A055514 -id:A055514 - cp's OEIS frontend

A055233 Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

10, 39, 155, 371, 2935561623745, 454539357304421

Offset: 1

Carlos Rivera, Jun 21 2000

Composite n such that n = p_1 + p_2 + ... + p_k where the p_i are consecutive primes, p_1 is the smallest prime factor of n and p_k is the largest.
Concerning a(6): 454539357304421 is the product of two primes, 3536123 * 128541727 and also the sum of these two plus all the primes in between: 3536123 + 3536129 + 3536131 + ... + 128541719 + 128541727. I do not know if there are any terms in A055233 between 2935561623745 and 454539357304421. (I have searched for values of N satisfying N=Pa*Pb=Pa+...+Pb as far as 5.98*10^16, but this is not quite the same as A055233 or A055514.) - Robert Munafo, Nov 20 2002
This is a subsequence of A055514 where the sum must be divisible by the smallest and largest term, but they need not be its smallest and largest prime factor. Without restriction to composite numbers, all primes would be trivially included: see A169802. - M. F. Hasler, Nov 21 2021

			10 = 2*5 = 2 + 3 + 5;
39 = 3*13 = 3 + 5 + 7 + 11 + 13;
371 = 7*53 = 7 + 11 + 13 + ... + 53.

Erich Friedman, What's Special About This Number?
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 39
Miroslav Kureš, Straddled numbers: numbers equal to the sum of powers of consecutive primes from the least prime factor to the largest prime factor, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 2, 8-15.
Robert Munafo, Notable Properties of Specific Numbers
Carlos Rivera, Puzzle 98. Curio 39, The Prime Puzzles and Problems Connection.

Subsequence of A055514.
Cf. A074036 (sum of primes from sfp(n) to gpf(n)), A169802 (n = A074036(n)).
Cf. A020639 (spf: smallest prime factor), A006530 (gpf: greatest prime factor).

Select[Range[2, 10^3], And[CompositeQ@ #1, #1 == #2] & @@ {#, Total@ Prime[Range @@ PrimePi@ {#[[1, 1]], #[[-1, 1]]} &@ FactorInteger[#]]} &] (* Michael De Vlieger, Sep 04 2019 *)

a(5) found by Jud McCranie, Jul 03 2000
454539357304421 confirmed to be the 6th term by Donovan Johnson, Aug 23 2010
Example: removed last (see A055514). - Manuel Valdivia, Nov 19 2011

A055513 Class number h = h- * h+ of cyclotomic field Q( exp(2 Pi / prime(n)) ).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 8, 9, 37, 121, 211, 695, 4889, 41241, 76301, 853513, 3882809, 11957417, 100146415, 838216959, 13379363737, 411322824001, 3547404378125, 9069094643165, 63434933542623, 161784800122409, 1612072001362952, 2604529186263992195, 28496379729272136525, 646901570175200968153, 1753848916484925681747, 687887859687174720123201, 2333546653547742584439257, 56234327700401832767069245, 10834138978768308207500526544

Offset: 1

N. J. A. Sloane, Jun 16 2001

Washington gives a very extensive table (but beware errors!).
From Jianing Song, Nov 10 2023: (Start)
h+(n) denotes the class number of Q(exp(2*Pi/n) + exp(-2*Pi/n)).
Primes p such that h+(p) != 1 are listed in A230869. As a result, if prime(n) is not in A230869, then a(n) = A000927(n), otherwise a(n) = A000927(n) * A230870(m) for prime(n) = A230869(m). (End)

			For n = 9, prime(9) = 23, a(9) = 3.
For n = 38, prime(38) = 163, a(38) = 4*2708534744692077051875131636 = 10834138978768308207500526544.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, p. 429.
L. C. Washington, Introduction to Cyclotomic Fields, Springer, pp. 353-360.

Jianing Song, Table of n, a(n) for n = 1..100 (b-file based on data of A000927, A230869 and A230870)
Hisanori Mishima, Factorizations of Cyclotomic Numbers
M. Newman, A table of the first factor for prime cyclotomic fields, Math. Comp., 24 (1970), 215-219.
Rene Schoof, Class numbers of real cyclotomic fields of prime conductor, Math. Comp., 72 (2002), 913-937.
M. A. Shokrollahi, Tables

For the relative class number h-, see A000927, which agrees for the first 36 terms, assuming the Generalized Riemann Hypothesis. See also A230869 and A230870.

Cf. A035115, A055514, A061653.

Washington incorrectly gives a(17) = 41421, a(25) = 411322842001.
Edited by Max Alekseyev, Oct 25 2012
a(1) = 1 prepended by Jianing Song, Nov 10 2023

A086447 a(n) = the least k such that prime(n+1)+prime(n+2)+...+prime(n+k) is a multiple of prime(n).

Original entry on oeis.org

2, 2, 6, 6, 6, 6, 4, 8, 4, 30, 7, 31, 37, 67, 22, 60, 46, 38, 69, 13, 65, 76, 19, 163, 9, 52, 100, 84, 66, 136, 66, 119, 33, 79, 47, 76, 187, 214, 37, 96, 461, 111, 62, 189, 510, 37, 256, 121, 130, 132, 144, 481, 64, 195, 53, 47, 136, 90, 194, 318, 526, 151, 788, 1542

Offset: 1

Zak Seidov, Jul 20 2003

Conjecture: a(n) exists for every n.

			a(3)=6 because prime(3)=5 divides 7+11+13+17+19+23 = 90.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A055233, A055514, A086448.

bb={}; Do[s0=Prime[n0]; s=0; Do[s+=Prime[n]; If[IntegerQ[s/s0], bb=Append[bb, n-n0]; Break[]], {n, n0+1, 8000}], {n0, 1, 100}]; bb
sncp[n_]:=Module[{p=Prime[n],k=n+1,t},t=Prime[k];While[!Divisible[ t, p], k++;t=t+Prime[k]];k-n]; Array[sncp,100]  (* Harvey P. Dale, May 21 2017 *)

a(n)=my(p = prime(n), sp = nextprime(p+1), lp = sp, nb = 1); while (sp % p, lp = nextprime(lp+1); nb++; sp += lp); nb; \\ Michel Marcus, May 21 2017

a(n, p=prime(n))=my(s, k); forprime(q=p+1, , s+=q; k++; if(s%p==0, return(k))) \\ Charles R Greathouse IV, May 21 2017

Edited by Don Reble, Nov 10 2005

A086448 a(n) = the least integer of the form [prime(n+1)+prime(n+2)+...+prime(n+k)]/prime(n).

Original entry on oeis.org

4, 4, 18, 16, 12, 12, 6, 16, 6, 102, 11, 93, 119, 345, 48, 240, 138, 100, 263, 19, 227, 282, 31, 1071, 11, 126, 386, 278, 184, 642, 164, 445, 55, 213, 89, 190, 895, 1120, 61, 258, 4629, 323, 122, 789, 5226, 59, 1292, 325, 364, 374, 430, 3939, 118, 695, 87, 73, 358

Offset: 1

Zak Seidov, Jul 20 2003

It seems that a(n) exists for any n.
Among first 1000 terms, the largest term is a(793) = 1807606, with p = prime(793) = 6079, and 6079*1807606 = the sum of 42840 consecutive primes after p. - Zak Seidov, Nov 07 2014
Among first 10000 terms, the largest term is a(9349) = 30376745, with p = prime(9349) = 97159, and p*(9349) = the sum of 629543 consecutive primes after p: 2951374167455 = sum(prime(k), k = 9349 + 1..9349 + 629543) - Zak Seidov, Feb 21 2015

			a(3)=18 because prime(3)=5 and (7+11+13+17+19+23)/5 = 18.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A055233, A055514, A086447.

bb={}; Do[s0=Prime[n0]; s=0; Do[If[IntegerQ[ss0=(s+=Prime[n])/s0], bb=Append[bb, ss0]; Break[]], {n, n0+1, 8000}], {n0, 1, 10}]; bb

Edited by Don Reble, Nov 10 2005

A074195 Sum of the primes from smallest prime factor of n to the largest prime factor of n = largest difference between consecutive divisors of n (ordered by size).

Original entry on oeis.org

4, 20, 1278, 1339, 11074308238, 19096347067

Offset: 1

Jason Earls, Sep 19 2002

10^11 < a(7) <= 8912510129422438. - Giovanni Resta, May 13 2016

Cf. A060681, A074036, A055233, A055514.

Select[Range[2, 2000], (p = First /@ FactorInteger[#]; #-#/p[[1]] == Sum[ Prime[i], {i, PrimePi@ p[[1]], PrimePi@ p[[-1]]}]) &] (* Giovanni Resta, May 13 2016 *)

isok(n) = {pf = factor(n)[,1]; my(pmin = vecmin(pf)); s = 0; forprime(p = pmin, vecmax(pf), s += p); s == n - n/pmin;} \\ Michel Marcus, Feb 03 2014

a(5)-a(6) from Giovanni Resta, May 13 2016

A200721 Product of two nonconsecutive primes p and q that divides the sum of primes between p and q, exclusively.

Original entry on oeis.org

26, 1133, 20309, 51159, 3246905, 28673661, 5201685791

Offset: 1

Manuel Valdivia, Nov 21 2011

nonn

Prime p is approximately q/((2*log(q)-1)*k), for k = 1, 1, 3, 307, 5041, 102378,..(quotients).

a(8) > 2*10^10. 3235398421447 is also a term. - Donovan Johnson, Nov 23 2011

			51159 = 3*17053, (5+ ... +17047)/51159 = 307.

Cf. A055514, A055233, A086447, A086448, A001358.

ss[n_] := Module[{f = Transpose[FactorInteger[n]], p, q, s}, If[f[[2]] == {1, 1}, {p, q} = PrimePi[f[[1]]]; s = Total[Table[Prime[i], {i, p + 1, q - 1}]]; s != 0 && Mod[s, n] == 0, False]]; Select[Range[2, 21000], ss] (* T. D. Noe, Nov 21 2011 *)

a(7) from Donovan Johnson, Nov 23 2011

cp's OEIS Frontend

A055233 Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.

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A055513 Class number h = h- * h+ of cyclotomic field Q( exp(2 Pi / prime(n)) ).

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A086447 a(n) = the least k such that prime(n+1)+prime(n+2)+...+prime(n+k) is a multiple of prime(n).

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A086448 a(n) = the least integer of the form [prime(n+1)+prime(n+2)+...+prime(n+k)]/prime(n).

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A074195 Sum of the primes from smallest prime factor of n to the largest prime factor of n = largest difference between consecutive divisors of n (ordered by size).

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A200721 Product of two nonconsecutive primes p and q that divides the sum of primes between p and q, exclusively.

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