sep 07 2001
a(16) and a(17) from Jud McCranie, Jun 22 2005
a(18)-a(21) from Donovan Johnson, May 11 2010
a(22)-a(26) from Donovan Johnson, Dec 09 2011

A085831 a(n) = Sum_{k=1..2^n} d(k) where d(n) = number of divisors of n (A000005).

Original entry on oeis.org

1, 3, 8, 20, 50, 119, 280, 645, 1466, 3280, 7262, 15937, 34720, 75108, 161552, 345785, 736974, 1564762, 3311206, 6985780, 14698342, 30850276, 64607782, 135030018, 281689074, 586636098, 1219788256, 2532608855, 5251282902, 10874696106, 22493653324, 46475828418
Offset: 0

Views

Author

Robert G. Wilson v, Sep 21 2000

Keywords

Crossrefs

Programs

  • Mathematica
    k = s = 0; Do[ While[ k < 2^n, k++; s = s + DivisorSigma[ 0, k ]]; Print[s], {n, 0, 29} ]
  • PARI
    a(n) = sum(k=1, 2^n, numdiv(k)); \\ Michel Marcus, Oct 10 2021
  • Python
    from math import isqrt
    def A085831(n): return (lambda m, r: 2*sum(r//k for k in range(1, m+1))-m*m)(isqrt(2**n),2**n) # Chai Wah Wu, Oct 08 2021
    
Showing 1-10 of 31 results. Next