A037074 -id:A037074 - cp's OEIS frontend

A071141 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n is neither a prime, nor a true power of prime and n is squarefree. Squarefree solutions of A071140.

30, 70, 286, 646, 1798, 3135, 3526, 3570, 6279, 7198, 8855, 8970, 10366, 10626, 10695, 11571, 15015, 16095, 16530, 17255, 17391, 20615, 20706, 20735, 20806, 23326, 24738, 24882, 26691, 28083, 31031, 36519, 36890, 38086, 38130, 41151, 41615, 44330, 44998

Offset: 1

Labos Elemer, May 13 2002

nonn

			n = 286 = 2*11*13 has a form of 2pq, where p and q are twin primes;
n = 5414430 = 2*3*5*7*19*23*59, sum = 2+3+5+7+19+23+59 = 118 = 2*59.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[lf[n], 1]&& !Equal[amo[n], 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 45000, Function[n, And[Length@ # > 1, SquareFreeQ@ n, Divisible[Total@ #, Last@ #]] &[FactorInteger[n][[All, 1]] ]]] (* Michael De Vlieger, Jul 18 2017 *)

A008472(n)/A006530(n) is an integer, n has at least 3 distinct prime factors and n is squarefree.

A097491 Primes which are two greater than the terms of A079164.

Original entry on oeis.org

5, 17, 21800053277, 72409291238312731227527, 86984485062381462583582279727, 21679097826151232817152558557032490897727272048343000297777, 107025222275017133994159705286756083545279583250537082122450588876727

Offset: 1

Cino Hilliard, Aug 24 2004

nonn

A097491(8) = 2948...794027 has 76 digits and A097491(9) = 152400...802327 has 288 digits. - Hartmut F. W. Hoft, Apr 27 2021

			a(3) = 21800053277 = A079164(17) + 2 = 3*5*5*7*11*13*17*19*29*31 + 2. - _Hartmut F. W. Hoft_, Apr 27 2021

Amiram Eldar, Table of n, a(n) for n = 1..9

Cf. A048599, A097490, A097493.

Cf. A037074, A077800, A079164.

step[{list_, q_}] := Module[{p=NextPrime[q]}, {Join[list, If[PrimeQ[p+2], {{p,p+2}}, {}]], p}]
pairList[n_] := First[NestWhile[step, {{{3, 5}}, 3}, Length[First[step[#]]]<=n&]]
a079164[n_] := Rest[FoldList[Times, 1, Take[Flatten[pairList[n]], n]]]
a097491[n_] := Select[Map[#+2&, a079164[n]], PrimeQ]
a097491[39] (* Hartmut F. W. Hoft, Apr 27 2021 *)

ft(n) = p=1;for(x=1,n,p*=twinl(x);if(isprime(p+2),print1(p+2", ")); p*=twinu(x);if(isprime(p+2),print1(p+2", ")))
twinl(n) = { local(c,x); c=0; x=1; while(c

Edited by Don Reble, Apr 16 2007

Name corrected by Hartmut F. W. Hoft, Apr 27 2021

A120875 Product of twin primes minus 1.

Original entry on oeis.org

14, 34, 142, 322, 898, 1762, 3598, 5182, 10402, 11662, 19042, 22498, 32398, 36862, 39202, 51982, 57598, 72898, 79522, 97342, 121102, 176398, 186622, 213442, 272482, 324898, 359998, 381922, 412162, 435598, 656098, 675682, 685582, 736162

Offset: 1

Lekraj Beedassy, Jul 09 2006

nonn

This sequence is a subsequence of A023515.

Jason Kimberley (using T. Noe's b037074), Table of n, a(n) for n = 1..10000

Cf. A002822, A023515, A037074, A075369, A120876.

Times[#, # + 2] - 1 & /@ Select[Prime@ Range@ 150, PrimeQ[# + 2] &] (* Michael De Vlieger, Oct 23 2015 *)

for(n=1, 200, if(prime(n+1)-prime(n)==2, print1(prime(n)*prime(n+1)-1", "))) \\ Altug Alkan, Oct 23 2015

a(n) = A037074(n)-1 = (A014574(n))^2 -2 = A075369(n)-2.
a(n) = 2*A120876(n). - Jason Kimberley, Oct 23 2015
a(n) = 36*A002822(n-1)^2-2, for n>1. - Jason Kimberley, Oct 23 2015
a(n) = A023515(A107770(n)). - Jason Kimberley, Oct 23 2015

A143202 Numbers having exactly two distinct prime factors p, q with q = p+2.

Original entry on oeis.org

15, 35, 45, 75, 135, 143, 175, 225, 245, 323, 375, 405, 675, 875, 899, 1125, 1215, 1225, 1573, 1715, 1763, 1859, 1875, 2025, 3375, 3599, 3645, 4375, 5183, 5491, 5625, 6075, 6125, 6137, 8575, 9375, 10125, 10403, 10935, 11663, 12005, 16875, 17303, 18225

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

Subsequence of A007774.

A037074 is a subsequence.

			a(1) = 15 = 3 * 5 = A001359(1) * A006512(1).
a(2) = 35 = 5 * 7 = A001359(2) * A006512(2).
a(3) = 45 = 3^2 * 5 = A001359(1)^2 * A006512(1).
a(4) = 75 = 3 * 5^2 = A001359(1) * A006512(1)^2.
a(5) = 135 = 3^3 * 5 = A001359(1)^3 * A006512(1).
a(6) = 143 = 11 * 13 = A001359(3) * A006512(3).
a(7) = 175 = 5^2 * 7 = A001359(2)^2 * A006512(2).
a(8) = 225 = 3^2 * 5^2 = A001359(1)^2 * A006512(1)^2.
a(9) = 245 = 5 * 7^2 = A001359(2) * A006512(2)^2.
a(10) = 323 = 17 * 19 = A001359(4) * A006512(4).
a(11) = 375 = 3 * 5^3 = A001359(1) * A006512(1)^3.
a(12) = 405 = 3^4 * 5 = A001359(1)^4 * A006512(1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Twin Primes.
Index entries for primes, gaps between.

Cf. A001221, A001359, A006512, A006530, A007774, A020639, A037074, A143201.

a143202 n = a143202_list !! (n-1)
a143202_list = filter (\x -> a006530 x - a020639 x == 2) [1,3..]
-- Reinhard Zumkeller, Sep 13 2011

tdpfQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==2]; Select[Range[20000],tdpfQ] (* Harvey P. Dale, Mar 04 2023 *)

A143201(a(n)) = 3.
A020639(a(n)) in A001359 and A006530(a(n)) in A006512.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/(A001359(n)^2-1) = 0.1812568234997... . - Amiram Eldar, Oct 26 2024

A163836 Composites whose largest prime factor is equal to the sum of all the other prime factors (with repetition).

Original entry on oeis.org

4, 9, 25, 30, 49, 70, 84, 121, 169, 286, 289, 308, 361, 440, 495, 528, 529, 594, 646, 728, 819, 841, 884, 961, 975, 1040, 1170, 1248, 1369, 1404, 1496, 1681, 1683, 1748, 1798, 1849, 1976, 2209, 2223, 2499, 2809, 2975, 3128, 3135, 3344, 3481, 3519, 3526, 3570

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2009

nonn

Sequence contains the square of every prime. - Sean A. Irvine, Oct 05 2009
Contains 4*A143206. - David A. Corneth, Apr 28 2020
Contains 2*A037074. - Bernard Schott, Apr 28 2020

			a(1) = 4 (2=2), a(2) = 9 (3=3), a(3) = 25 (5=5), a(4) = 30 (5=3+2), a(5) = 49 (7=7), a(6) = 70 (7=5+2), a(7) = 84 (7=3+2+2), a(8) = 121 (11=11), a(9) = 169 (13=13), a(10) = 286 (13=11+2), a(11) = 289(17=17), a(12) = 308 (11=7+2+2), ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002808, A037074, A143206.

A002808 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do: end if; end proc: A006530 := proc(n) if n = 1 then 1; else numtheory[factorset](n) ; max(op(%)) ; end if; end: A001414 := proc(n) ifactors(n)[2] ; add( op(1,p)*op(2,p),p=%) ; end: A163836 := proc(n) option remember; local a,lpf; if n =1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then lpf := A006530(a) ; if 2*lpf = A001414(a) then return a; end if; end if; od: end if; end: seq(A163836(n),n=1..80) ; # R. J. Mathar, Oct 10 2009

seqQ[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 2]] == 2, f[[-1, 2]] == 1 && f[[-1, 1]] == Plus @@ Times @@@ Most[f]]]; Select[Range[4000], seqQ] (* Amiram Eldar, Apr 28 2020 *)

from sympy import factorint
def ok(n):
  f = factorint(n)
  return sum(f[p] for p in f) > 1 and 2*max(f) == sum(p*f[p] for p in f)
print(list(filter(ok, range(3571)))) # Michael S. Branicky, Apr 09 2021

Corrected and extended by Sean A. Irvine and R. J. Mathar, Oct 05 2009

A063530 Numbers k such that phi(k)+1 is a square.

Original entry on oeis.org

15, 16, 20, 24, 30, 35, 39, 45, 52, 56, 65, 70, 72, 78, 84, 90, 104, 105, 112, 123, 130, 140, 143, 144, 155, 156, 164, 165, 168, 175, 176, 180, 183, 200, 203, 210, 215, 220, 225, 231, 244, 245, 246, 248, 261, 264, 286, 300, 308, 310, 323, 330, 339, 344, 350

Offset: 1

Labos Elemer, Aug 02 2001

nonn

Numbers k such that A000010(k) = -1 + m^2 for some m.

			If k = p*(p+2), a product of twin primes (from A037074), then k is in the sequence. The corresponding square is p^2. Other solutions are k = {56,72,78,84}, since phi(k) + 1 = 25 for all. Also phi(123) + 1 = 9^2, the square of a composite.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010, A037074, A039649.

Select[Range[400],IntegerQ[Sqrt[1+EulerPhi[#]]]&] (* Harvey P. Dale, Jul 31 2020 *)

{ n=0; for (a=1, 10^9, if (issquare(eulerphi(a) + 1), write("b063530.txt", n++, " ", a); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

A070222 Numbers k such that the sum of prime divisors of k divides the sum of divisors of k.

Original entry on oeis.org

15, 20, 24, 35, 42, 54, 66, 72, 95, 98, 100, 104, 108, 110, 114, 119, 120, 126, 132, 135, 140, 143, 160, 168, 189, 195, 207, 209, 216, 220, 224, 258, 264, 270, 276, 287, 290, 294, 319, 322, 323, 351, 360, 363, 375, 377, 378, 384, 392, 432, 440, 456, 459, 464

Offset: 1

Benoit Cloitre, May 07 2002

			The sum of divisors of 132 is sigma(132) = 336; prime divisors of 132 are 2,3,11 and (336)/(2+3+11) = 336/16 = 21 hence 132 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A008472.

A037074 is a subsequence. - Amiram Eldar, Aug 08 2020

Select[Range[2,464], IntegerQ[DivisorSigma[1,#]/Total[First/@FactorInteger[#]]] &] (* Jayanta Basu, May 16 2013 *)

for(n=2,700,if(sigma(n)%sumdiv(n,d,isprime(d)*d)==0,print1(n,",")))

A072027 Swap (2,3) and all twin prime pairs >(3,5) in prime factorization of n.

Original entry on oeis.org

1, 3, 2, 9, 7, 6, 5, 27, 4, 21, 13, 18, 11, 15, 14, 81, 19, 12, 17, 63, 10, 39, 23, 54, 49, 33, 8, 45, 31, 42, 29, 243, 26, 57, 35, 36, 37, 51, 22, 189, 43, 30, 41, 117, 28, 69, 47, 162, 25, 147, 38, 99, 53, 24, 91, 135, 34, 93, 61, 126, 59, 87, 20, 729, 77

Offset: 1

Reinhard Zumkeller, Jun 07 2002

			a(143) = a(11*13) = a(11)*a(13) = 13*11 = 143.
a(77) = a(7*11) = a(7)*a(11) = 5*13 = 65.

Cf. A001359, A006512, A007510, A037074, A061898, A064505, A072026, A072028, A072029.

f[p_, e_] := If[p < 5, 5 - p, If[PrimeQ[p + 2], p + 2, If[PrimeQ[p - 2], p - 2, p]]]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 26 2024 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; if(p < 5, 5-p, if(isprime(p+2), p+2, if(isprime(p-2), p-2, p)))^f[i,2]);} \\ Amiram Eldar, Feb 26 2024

Multiplicative with a(p) = (if p<=3 then 5-p else (if p+2 is prime then p+2 else (if p-2 is prime then p-2 else p))), p prime.
a(a(n)) = n, self-inverse permutation of natural numbers.
a(n) = n for single primes (A007510) and products of twin prime pairs (A037074).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{(p < q) swapped pair} ((p^2-p)*(q^2-q)/((p^2-q)*(q^2-p))) = 1.832194438922717... . - Amiram Eldar, Feb 26 2024

A063532 Numbers k such that phi(k) + 1 = x^2 and sigma(k) + 1 = y^2 for some x and y.

Original entry on oeis.org

15, 35, 56, 72, 78, 84, 123, 143, 165, 323, 543, 627, 678, 728, 814, 836, 899, 1350, 1484, 1535, 1683, 1763, 1846, 2296, 2967, 3288, 3444, 3599, 3784, 4103, 4620, 5084, 5183, 5964, 6580, 6693, 6820, 7150, 7626, 7806, 9096

Offset: 1

Labos Elemer, Aug 02 2001

nonn

			If k = p(p+2) is a product of twin primes then phi(k) + 1 = p^2, sigma(k) + 1 = (p+2)^2, so k is in the sequence, A037074 a proper subset. There are many solutions not of this form, such as 72, 123, and 165.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A000010, A000203, A037074.

{ n=0; for (a=1, 10^9, if (issquare(eulerphi(a) + 1) && issquare(sigma(a) + 1), write("b063532.txt", n++, " ", a); if (n==500, break)) ) } \\ Harry J. Smith, Aug 25 2009

A063533 Hypotenuses of special Pythagorean triples constructed from twin primes as follows: {u, w}={p,p+2}; side a=2p(p+2), side b=(p+2)^2-p^2 and the terms of sequence are values of c=a(n)=p^2+(p+2)^2=phi(a/2)+1+sigma(a/2)+1.

Original entry on oeis.org

34, 74, 290, 650, 1802, 3530, 7202, 10370, 20810, 23330, 38090, 45002, 64802, 73730, 78410, 103970, 115202, 145802, 159050, 194690, 242210, 352802, 373250, 426890, 544970, 649802, 720002, 763850, 824330, 871202, 1312202, 1351370

Offset: 1

Labos Elemer, Aug 02 2001

nonn

Sum of the numbers on the corners of the square array that lists the numbers from 1..A014574(n)^2 in increasing order by rows. - Wesley Ivan Hurt, May 27 2023

			a(6) is obtained as follows: u = p = 41, w = p+2 = 43; a = 2*41*43 = 2*1763 = 3526; b = 43*2-41^2 = 1849-1681 = 168; c = 43^2+41^2 = 1849+1681 = 3530 = 1+phi(1763)+1+sigma(1763) = 1680+1848+2 = a(6); and 3526^2+168^2 = 3530^2.

Cf. A001359, A006512, A000010, A000203, A014574, A037074, A063530-A063532.

a(n) = 2 + A000203(A037074(n)) + A000010(A037074(n)) = A001359(n)^2 + A006512(n)^2.

a(n) = 2*(A014574(n)^2 + 1). - Wesley Ivan Hurt, May 27 2023

cp's OEIS Frontend

A071141 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n is neither a prime, nor a true power of prime and n is squarefree. Squarefree solutions of A071140.

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A097491 Primes which are two greater than the terms of A079164.

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A120875 Product of twin primes minus 1.

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A143202 Numbers having exactly two distinct prime factors p, q with q = p+2.

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Haskell

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A163836 Composites whose largest prime factor is equal to the sum of all the other prime factors (with repetition).

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A063530 Numbers k such that phi(k)+1 is a square.

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A070222 Numbers k such that the sum of prime divisors of k divides the sum of divisors of k.

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