A037074 -id:A037074 - cp's OEIS frontend

A191746 Partial sums of product of twin primes.

15, 50, 193, 516, 1415, 3178, 6777, 11960, 22363, 34026, 53069, 75568, 107967, 144830, 184033, 236016, 293615, 366514, 446037, 543380, 664483, 840882, 1027505, 1240948, 1513431, 1838330, 2198329, 2580252, 2992415, 3428014, 4084113, 4759796, 5445379, 6181542

Offset: 1

Harvey P. Dale, Jun 14 2011

nonn

Cf. A037074.

Accumulate[Times@@@Select[Partition[Prime[Range[500]],2,1], Last[#]- First[#] ==2&]]

A284203 Number of twin prime (A001097) divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 0, 0, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 0, 2, 2, 0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 2, 2, 0, 1, 1, 1, 0, 2, 2, 1, 2, 1, 0, 2, 2, 0, 2, 0, 2, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 0, 1, 1, 1, 2

Offset: 1

Ilya Gutkovskiy, Mar 22 2017

nonn

			--------------------------------------------
| n | divisors of n | twin prime    | a(n) |
|   |               | divisors of n |      |
|------------------------------------------
| 1 | {1}           |      {-}      |  0   |
| 2 | {1, 2}        |      {-}      |  0   |
| 3 | {1, 3}        |      {3}      |  1   |
| 4 | {1, 2, 4}     |      {-}      |  0   |
| 5 | {1, 5}        |      {5}      |  1   |
| 6 | {1, 2, 3, 6}  |      {3}      |  1   |
| 7 | {1, 7}        |      {7}      |  1   |
| 8 | {1, 2, 4, 8}  |      {-}      |  0   |
| 9 | {1, 3, 9}     |      {3}      |  1   |
--------------------------------------------

Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A001097, A001221, A005087, A037074, A062729, A065421, A284599.

Cf. A048599 (positions of records).

nmax = 110; Rest[CoefficientList[Series[Sum[Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Length[Select[Divisors[n], PrimeQ[#] && (PrimeQ[# - 2] || PrimeQ[# + 2]) &]], {n, 110}]

concat([0, 0],Vec(sum(k=1, 110, (isprime(k) && (isprime(k - 2) || isprime(k + 2)))* x^k/(1 - x^k)) + O(x^111))) \\ Indranil Ghosh, Mar 22 2017

a(n) = sumdiv(n, d, isprime(d) && (isprime(d-2) || isprime(d+2))); \\ Amiram Eldar, Jun 03 2024

from sympy import isprime, divisors
print([len([i for i in divisors(n) if isprime(i) and (isprime(i - 2) or isprime(i + 2))]) for n in range(1, 111)]) # Indranil Ghosh, Mar 22 2017

G.f.: Sum_{k>=1} x^A001097(k)/(1 - x^A001097(k)).
a(A062729(n)) = 0. - Ilya Gutkovskiy, Apr 02 2017
From Amiram Eldar, Jun 03 2024: (Start)
a(A048599(n)) = n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065421 - 1/5 = 1.7021605... . (End)
Additive with a(p^e) = 1 if p is in A001097, and 0 otherwise. - Amiram Eldar, May 15 2025
a(A037074(n)) = 2. - Michel Marcus, May 15 2025

A329229 Numbers that are the product of two odd prime powers with Euler phi-functions having solely a single 2 as a common prime factor.

Original entry on oeis.org

15, 21, 33, 35, 39, 45, 51, 55, 57, 69, 75, 77, 87, 93, 95, 99, 111, 115, 119, 123, 129, 135, 141, 143, 147, 153, 155, 159, 161, 175, 177, 183, 187, 201, 203, 207, 209, 213, 215, 219, 225, 235, 237, 245, 249, 253, 261, 267, 287, 291, 295, 297, 299, 303, 309, 319

Offset: 1

Gerold Brändli, Nov 08 2019

nonn

Numbers p^j*q^k, denoted "cyclic semiprimes", such that gcd(phi(p^j), phi(q^k)) = 2, p and q odd primes, j and k positive integers (Brändli and Beyne, 2016, def.4 and Lee et al., 2013, theo.1).
The products of twin primes (A037074), and odd composite numbers with a single pes-sequence, i.e. parameter B = 1, are a subset of this sequence (Schick 2003, eq.1.6.2).
Any element x in Zs* is said to be a "semi-primitive root", if the order of x modulo s is phi(s)/2, where phi(s) is the Euler phi-function (Lee 2013, def.1).
If s is a cyclic semiprime, x is a generating element and k an integer, then the following reduced modulus denoted mod* returns all elements of Zs* in the interval ]0,s/2[, with mod* defined by x^k mod* s = min(+-x^k mod s) (Lee et al., 2018, def.2.3).
Trivially, the number of cyclic semiprimes of the form 3*p is infinite.

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Selbstverlag, Zürich, 2003, ISBN 3-9522917-0-6. See p. 15.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half The Amount of Residues, arXiv:1504.02757v2 [math.NT], 7 Feb 2016. See p. 10.
Ki-Suk Lee, Miyeon Kwon and GiCheon Shin, Multiplicative Groups of Integers with Semi-primitive Roots Modulo n, Commun. Korean Math. Soc., Vol. 28, No. 1 (2013), pp. 71-77.
Ki-Suk Lee, Ji-Eun Lee, Gerold Brändli and Tim Beyne, Galois Polynomials from Quotient Groups, Journal Chungcheong Math. Soc., Vol. 31, No. 3 (2018), pp. 309-319. See p. 311.

Cf. A037074.

with(NumberTheory, Totient, PrimitiveRoot, Divisors, tau, phi, lambda); K := {}; for i from 3 by 2 to 100 do for j from i+2 by 2 to 100 do if numelems(ifactors(i*j)[2]) = 2 and gcd(phi(i), phi(j)) = 2 and gcd(i, j) = 1 then K := K union {i*j} end if end do end do; print(K)

Select[Range[5, 320, 2], (f = FactorInteger[#]; Length[f] == 2 && GCD[ EulerPhi[ f[[1, 1]]^f[[1, 2]]], EulerPhi[f[[2, 1]]^f[[2, 2]]]] == 2) &] (* Giovanni Resta, Dec 01 2019 *)

More terms from Giovanni Resta, Dec 01 2019

A343778 Primes which are two greater than A074040 terms.

Original entry on oeis.org

17, 21800053277, 86984485062381462583582279727, 2948338207972508983453357158259221375675126583677039825367935271466652794027

Offset: 1

Hartmut F. W. Hoft, Apr 29 2021

nonn

a(5) = 3052230...330677 has 17332 digits, the only prime larger than a(4) and among the cumulative products of the first 2000 twin primes pairs plus two.

			a(1) = 17 = A074040(1) + 2 = 3*5 + 2.

Cf. A037074, A077800, A074040, A097490, A097491, A097493.

(* function a074040[ ] is defined in A074040 *)
a343778[n_] := Select[Map[#+2&, a074040[n]], PrimeQ]
a343778[30]

A344147 Primes in A191746.

Original entry on oeis.org

193, 53069, 58422233, 1348470667, 2847740783, 3237916229, 5029745827, 7643871979, 15107731019, 17902513283, 21052092827, 22187962591, 28412311451, 59363922119, 81459096899, 85780812149, 102742076659, 123894775231, 137692362377, 143889901511, 170038274723, 174648621811

Offset: 1

Hartmut F. W. Hoft, May 10 2021

nonn

			a(1)=193=A191746(3) is the first prime in A191746 and a(2)=53069=A191746(11) is the second.

Cf. A037074, A074040, A077800, A079164, A191746.

(* function a037074[ ] and support functions are defined in A074040 *)
a191746[n_] := Rest[FoldList[Plus, 0, a037074[n]]]
a344147x[n_] := Select[a191746[n], PrimeQ]
a344147[550]

A071143 Numbers n such that (i) the sum of the distinct primes dividing n is divisible by the largest prime dividing n and (ii) n has exactly 4 distinct prime factors and (iii) n is squarefree.

Original entry on oeis.org

3135, 6279, 8855, 10695, 11571, 16095, 17255, 17391, 20615, 20735, 26691, 28083, 31031, 36519, 41151, 41615, 45695, 46655, 47859, 48495, 50439, 54131, 56823, 57239, 59295, 61295, 66215, 72611, 76055, 76479, 80135, 84135, 88595, 89999, 90951, 93651, 94611

Offset: 1

Labos Elemer, May 13 2002

nonn

			n = pqrs, p<q<r<s, p+q+r+s = ks; n = 6279 = 3*7*13*23, sum = 3+7+13+23 = 2*23

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]&&Equal[lf[n], 4]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]
s = {}; Do[Length[f=FactorInteger@n] == 4 && Max[(t = Transpose@f)[[2]]] == 1 && Mod[Plus @@ t[[1]], t[[1,-1]]] == 0 && AppendTo[s,n], {n, 3, 10^6, 2}]; s (* 12 times faster, Giovanni Resta, Apr 10 2013 *)
sdpQ[n_]:=Module[{fi=FactorInteger[n][[All,1]]},Divisible[Total[fi], Last[ fi]] &&Length[fi]==4&&SquareFreeQ[n]]; Select[Range[100000],sdpQ] (* Harvey P. Dale, May 01 2018 *)

A008472(n)/A006530(n) is an integer; A001221(n) = 4, n is squarefree.

Definition clarified by Harvey P. Dale, May 01 2018

A071144 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 5 distinct prime factors and n is squarefree.

Original entry on oeis.org

3570, 8970, 10626, 15015, 16530, 20706, 24738, 24882, 36890, 38130, 44330, 49938, 51051, 52170, 54834, 55986, 59570, 62985, 68370, 73554, 74613, 77330, 79458, 81770, 87290, 91266, 96162, 96866, 103730, 106314, 116466, 123234, 128570, 129426, 129930, 138890

Offset: 1

Labos Elemer, May 13 2002

nonn

			n = pqrst, p<q<r<s<t, primes, p+q+r+s+t = kt; n = 8970 = 2*3*5*13*23, sum = 46 = 2*23.

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]&&Equal[lf[n], 5]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]
sdpQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==5&&SquareFreeQ[n]&&Mod[Total[ fi],Max[fi]]==0]; Select[Range[150000],sdpQ] (* Harvey P. Dale, May 04 2023 *)

A008472(n)/A006530(n) is an integer; A001221(n) = 5, n is squarefree.

A071145 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 6 distinct prime factors and n is squarefree.

Original entry on oeis.org

72930, 106590, 190190, 222870, 335478, 397670, 620310, 836418, 844305, 884442, 1008678, 1195670, 1218945, 1247290, 1704794, 1761110, 1799798, 2086238, 2206022, 2328410, 2485830, 2496585, 2517258, 2863718, 2903538, 3024021, 3157665, 3172785, 3291890

Offset: 1

Labos Elemer, May 13 2002

nonn

			n = pqrstw, p<q<r<s<t<w, primes, p+q+r+s+t+w = kt; n = 106590 = 2*3*5*11*17*19; sum = 2+3+5+11+17+19 = 57 = 3*19 (quotient=3) (Corrected Mar 06 2006.)

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]&&Equal[lf[n], 6]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

A008472(n)/A006530(n) is an integer; A001221(n) = 6, n is squarefree.

A079877 Numbers that are divisible by at least one pair of twin primes (A001097).

Original entry on oeis.org

15, 30, 35, 45, 60, 70, 75, 90, 105, 120, 135, 140, 143, 150, 165, 175, 180, 195, 210, 225, 240, 245, 255, 270, 280, 285, 286, 300, 315, 323, 330, 345, 350, 360, 375, 385, 390, 405, 420, 429, 435, 450, 455, 465, 480, 490, 495, 510, 525, 540, 555, 560, 570, 572

Offset: 1

Reinhard Zumkeller, Feb 20 2003

Numbers that are divisible by both primes of some twin-prime pair. Harvey P. Dale, Aug 18 2017

By definition, if k is in the sequence, then so is every positive multiple of k. - Richard Locke Peterson, Aug 17 2017

			429 = 3*11*13 = 3*A001359(3)*A006512(3), therefore 429 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001097, A001359, A006512, A037074, A062505, A074480, A289484.

Select[Range[600],MemberQ[Differences[Transpose[FactorInteger[#]][[1]]], 2]&] (* Harvey P. Dale, Sep 19 2011 *)

Definition clarified by N. J. A. Sloane, Aug 18 2017

Definition further clarified by Sean A. Irvine, Aug 29 2025

A094949 Phi(m)*sigma(m), where m is the product of exactly two primes that differ by 2, where phi=A000010, sigma=A000203.

Original entry on oeis.org

192, 1152, 20160, 103680, 806400, 3104640, 12945600, 26853120, 108201600, 136002240, 362597760, 506160000, 1049630400, 1358807040, 1536796800, 2702128320, 3317529600, 5314118400, 6323748480, 9475464960, 14665694400

Offset: 1

Lekraj Beedassy, Jun 19 2004

nonn

If m=p*q for the twin prime pair (p, q), then the relation p^2 + q^2 = 2*(m+2) is evident from equations p*(p+2)=m=q*(q-2). Now phi(m)=(p-1)*(q-1)=p^2 - 1 and sigma(m)=(p+1)*(q+1)=q^2 - 1, so that phi(m)*sigma(m)=(p*q)^2 -(p^2 + q^2)+1=m^2-2*(m+2)+1=(m-3)*(m+1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001359, A006512, A037074.

EulerPhi[#]DivisorSigma[1,#]&/@Times@@@Select[Partition[Prime[ Range[ 200]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Apr 13 2017 *)

PARI
```
{m=400;p=1;while(p
				
```

a(n)=(m-3)*(m+1), where m=A037074(n).
a(n)=192*A002415(k), where k=A040040(n-1).
a(n) = (A120875(n))^2 - 4 = 4*((A120876(n))^2 - 1). - Lekraj Beedassy, Jul 09 2006

Corrected and extended by Jason Earls, Rick L. Shepherd, Vladeta Jovovic and Klaus Brockhaus, Jun 20 2004

cp's OEIS Frontend

A191746 Partial sums of product of twin primes.

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A284203 Number of twin prime (A001097) divisors of n.

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A329229 Numbers that are the product of two odd prime powers with Euler phi-functions having solely a single 2 as a common prime factor.

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A343778 Primes which are two greater than A074040 terms.

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A344147 Primes in A191746.

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A071143 Numbers n such that (i) the sum of the distinct primes dividing n is divisible by the largest prime dividing n and (ii) n has exactly 4 distinct prime factors and (iii) n is squarefree.

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A071144 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 5 distinct prime factors and n is squarefree.

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A071145 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 6 distinct prime factors and n is squarefree.

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