A025281 -id:A025281 - cp's OEIS frontend

A343197 Numbers k such that A025281(k) is prime.

2, 3, 6, 16, 29, 30, 34, 35, 36, 39, 57, 59, 76, 77, 88, 94, 101, 112, 126, 166, 177, 192, 206, 228, 238, 248, 251, 258, 259, 260, 271, 275, 276, 282, 299, 317, 318, 333, 345, 347, 353, 354, 370, 378, 386, 391, 402, 407, 417, 437, 445, 452, 455, 466, 470, 475, 478, 489, 494, 499, 508, 521, 530

Offset: 1

J. M. Bergot and Robert Israel, Apr 07 2021

nonn

Numbers k such that A343196(k) = 1.

			a(4) = 16 is a term because A025281(16) = A001414(16!) = 101.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001414, A025281, A343196.

sopf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
R:= NULL: count:= 0: T:= 0:
for n from 2 while count < 100 do
T:= T + sopf(n);
if isprime(T) then R:= R, n; count:= count+1 fi;
od:
R;

Select[Range[2, 530], PrimeQ@ Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]] &[#!] &] (* Michael De Vlieger, Apr 07 2021 *)

from sympy import isprime, factorint
A343197_list = [n for n in range(2,10**6) if isprime(sum(sum(p*e for p, e in factorint(i).items()) for i in range(2,n+1)))] # Chai Wah Wu, Apr 09 2021

A072692 Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.

Original entry on oeis.org

1, 87, 8299, 823081, 82256014, 8224740835, 822468118437, 82246711794796, 8224670422194237, 822467034112360628, 82246703352400266400, 8224670334323560419029, 822467033425357340138978, 82246703342420509396897774, 8224670334241228180927002517

Offset: 0

Rick L. Shepherd, Jul 02 2002

nonn

			For n=1, the sum of sigma(j) for j<=10 is 1+3+4+7+6+12+8+15+13+18=87, so a(1)=87 (=69+18=A049000(1)+A046915(1)).

P. L. Patodia, Seth Troisi and Hiroaki Yamanouchi, Table of n, a(n) for n = 0..36 (terms a(0)-a(18) by P. L. Patodia and a(19)-a(24) by Seth Troisi)
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.
P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916

Compare with A049000. Note that a(n) = A049000(n) + A046915(n).

Cf. A000203 (sigma(n)), A072691 (Pi^2/12), A049000, A046915, A024916, A025281.

for(m=0,10,print1(sum(n=1,k=10^m,n*(k\n)),",")) \\ Improved by M. F. Hasler, Apr 18 2015

A072692(n)=A024916(10^n) \\ This is very efficient, using efficient code of A024916. - M. F. Hasler, Apr 18 2015

[(i, sum([d*(10**i//d) for d in range(1,10**i+1)])) for i in range(8)] # Seth A. Troisi, Jun 27 2010

from math import isqrt
def A072692(n): return -(s:=isqrt(m:=10**n))**2*(s+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

Asymptotic formula: a(n) ~ Pi^2/12 * 10^2n. See A072691 for Pi^2/12. Observe that A025281 also contains that constant in its asymptotic formula.

More terms from P L Patodia (pannalal(AT)usa.net), Jan 11 2008, Jun 25 2008

Corrected by N. J. A. Sloane, Jun 08 2008, following suggestions from Don Reble and David W. Wilson

A343196 a(n) is the least positive number k such that Sum_{k<=j<=k+n-1} A001414(j) is prime.

Original entry on oeis.org

2, 1, 1, 3, 2, 1, 5, 3, 5, 3, 9, 3, 5, 4, 2, 1, 4, 8, 5, 13, 9, 6, 4, 6, 3, 15, 3, 2, 1, 1, 8, 22, 2, 1, 1, 1, 3, 2, 1, 9, 5, 9, 5, 3, 6, 3, 3, 5, 8, 5, 6, 31, 11, 9, 4, 2, 1, 2, 1, 3, 5, 4, 9, 9, 5, 5, 8, 9, 7, 3, 5, 3, 6, 10, 2, 1, 1, 3, 3, 6, 7, 10, 44, 17, 51, 4, 2, 1, 3, 8, 12, 16, 2, 1, 8

Offset: 1

J. M. Bergot and Robert Israel, Apr 07 2021

nonn

a(n) is the first k such that the sum of primes, with repetition, dividing (k+n-1)!/(k-1)! is prime.

			a(4) = 3 because A001414(3) + A001414(4) + A001414(5) + A001414(6) = 17 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001414, A025281, A343197.

sopf:= proc(n) option remember; local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
f:= proc(n) local j,t,i;
  t:= add(sopf(i),i=1..n);
  for j from 1  do
    if isprime(t) then return j fi;
    t:= t + sopf(j+n)-sopf(j)
  od
end proc:
map(f, [$1..100]);

A356646 Numbers k such that the integer log of k! is a perfect power.

Original entry on oeis.org

4, 8, 27, 31, 575, 669, 1201, 2505, 4784, 7618, 35710, 65005, 166422, 870062, 994086, 1105670, 1209538, 2140133, 3020610, 9147713, 9404277, 14492743, 16792162, 18566766, 19445469, 21264479, 46483343, 109424090, 292374429, 293351547, 362681674, 399576585, 450622855

Offset: 1

J. M. Bergot and Robert Israel, Aug 19 2022

nonn

Numbers k such that A025281(k) is a perfect power.

Numbers k such that A356631(k) = 1.

			a(2) = 8 because the integer log of 8! = 2^7 * 3^2 * 5 * 7 is 2*7 + 3*2 + 5 + 7 = 32 = 2^5 is a perfect power.

Chai Wah Wu, Table of n, a(n) for n = 1..38

Cf. A001414, A001597, A025281, A356631.

spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:ispow:= proc(n) igcd(map(t -> t[2], ifactors(n)[2]))>1 end proc:s:= 0: R:= NULL: count:= 0:
for i from 1 while count < 27 do
  s:= s+spf(i);
  if ispow(s) then
    count:= count+1; R:= R,i;
  fi
od:
R;

Select[Range[8000], GCD @@ FactorInteger[Plus @@ Times @@@ FactorInteger[#!]][[;; , 2]] > 1 &] (* Amiram Eldar, Aug 26 2022 *)

from itertools import count, islice, accumulate
from math import prod
from sympy import perfect_power, factorint
def A356646_gen(): # generator of terms
    return (a+2 for a, b in enumerate(accumulate(sum(prod(d) for d in factorint(n).items()) for n in count(2))) if perfect_power(b))
A356646_list = list(islice(A356646_gen(),10)) # Chai Wah Wu, Aug 28 2022

a(28)-a(33) from Chai Wah Wu, Aug 28 2022

A061708 Smallest number whose square has (2n - 1)^2 divisors.

Original entry on oeis.org

1, 6, 36, 216, 210, 7776, 46656, 1260, 1679616, 10077696, 7560, 362797056, 44100, 18480, 78364164096, 470184984576, 272160, 264600, 101559956668416, 1632960, 3656158440062976, 21936950640377856, 180180, 789730223053602816, 9261000, 58786560, 170581728179578208256

Offset: 1

Labos Elemer, Jun 19 2001

nonn

a(n) <= 6^(n-1); 36^(n-1) has (2n-1)^2 divisors for all n.

			For n = 8, a(8) = 1260 = 2*2*3*3*5*7 and d(1260^2) = d(2*2*2*2*3*3*3*3*5*5*7*7) = 225 = (2*8-1)^2.
For n = 14, a(14) = 18480 and d((2*2*2*2*2*2*2*2*3*5*7*11)^2) = 729 = (2*14-1)^2.

Amiram Eldar, Table of n, a(n) for n = 1..68

Cf. A000005, A000290, A005179, A005408, A016017, A025281, A048691.

a(n) = Min_{x : d(x^2) = (2n-1)^2};

a(n) = Min_{x : A000005(A000290(x)) = A000290(A005408(n))}.

More terms from David Wasserman, Jun 24 2002
Edited by Charlie Neder, Jun 03 2019
a(26)-a(27) from Amiram Eldar, Dec 03 2023

A343424 Numbers k such that sopfr((k-1)!) is divisible by k, where sopfr(k) = A001414(k) = sum of primes, with repetition, dividing k.

Original entry on oeis.org

1, 2, 45, 53, 177, 436, 1239, 3651, 6463, 6869, 10753, 19450, 29721, 33289, 88907, 93682, 1137232, 1516121, 4361271, 9428534, 43778664, 74738670, 271442366, 775223371, 835126289, 1736463189, 3088442241, 5054888590, 11184483614, 16993011938, 30788570768, 33342871740

Offset: 1

Scott R. Shannon, Apr 15 2021

nonn

See A025281(k-1) for the values of sopfr((k-1)!).

			45 is a term as sopfr(44!) = 585 which is divisible by 45.

Martin Ehrenstein, Table of n, a(n) for n = 1..35

Cf. A001414, A025281, A027746.

sopfr[0] = sopfr[1] = 0; sopfr[n_] := Plus @@ Times @@@ FactorInteger[n]; sum = 0; s = {}; Do[sum += sopfr[n]; If[Divisible[sum, n + 1], AppendTo[s, n + 1]], {n, 0, 10^6}]; s (* Amiram Eldar, May 06 2021 *)

sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
isok(k) = !(sopfr((k-1)!) % k); \\ Michel Marcus, May 06 2021

a(27) from Amiram Eldar, May 06 2021

a(28) and beyond from Martin Ehrenstein, May 16 2021

A065131 Arithmetic mean of first n terms of A001414 is an integer.

Original entry on oeis.org

2, 8, 18, 26, 27, 53, 344, 539, 1221, 6869, 7271, 10753, 17742, 33289, 59688, 156649, 949350, 2291181, 3115026, 3283468, 4148225, 4493333, 18012397, 80433020, 218082214, 268274214, 639257836, 4081484513, 9373412685, 28315174578, 43438521639, 208170358834

Offset: 1

Labos Elemer, Oct 15 2001

nonn

			Sum of first 18 terms of A001414 gives A025281(18)=126 which is divisible by n=18, so 18 is a term: 0+2+3+4+5+5+7+6+6+7+11+7+13+9+8+8+17+8=126=7*18,

Cf. A025281, A001414.

s=0; Do[s=s+lg[n]; If[IntegerQ[n/25000], Print[n]]; If[IntegerQ[s/n], Print[{n, s, s/n}]], {n, 2, 2000000}] where lg[n]=A001414(n).

A025281(n)/n is an integer, where A025281(n+1)=A025281(n)+A001414(n) and A001414 is sum of primes dividing n (with repetition).

a(21)-a(29) from Donovan Johnson, Apr 25 2010
a(30)-a(31) from Donovan Johnson, Jun 11 2011
a(32) from Donovan Johnson, Feb 26 2014

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A343197 Numbers k such that A025281(k) is prime.

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A072692 Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.

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A343196 a(n) is the least positive number k such that Sum_{k<=j<=k+n-1} A001414(j) is prime.

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A356646 Numbers k such that the integer log of k! is a perfect power.

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A061708 Smallest number whose square has (2n - 1)^2 divisors.

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A343424 Numbers k such that sopfr((k-1)!) is divisible by k, where sopfr(k) = A001414(k) = sum of primes, with repetition, dividing k.

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A065131 Arithmetic mean of first n terms of A001414 is an integer.

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