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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A217601 Integer averages of squares of first primes.

Original entry on oeis.org

4, 1314, 7128, 3081302, 4009568, 5312966, 16834447856, 2462344442400, 289274033242208, 46671783125431818542, 221000817555367050608, 618811172463743796896678, 13954866972387224169218132, 176536110349401666017009273532, 996528450408723697487070591774
Offset: 1

Views

Author

Robert Price, Mar 19 2013

Keywords

Examples

			a(2) = 1314 is the average of squares of first 19 primes (24966/19=1314).

Links

Crossrefs

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248, A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

Formula

a(n) = A217600(n)/A111441(n).

Extensions

a(13) from Karl-Heinz Hofmann, Dec 08 2020
a(14) from Karl-Heinz Hofmann, Dec 26 2020
a(15) from Karl-Heinz Hofmann, Dec 27 2020

A233523 Prime(n), where n is such that (1+sum_{i=1..n} prime(i)) / n is an integer.

Original entry on oeis.org

2, 3, 13, 29, 71, 79, 107, 907, 3491, 4967, 7853, 61223, 80051, 81547, 90901, 211811, 381629, 1990007, 3220793, 4749637, 6725027, 6784937, 34463699, 143691323, 185831033, 213609173, 285336497, 442634651, 911588849, 953122843, 1548789581, 2153787017
Offset: 1

Views

Author

Robert Price, Dec 15 2013

Keywords

Comments

a(50) > 3475385758524527. - Bruce Garner, Jun 05 2021

Examples

			a(3) = 13, because 13 is the 6th prime and the sum of the first 6 primes+1 = 42 when divided by 6 equals 7 which is an integer.

Links

Crossrefs

Cf. A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    t = {}; sm = 1; Do[sm = sm + Prime[n]; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A233862 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^2) / k is an integer.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 37, 41, 101, 107, 197, 317, 1033, 2029, 2357, 2473, 2879, 5987, 6173, 35437, 56369, 81769, 195691, 199457, 793187, 850027, 1062931, 1840453, 2998421, 4217771, 6200923, 9914351, 10153807, 13563889, 18878099, 60767923, 118825361, 170244929
Offset: 1

Views

Author

Robert Price, Dec 16 2013

Keywords

Comments

a(51) > 1428199016921.
a(67) > 2407033812270611. - Bruce Garner, May 05 2021

Examples

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^2+1 = 378 when divided by 6 equals 63 which is an integer.

Links

Crossrefs

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    t = {}; sm = 1; Do[sm = sm + Prime[n]^2; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Module[{nn=9600000},Prime[#]&/@Transpose[Select[Thread[{Range[nn], 1+ Accumulate[ Prime[Range[nn]]^2]}],IntegerQ[Last[#]/First[#]]&]][[1]]] (* Harvey P. Dale, Sep 09 2014 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^2); s==0 \\ Charles R Greathouse IV, Nov 30 2013

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

Cf. A001157, A064602, A050226, A056650, A064606, A064607, A064610, A064611, A064612, A048290, A062982, A045345.

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.

Links

Crossrefs

Cf. A034448, A064609, A050226, A056550, A064605, A064606, A064607, A064610, A064612, A048290, A062982, A045345.

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

A128169 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^5 = 1 + A122103(k).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 22, 58, 155, 363, 464, 665, 1146, 2870, 3048, 4019, 5931, 8724, 21503, 50439, 67560, 476281, 705570, 4050684, 6956459, 7443590, 10449928, 10799546, 15385564, 17735139, 83325458, 245271750, 255583775, 1395860516, 2921734534, 6255577368, 9050771725, 12062893218, 13689205205, 42254229197, 46440930382
Offset: 1

Views

Author

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

Keywords

Comments

a(52) > 3*10^13. - Bruce Garner, Jun 05 2021
a(53) > 1.2*10^14. - Bruce Garner, Mar 28 2022

Links

Crossrefs

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    p = 2; k = 0; s = 1; lst = {}; While[k < 521330000, s = s + p^5; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p]

Extensions

a(31) from Sean A. Irvine, Jan 19 2011
a(32)-a(33) from Robert G. Wilson v, Jan 20 2011
a(34)-a(41) from Robert Price, Dec 18 2013

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

Cf. A001159, A064604, A050226, A056650, A064605, A064606, A064610-A064612, A048290, A062982, A045345.

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

A001222, A022559, A050226, A056650, A064602-A064611, A048290, A062982, A045345.

Formula

Mod[A022559(n), n]=0

Extensions

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

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

Cf. A001158, A064603, A050226, A056650, A064605, A064607, A064610-A064612, A048290, A062982, A045345.

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

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