cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 145 results. Next

A116360 Smallest number having exactly n partitions into products of two successive primes (A006094), or -1 if no such number exists.

Original entry on oeis.org

1, 6, 30, 60, 90, 105, 120, 135, 143, 158, 155, 167, 173, 182, 185, 207, 197, 203, 212, 215, 221, 231, 227, 233, 239, 242, 256, 245, 251, 261, 257, 260, 263, 266, 282, 272, 275, 278, 281, 291, -1, 287, 290, 293, 296, 309, 312, 302, 305, 319, 308, 314, -1, 317, 322, 320
Offset: 0

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

If a(n) <> -1: A116357(a(n))=n and A116357(m)<>n for m
From David A. Corneth, Sep 11 2024: (Start)
To prove a value -1 we need two facts:
1. For some k we have A116357(k), A116357(k+1), A116357(k+2), A116357(k+3), A116357(k+4), A116357(k+5) > n as A116357(k + 6) >= A116357(k) for all k.
2. A116357(m) != n for 1 <= m < k. (End)

Examples

			Without proof: a(40) = -1 and a(52) = -1.
a(40) = -1 as A116357(296) through A116357(296+5) are larger than 40 and for 1 <= m < 296 we have A116357(m) != 40. - _David A. Corneth_, Sep 11 2024

Links

Crossrefs

Cf. A006094, A116357.

Extensions

Edited by D. S. McNeil, Sep 06 2010

A116357 Number of partitions of n into products of two successive primes (A006094).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 2, 0, 2, 3, 0, 1, 2, 1, 2, 3, 0, 1, 3, 1, 3, 3, 0, 2, 3, 1, 3, 3, 1, 2, 3, 1, 3, 4, 1, 3, 3, 1, 4, 4, 1, 3, 3, 2, 4, 4, 1, 3, 5
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 12 2006

Keywords

Comments

a(A116358(n)) = 0; a(A116359(n)) > 0;
a(n) < A101048(n).

Examples

			a(41) = #{2*3 + 5*7} = 1;
a(42) = #{2*3+2*3+2*3+2*3+2*3+2*3+2*3, 2*3+2*3+3*5+3*5} = 2.

Links

Crossrefs

Cf. A006094, A101048, A116358, A116359, A116360.

Programs

  • Maple
    N:= 200: # to get a(1) to a(N)
Primes:= select(isprime,[2,seq(i,i=3..1+floor(sqrt(N)),2)]):
G:= mul(1/(1 - x^(Primes[i]*Primes[i+1])), i=1..nops(Primes)-1):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Dec 09 2016
  • Mathematica
    m = 105; kmax = PrimePi[Sqrt[m]]; Product[1/(1-x^(Prime[k]*Prime[k+1])), {k, 1, kmax}] + O[x]^(m+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Mar 09 2019, after Robert Israel *)

Formula

G.f.: Product_{k >= 1} 1/(1 - x^(prime(k)*prime(k+1))). - Robert Israel, Dec 09 2016

A245630 Products of terms of A006094 (products of 2 successive primes).

Original entry on oeis.org

1, 6, 15, 35, 36, 77, 90, 143, 210, 216, 221, 225, 323, 437, 462, 525, 540, 667, 858, 899, 1147, 1155, 1225, 1260, 1296, 1326, 1350, 1517, 1763, 1938, 2021, 2145, 2491, 2622, 2695, 2772, 3127, 3150, 3240, 3315, 3375, 3599, 4002, 4087, 4757, 4845, 5005, 5148, 5183
Offset: 1

Views

Author

Robert Israel, Jul 27 2014

Keywords

Comments

Multiplicative monoid generated by products of two successive primes.
All positive integers of the form Product_{i>=1} (prime(i)*prime(i+1))^m_i for integers m_i >= 0 (all but finitely many m_i = 0).
The smallest subset A of the positive integers such that
1) 1 is in A
2) if n is in A then so is n * prime(i) * prime(i+1) for all i.
Subsequence of A028260.
If A059897(.,.) is used as multiplicative operator in place of standard integer multiplication, A006094 generates A030229 (products of an even number of distinct primes). - Peter Munn, Oct 04 2019

Examples

			1 is in the sequence.
6 = 2*3 is in the sequence.
36 = (2*3)^2 is in the sequence.
90 = (2*3) * (3*5) is in the sequence.

Links

Crossrefs

Cf. A006094, A030229, A059897, A245636, A267251.
Subsequence of: A028260, A325698.

Programs

  • Maple
    N:= 10^6: # to get all terms <= N
PP:= [seq(ithprime(i)*ithprime(i+1),i=1.. numtheory[pi](floor(sqrt(N)))-1)]:
ext:= (x,p) -> seq(x*p^i,i=0..floor(log[p](N/x))):
S:= {1}:
for i from 1 to nops(PP) do S:= map(ext,S,PP[i]) od:
S;
  • Mathematica
    M = 10^6;
T = Table[Prime[n] Prime[n + 1], {n, 1, PrimePi[Sqrt[M]]}];
T2 = Select[Join[T, T^2], # <= M &];
Join[{1}, T2 //. {a___, b_, c___, d_, e___} /; b*d <= M && FreeQ[{a, b, c, d, e}, b*d] :> Sort[{a, b, c, d, e, b*d}]] (* Jean-François Alcover, Apr 12 2019 *)
  • PARI
    f(n) = prime(n)*prime(n+1); \\ A006094
mul(x,y) = x*y;
lista(nn) = {my(v = vector(nn, k, f(k)), lim = f(nn+1), ok = 0, nv); while (!ok,  nv = select(x->(xMichel Marcus, May 10 2021

Formula

Limit_{n->oo} a(n)/n^2 = Product_{i>=1} (1 - 1/sqrt(prime(i)*prime(i+1)))^2 / (1 - 1/prime(i))^2 = 1/A267251^2 (see Erdős reference).

A157742 A006094(n+3) mod 9.

Original entry on oeis.org

8, 5, 8, 5, 8, 5, 1, 8, 4, 5, 8, 5, 7, 4, 8, 1, 5, 8, 7, 5, 7, 2, 5, 8, 5, 8, 5, 5, 5, 1, 8, 2, 8, 1, 4, 5, 1, 7, 8, 2, 8, 5, 8, 4, 1, 5, 8, 5, 4, 8, 2, 4, 1, 7, 8, 7, 5, 8, 2, 5, 5, 8, 5, 5, 1, 2, 8, 5, 7, 2, 1, 4, 5, 1, 2, 5, 2, 2, 8, 2, 8, 7, 5, 7, 2, 5, 8, 5, 7, 2, 5, 2, 5, 4, 4, 8, 1, 7, 2, 4, 1, 8, 4, 2, 7
Offset: 0

Views

Author

Paul Curtz, Mar 05 2009

Keywords

Comments

Values of 0, 3 or 6 are obviously absent, because that would indicate divisibility by 3 which does not happen for prime products involving primes >3.

Formula

a(n) = A038194(n+3)*A038194(n+4) mod 9.

Extensions

Edited by R. J. Mathar, Apr 01 2009

A088328 Table read by rows where row n contains lower twin primes of the form k*A002110(n)-1 in the range 0 < k < A006094(n+1).

Original entry on oeis.org

3, 5, 11, 17, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 29, 59, 149, 179, 239, 269, 419, 569, 599, 659, 809, 1019, 1049, 1229, 1319, 1619, 1949, 2129, 419, 1049, 2309, 2729, 3359, 5879, 6089, 6299, 7349, 7559, 8819, 9239, 10499, 10709
Offset: 1

Views

Author

Pierre CAMI, Nov 06 2003

Keywords

Comments

Some of the values k*primorial(n)-1 generated by k in the range 1 to prime(n+1)*prime(n+2)-1 are not lower twin primes, A001359, so the list of k that produces the n-th row of the irregular table, as shown in A088329, is not a list of necessarily consecutive integers.
If n>2 the count of k values is near or greater than 4*log(4*p(n+1)); is this related to a proof of the infinity of twin prime pairs?

Examples

			2*2 -1 = 3, k=2, n=1
3*2 -1 = 5, k=3, n=1
6*2 -1 = 11, k=6, n=1
9*2 -1 = 17, k=9, n=1
The first three rows are:
3,5,11,17; generated by k=2, 3, 6, 9
5,11,17,29,41,59,71,101,107,137,149,179,191,197;
29,59,149,179,239,269,419,569,599,659,809,1019,1049,1229,1289,1319,1619,1949,2129;

Programs

  • Maple
    isA001359 := proc(n) option remember ; return isprime(n) and isprime(n+2) ; end:
A002110 := proc(n) local i ; if n = 0 then 1; else mul(ithprime(i),i=1..n) ; end if; end proc:
A006094 := proc(n) return ithprime(n)*ithprime(n+1) ; end proc:
A088328 := proc(n,k) option remember; for j from 1 to A006094(n+1)-1 do a := j*A002110(n)-1 ; if isA001359(a) and k =1 then return a ; elif isA001359(a) and a > procname(n,k-1) then return a ; end if; end do; return -1 ; end proc:
for n from 1 to 10 do for k from 1 do T := A088328(n,k) ; if T < 0 then break; else printf("%d,",T) ; end if; end do; printf("\n") ; od: # R. J. Mathar, Oct 30 2009

Extensions

Edited by R. J. Mathar, Oct 30 2009

A157940 Numbers n divisible by the largest prime <= sqrt(n) which are not in A001248 (primes squared) or A006094 (product of two consecutive primes).

Original entry on oeis.org

8, 12, 18, 24, 30, 40, 45, 56, 63, 70, 84, 98, 105, 112, 132, 154, 165, 182, 195, 208, 234, 260, 273, 286, 306, 340, 357, 380, 399, 418, 456, 475, 494, 513, 552, 575, 598, 621, 644, 690, 736, 759, 782, 805, 828, 870, 928, 957, 992, 1023, 1054, 1085, 1116, 1178
Offset: 1

Views

Author

M. F. Hasler, Mar 10 2009

Keywords

Comments

A subsequence of A157941.

Examples

			For numbers less than 4 the definition does not make sense, since there's no prime < 2=sqrt(4).
a(1)=8 which is divisible by 2 = precprime(sqrt(8)) and neither a prime squared (as would be 4 and 9) nor product of consecutive primes.
5 and 7 are not in this sequence, since not a multiple of 2=precprime(sqrt(5)) =precprime(sqrt(7)).
6 is not in the sequence, since it is the product of 2=precprime(sqrt(6)) and the following prime, 3. For the same reason, 15 is excluded.

Programs

  • PARI
    for( n=5,1999, n % precprime(sqrtint(n)) & next; n % nextprime(sqrtint(n-1)+1) & print1(n","))

Formula

A157940 = A157942 \ A157936 = A157941 \ A006094, where A157941 = A157942 \ A001248.

A240052 2nd arithmetic derivative of products of 2 successive prime numbers (A006094).

Original entry on oeis.org

1, 12, 16, 21, 44, 31, 60, 41, 56, 92, 72, 71, 124, 123, 140, 240, 244, 448, 121, 384, 236, 297, 176, 161, 249, 284, 247, 540, 191, 608, 221, 272, 380, 912, 520, 380, 1024, 371, 428, 912, 852, 508, 1472, 433, 696, 297, 293, 705, 860, 493, 716, 1456, 668, 512, 924, 636, 1188, 552, 669, 764, 2112, 1340, 521, 1504, 951, 1836, 672, 1176, 1300, 1107, 1076, 737, 908, 1520, 641, 776, 661, 821, 1647, 1416, 1828
Offset: 1

Views

Author

Freimut Marschner, Mar 31 2014

Keywords

Comments

The first arithmetic derivative of products of 2 successive prime numbers (A006094) is the sum of 2 successive prime numbers (A001043). A001043 = (A006094)’. The second arithmetic derivative is a(n)=( A001043)’ = (A006094)’’.

Examples

			(2*3)’ = 1*3+2*1 = 5; (5)’ = 1; (2^2)’ = 2*2^1 = 2*2 = 4.

Links

Crossrefs

Cf. A006094, A001043.
Cf. A003415 (1st derivative), A068346(2nd derivative).

Programs

  • Haskell
    a240052 = a068346 . a006094  -- Reinhard Zumkeller, Apr 15 2014
  • Maple
    with(numtheory); P:=proc(q) local a,b,c,p,n;
for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]);
c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
print(c); od; end: P(10^3); # Paolo P. Lava, Apr 01 2014

Formula

a(n) = (A006094(n))''.
a(n) = A068346(A006094(n))= A003415(A003415(A006094(n))).

A240053 3rd Arithmetic derivation of products of 2 successive prime numbers (A006094).

Original entry on oeis.org

0, 16, 32, 10, 48, 1, 92, 1, 92, 96, 156, 1, 128, 44, 188, 608, 248, 1408, 22, 1472, 240, 324, 368, 30, 86, 288, 32, 1188, 1, 1552, 30, 560, 476, 2176, 924, 476, 5120, 60, 432, 2176, 1148, 512, 4480, 1, 1300, 324, 1, 391, 1052, 46, 720, 3232, 672, 2304, 1448, 860, 2484, 1036, 226, 768, 7232, 1628
Offset: 1

Views

Author

Freimut Marschner, Mar 31 2014

Keywords

Comments

The first arithmetic derivation of products of 2 successive prime numbers (A006094) is the sum of 2 successive prime numbers (A001043). A001043 = (A006094)’. The second arithmetic derivation is (A240052) = (A001043)’ = (A006094)’’. The third arithmetic derivation of products of 2 successive prime numbers (A006094) is a(n) = (A240052)’ = (A001043)’’ = (A006094)’’’.

Examples

			a(12)=(A006094(12))'''=(37*41)'''=(A001043(12))''=(78)''=(71)'=1;
a(14)=(A006094(14))'''=(43*47)'''=(A001043(12))''=(90)''=(123)'=44.

Links

Crossrefs

Cf. A006094, A001043, A240052.
Cf. A003415 (1st derivative), A068346 (2nd derivative), A099306 (3rd derivative).

Programs

  • Maple
    with(numtheory); P:= proc(q) local a,b,c,d,n,p;  a:=ithprime(n)*ithprime(n+1);
for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]); c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
d:=c*add(op(2,p)/op(1,p),p=ifactors(c)[2]); print(d);
od; end: P(10^4); # Paolo P. Lava, Apr 07 2014

Formula

a(n) = (A006094(n))’’’.
a(n) = A099306(A006094(n)).
a(n) = A003415(A240052(n)).

A088329 Table read by rows where row n consists of all k in the range 0A006094(n+1) such that k*A002110(n)-1 are lower twin primes.

Original entry on oeis.org

2, 3, 6, 9, 1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 1, 2, 5, 6, 8, 9, 14, 19, 20, 22, 27, 34, 35, 41, 43, 44, 54, 65, 71, 2, 5, 11, 13, 16, 28, 29, 30, 35, 36, 42, 44, 50, 51, 55, 57, 69, 73, 86, 95, 104, 121, 125, 128, 135, 140, 1, 4, 5
Offset: 1

Views

Author

Pierre CAMI, Nov 06 2003

Keywords

Comments

The lower twin primes generated by this kind of multiplication of k and primorials are in A088328.

Examples

			The first 4 rows are:
2,3,6,9 ;
1,2,3,5,7,10,12,17,18,23,25,30,32,33;
1,2,5,6,8,9,14,19,20,22,27,34,35,41,43,44,54,65,71 ;
2,5,11,13,16,28,29,30,35,36,42,44,50,51,55,57,69,73,86,95,104,121,125,128,135,140;

Crossrefs

Cf. A088328, A001359.

Extensions

Edited by R. J. Mathar, Oct 30 2009

A240054 4th arithmetic derivative of products of 2 successive prime numbers (A006094).

Original entry on oeis.org

0, 32, 80, 7, 112, 0, 96, 0, 96, 272, 220, 0, 448, 48, 192, 1552, 380, 5056, 13, 4480, 608, 756, 752, 31, 45, 912, 80, 2484, 0, 3120, 31, 1312, 572, 7744, 1448, 572, 26624, 92, 1296, 7744, 1340, 2304, 17216, 0, 1920, 756, 0, 40, 1056, 25, 2064, 8112, 2000, 10752, 2180, 1052, 5076, 1212, 115, 3328, 21760, 1820
Offset: 1

Views

Author

Freimut Marschner, Mar 31 2014

Keywords

Comments

Let a'=a1 be the first arithmetic derivative, then a2 is the second and so on. It is interesting to examine the length of successive arithmetic derivatives ending with 1. For example, a(168) = 445 is the 4th arithmetic derivative of prime(168)*prime(169) = 997*1009 = 1005973. The example given here is of length 11; that means that the 11th arithmetic derivative of 1005973 is 1.

Examples

			(997*1009)' = a, a' = a1 = 2006, a2 = 1155, a3 = 886, a4 = 445, a5 = 94, a6 = 49, a7 = 14, a8 = 9, a9 = 6, a10 = 5, a11 = 1.

Links

Crossrefs

Cf. A006094, A001043, A068346, A003415, A240052, A240053.

Programs

  • Maple
    with(numtheory); P:= proc(q) local a,b,c,d,f,n,p;  a:=ithprime(n)*ithprime(n+1);
for n from 1 to q do a:=ithprime(n)*ithprime(n+1);
b:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]); c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
d:=c*add(op(2,p)/op(1,p),p=ifactors(c)[2]); f:=d*add(op(2,p)/op(1,p),p=ifactors(d)[2]);
print(d); od; end: P(10^4); # Paolo P. Lava, Apr 07 2014

Formula

a(n) = (A006094(n))''''.
a(n) = A068346(A068346(n)) = A003415(A003415(A003415(A003415(n)))).
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