A051838 -id:A051838 - cp's OEIS frontend

A121755 Numerator of Sum/Product of first n primes = Numerator[ A007504[n] / A002110[n] ].

1, 5, 1, 17, 2, 41, 29, 1, 10, 43, 16, 197, 1, 281, 4, 127, 4, 167, 284, 3, 356, 113, 1, 321, 2, 9, 8, 457, 4, 9, 4, 617, 2, 709, 1138, 809, 4, 1, 1, 147, 1, 1149, 1, 1277, 2, 1409, 317, 1, 4, 1, 5, 81, 1, 2027, 3169, 1, 1, 1, 3709, 7699, 307, 1655, 613, 8893, 4603, 1, 379, 1

Offset: 1

Alexander Adamchuk, Aug 19 2006

Many a(n) are equal to 1. The indices n such that a(n) = 1 are listed in A051838[n] = {1,3,8,13,23,38,39,41,43,48,50,53,56,57,58,66,68,70,73,77,84,90,94,98,...}. Primes p such that a(p) = 1 are listed in A121756[n] = {3,13,23,41,43,53,73,149,151,157,167,191,229,269,293,373,521,557,569,607,691, 701,829,853,863,887,947,991,...}. Many a(n) are primes. It appears that all prime a(n) {5,17,2,41,29,43,197,281,127,167,3,113,457,617,709,809,1277,1409,317,2027,307,...} and all prime divisors of composite a(n) {2,5,71,89,3,107,569,7,383,331,457,...} belong to A111267[n].

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A007504, A002110, A121744, A051838, A121756, A120291, A111267.

Table[Numerator[Sum[Prime[k],{k,1,n}]/Product[Prime[k],{k,1,n}]],{n,1,100}]
Module[{prs=Prime[Range[70]]},Flatten[Numerator[Thread[ {Accumulate[ prs]/ Rest[ FoldList[Times,1,prs]]}]]]] (* This is several hundred times faster than the first Mathematica program in generating 5000 terms of the sequence *) (* Harvey P. Dale, Dec 29 2012 *)

a(n) = Numerator[ Sum[ Prime[k], {k,1,n} ] / Product[ Prime[k], {k,1,n}] ]. a(n) = Numerator[ A007504[n] / A002110[n] ].

A159581 First divisors at which integral quotients occur consecutively in A116536 (and associated with A159580).

Original entry on oeis.org

2747, 6601, 75130, 133386, 148827, 208385, 255445, 799846, 814006, 890299, 993730, 1037571, 1049698, 1382738, 1723170, 1869711, 2168747, 2530165, 2569285, 2615298, 2838135, 2963531, 3020151, 3434151, 3510885, 3627674, 3706710, 3941521, 3999326, 4434411, 4700809, 5887533, 6133305

Offset: 1

Enoch Haga, Apr 16 2009

			The first run of consecutive integers in A051838 is A051838(6)=38 and A051838(7)=39, therefore a(1) = A140763(6)= 2747.
The second run of consecutive integers in A051838 is A051838(13)=56, A051838(14)= 57, A051838(15)=58, therefore a(2) = A140763(13) = 6601.

Cf. A051838, A116536, A140763, A159578, A159580.

Recomputed by R. J. Mathar, Aug 28 2018

A121756 Primes p such that sum of first p primes divides product of first p primes.

Original entry on oeis.org

3, 13, 23, 41, 43, 53, 73, 149, 151, 157, 167, 191, 229, 269, 293, 373, 521, 557, 569, 607, 691, 701, 829, 853, 863, 887, 947, 991, 1019, 1033, 1039, 1051, 1087, 1091, 1303, 1321, 1367, 1429, 1483, 1493, 1543, 1667, 1697, 1709, 1723, 1733, 1777, 1811, 1831

Offset: 1

Alexander Adamchuk, Aug 19 2006

nonn

All a(n) belong to A051838[n] = {1,3,8,13,23,38,39,41,43,48,50,53,56,57,...} Sum of first n primes divides product of first n primes.

Cf. A121755, A051838, A007504, A002110.

Select[Prime[Range[300]],IntegerQ[Product[Prime[k],{k,1,#1}]/Sum[Prime[k],{k,1,#1}]]&]

A259643 Numbers n such that sum of first n odd primes divides product of first n odd primes.

Original entry on oeis.org

1, 3, 5, 11, 25, 29, 41, 49, 51, 59, 69, 81, 99, 103, 113, 131, 133, 135, 147, 149, 153, 181, 187, 193, 197, 199, 205, 211, 213, 217, 219, 229, 235, 239, 243, 255, 271, 277, 281, 287, 289, 303, 309, 313, 323, 333, 335, 343, 347, 357, 359, 365, 367, 381, 383, 389

Offset: 1

Altug Alkan, Oct 02 2015

Obviously, a(n) is always an odd number.

			a(1) = 1 because prime(2) mod prime(2) = 3 mod 3 = 0.
a(2) = 3 because (prime(2) * prime(3) * prime(4)) mod (prime(2) + prime(3) + prime(4)) = 105 mod 15 = 0.
a(3) = 5 because (prime(2) * prime(3) * prime(4) * prime(5) * prime(6)) mod (prime(2) + prime(3) + prime(4) + prime(5) + prime(6)) = 15015 mod 39 = 0.

Cf. A051838, A065091, A070826, A071089, A071148, A262807.

Module[{nn=400,op},op=Prime[Range[2,nn+1]];Select[Range[nn],Divisible[ Times@@ Take[op,#],Total[Take[op,#]]]&]] (* Harvey P. Dale, Nov 16 2022 *)

A118219 Smallest number k>1 such that Sum_{i=1..k} Prime[i]^n divides Product_{i=1..k} Prime[i]^n.

Original entry on oeis.org

3, 30, 17, 248, 515, 49682

Offset: 1

Alexander Adamchuk, Feb 24 2007

a(7)>991430. - Robert G. Wilson v, Mar 02 2007

			a(1) = 3 because 2 + 3 + 5 = 10 divides 2*3*5 = 30 but 2 + 3 = 5 does not divide 2*3 = 6.

Cf. A051838 = Sum of first n primes divides product of first n primes. Cf. A125314 = Smallest number k>1 such that Sum_{i=1..k} i^n divides Product_{i=1..k} i^n. Cf. A007504, A002110, A024450, A098999, A122102, A122103.

f[n_] := Block[{k = 2, p = 2, s = 2^n}, While[p = p*Prime@ k; s = s + Prime@ k^n; PowerMod[p, n, s] != 0, k++ ]; k]; Do[ Print@ f@n, {n, 10}] (* Robert G. Wilson v *)

a(6) from Robert G. Wilson v, Mar 02 2007

A264897 Integers n such that A002110(n) is divisible by A098999(n).

Original entry on oeis.org

138, 163, 873, 1054, 1079, 1604, 1825, 1990, 2079, 2493, 2509, 2810, 2950, 3494, 3800, 3910, 4300, 4462, 4470, 4564, 4593, 4957, 5140, 5450, 5558, 5572, 5581, 5834, 6391, 6792, 6969, 7444, 7892, 8321, 8530, 8581, 9254, 9299, 9522, 9832, 9847, 10082, 10850

Offset: 1

Altug Alkan, Nov 27 2015

nonn

A002110(138) has 327 digits.

What is the minimum value of a(n) - a(n-1)?

Cf. A002110, A051838, A098999, A122137.

Select[Range@ 10000, Divisible[Product[Prime@ k, {k, #}], Sum[Prime[k]^3, {k, #}]] &] (* Michael De Vlieger, Nov 28 2015 *)

for(n=1, 11000, if(prod(k=1, n, prime(k)) % sum(k=1, n, prime(k)^3) == 0, print1(n, ", ")))

A318171 Least prime p such that Sum_{q prime <= p} q is divisible by the first n primes.

Original entry on oeis.org

2, 269, 269, 3823, 8539, 729551, 1416329, 23592593, 1478674861, 20458458289, 7558026467353, 201008815538749

Offset: 1

Amiram Eldar, Aug 20 2018

a(1)-a(9) are taken from De Koninck's book.

The sequence of indices of these primes is 1, 57, 57, 531, 1065, 58751, 108243, 1483151, 73716417, 901526695, 264119914199, 6301058125383.

			2 + 3 + ... + 269 = 2 * 3 * 1145
2 + 3 + ... + 269 = 2 * 3 * 5 * 229
2 + 3 + ... + 3823 = 2 * 3 * 5 * 7 * 4473
2 + 3 + ... + 8539 = 2 * 3 * ... * 11 * 1826
2 + 3 + ... + 729551 = 2 * 3 * ... * 13 * 682263
2 + 3 + ... + 1416329 = 2 * 3 * ... * 17 * 143884
2 + 3 + ... + 23592593 = 2 * 3 * ... * 19 * 1742804
2 + 3 + ... + 1478674861 = 2 * 3 * ... * 23 * 237859969
2 + 3 + ... + 20458458289 = 2 * 3 * ... * 29 * 1392427664
2 + 3 + ... + 7558026467353 = 2 * 3 * ... * 31 * 4886311486119
2 + 3 + ... + 201008815538749 = 2 * 3 * ... * 37 * 83956482342243

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 66.

Cf. A051838.

c=0; pr=2; p=2; s=2; q=2; While[c<6, While[!Divisible[s, pr], q = NextPrime[q]; s+=q]; Print[ q]; c++; p = NextPrime[p]; pr*=p]

my(c=0, pr=2, p=2, s=2, q=2); while(c<6, while(s%pr!=0, q = nextprime(q+1); s+=q); print1(q,", "); c++; p = nextprime(p+1); pr*=p)

a(11) from Giovanni Resta, Aug 20 2018

a(12) from Giovanni Resta, Aug 22 2018

A340534 a(n) is the least product of n consecutive primes that is divisible by the sum of those primes, or 0 if there is no such product.

Original entry on oeis.org

2, 0, 30, 0, 15015, 0, 37182145, 9699690, 33426748355, 0, 3710369067405, 0, 304250263527210, 0, 37420578814667938361329, 0, 18598027670889965365580513, 0, 107254825578022430263302818471, 0, 44510752614879308559270669665465, 0, 267064515689275851355624017992790, 0, 116431182179248680450031658440253681535, 0

Offset: 1

J. M. Bergot and Robert Israel, Jan 10 2021

nonn

a(27) > 10^225 if it is not 0.
If n is even, a(n) is either A002110(n) or 0.
a(n) = A002110(n) for n in A051838.

			a(5) = 15015 = 3*5*7*11*13 is the product of 5 consecutive primes and is divisible by 3+5+7+11+13 = 39.

Cf. A000210, A051838, A086487.

f:= proc(n) local L,i,p;
   L:= [seq(ithprime(i),i=1..n)]:
   p:= convert(L,`*`);
   if n::even then
     if p mod convert(L,`+`) = 0 then return p else return 0 fi
   else
     do
       p:= convert(L,`*`);
       if p mod convert(L,`+`) = 0 then return p fi;
       if p > 10^225 then return FAIL fi;
       L:= [op(L[2..-1]),nextprime(L[-1])];
     od
   fi;
end proc:
map(f, [$1..26]);

cp's OEIS Frontend

A121755 Numerator of Sum/Product of first n primes = Numerator[ A007504[n] / A002110[n] ].

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A159581 First divisors at which integral quotients occur consecutively in A116536 (and associated with A159580).

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A121756 Primes p such that sum of first p primes divides product of first p primes.

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A259643 Numbers n such that sum of first n odd primes divides product of first n odd primes.

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A118219 Smallest number k>1 such that Sum_{i=1..k} Prime[i]^n divides Product_{i=1..k} Prime[i]^n.

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A264897 Integers n such that A002110(n) is divisible by A098999(n).

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A318171 Least prime p such that Sum_{q prime <= p} q is divisible by the first n primes.

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A340534 a(n) is the least product of n consecutive primes that is divisible by the sum of those primes, or 0 if there is no such product.

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