A122142 - OEIS

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A122142

Numbers m such that m divides sum of 5th powers of the first m primes A122103(m).

1, 25, 837, 5129, 94375, 271465, 3576217, 3661659, 484486719, 2012535795, 31455148645, 95748332903, 145967218799, 165153427677, 21465291596581, 97698929023845

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OFFSET

1,2

COMMENTS

No other terms up to 10^8. - Stefan Steinerberger, Jun 06 2007

a(11) > 6*10^9. - Donovan Johnson, Oct 15 2012

a(13) > 10^11. - Robert Price, Mar 30 2013

a(15) > 10^12. - Paul W. Dyson, Jan 04 2021

a(16) > 2.2*10^13. - Bruce Garner, May 09 2021

a(17) > 10^14. - Paul W. Dyson, Feb 04 2022

a(17) > 10^15. - Paul W. Dyson, Nov 19 2024

LINKS

Table of n, a(n) for n=1..16.

OEIS Wiki, Sums of powers of primes divisibility sequences

EXAMPLE

a(2) = 25 because 25 is the first number n>1 that divides A122103[n] = Sum[ Prime[k]^5, {k,1,n} ].

Mod[ A122103[25], 25] = Mod[ 2^5 + 3^5 + 5^5 + ... + 89^5 + 97^5, 25 ] = 0.

MATHEMATICA

s = 0; t = {}; Do[s = s + Prime[n]^5; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t

Module[{nn = 4*10^6}, Select[Thread[{Range[nn], Accumulate[ Prime[ Range[ nn]]^5]}], Divisible[#[[2]], #[[1]]] &]][[All, 1]] (* Generates the first 8 terms; to generate more, increase the value of nn, but the program may take a long time to run. *) (* Harvey P. Dale, Aug 26 2019 *)

CROSSREFS

Cf. A122103, A098999, A007504, A045345, A024450, A111441, A122102, A122140.

Sequence in context: A218479 A183879 A246761 * A151557 A008844 A251925

Adjacent sequences: A122139 A122140 A122141 * A122143 A122144 A122145

KEYWORD

nonn,hard,changed

AUTHOR

Alexander Adamchuk, Aug 21 2006

EXTENSIONS

2 more terms from Stefan Steinerberger, Jun 06 2007

a(9)-a(10) from Donovan Johnson, Oct 15 2012

a(11)-a(12) from Robert Price, Mar 30 2013

a(13)-a(14) from Paul W. Dyson, Jan 04 2021

a(15) from Bruce Garner, May 09 2021

a(16) from Paul W. Dyson, Feb 04 2022

STATUS

approved