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 1-5 of 5 results.

A091350 First occurrence (*2) of n in A088627 - or - least number that yields n different primes if you factorize it in all possible ways in two factors and add these factors.

Original entry on oeis.org

8, 2, 6, 90, 30, 390, 690, 420, 210, 4290, 3990, 8778, 2310, 3570, 4830, 11550, 38850, 84630, 66990, 79170, 39270, 30030, 51870, 46410, 43890, 111930, 163020, 221340, 419430, 131670, 1902810, 1385670, 1009470, 1452990, 746130, 903210, 570570, 1067430, 1531530
Offset: 0

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Author

Matthias Engelhardt, Jan 05 2004

Keywords

Comments

a(0) .. a(29) are in the list; additional know values are a(34) = 746130, a(35) = 903210, a(36) = 570570, a(40) = 510510, a(41) = 690690 and a(46) = 870870. If n in { 30, 31, 32, 33, 37, 38, 39, 42, 43, 44, 45}, or if n > 46, then a(n) > 10^6.
a(258) > 10^11. - Donovan Johnson, Oct 15 2013

Examples

			Sequence A088627 starts with 1,1,2,0, meaning that 2 and 4 yield 1 prime, 6 yields 2 and 8 yields 0 primes; therefore a(0) = 8, a(1) = 2 and a(2) = 6.

Links

Crossrefs

Cf. A088627.

Programs

  • Java
    // Programs available from Matthias Engelhardt.
  • Mathematica
    DivPrimes[n_Integer] := Length[Select[Union[Divisors[n]+Reverse[Divisors[n]]], PrimeQ]]; nn=40; t=Table[0,{nn}]; cnt=0; k=0; While[cntT. D. Noe, Aug 02 2010 *)

Extensions

Extended by T. D. Noe, Aug 02 2010

A103787 a(n) = number of k's that make primorial P(n)/A019565(k)+A019565(k) prime, A019565(k)^2<=P(n).

Original entry on oeis.org

1, 2, 4, 8, 12, 21, 40, 70, 117, 263, 450, 703, 1385, 2423, 5501, 8617, 18249, 29352, 61970, 103568, 209309, 404977, 853279, 1609502, 3008915, 5342983, 10287184, 19087437, 38498011, 78520137, 145642314
Offset: 1

Views

Author

Lei Zhou, Feb 15 2005

Keywords

Comments

If we remove the restriction A019565(k)^2<=P(n), every term gets doubled.
Number of distinct primes of the form d + P(n)/d, where P(n) is the n-th primorial A002110(n) and d is a divisor of P(n).

Examples

			P(1)=2, A019565(0)=1, 2/1+1=3 is prime, a(1)=1;
P(2)=6, A019565(0)=1, 6/1+1=7; A019565(1)=2, 6/2+2=5; so a(2)=2.

Crossrefs

Cf. A002110, A019565, A103785, A103786.

Programs

  • Mathematica
    npd = 1; Do[npd = npd*Prime[n]; tn = 0; tt = 1; cp = npd/tt + tt; ct = 0; While[IntegerQ[cp], If[(cp >= (tt*2)) && PrimeQ[cp], ct = ct + 1]; tn = tn + 1; tt = 1; k1 = tn; o = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; cp = npd/tt + tt]; Print[ct], {n, 1, 22}]
Table[ps=Prime[Range[n]]; cnt=0; Do[b=IntegerDigits[i,2,n]; p=Times@@(ps^b) + Times@@(ps^(1-b)); If[PrimeQ[p], cnt++], {i,0,2^(n-1)-1}]; cnt, {n,22}]

Formula

a(n) = A088627(A002110(n)/2).

Extensions

a(28)-a(31) from James G. Merickel, Aug 07 2015

A282849 Number of divisors k of n such that (n + k^2)/k is a prime.

Original entry on oeis.org

1, 2, 0, 2, 0, 4, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 8, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 8, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 8, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 4, 0, 0, 0, 2, 0, 8, 0, 0, 0, 4, 0, 4, 0, 0, 0, 4, 0, 6
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Feb 24 2017

Keywords

Comments

Except for the single case of a(1)=1 all terms are even. - Robert G. Wilson v, Feb 25 2017
First occurrence of 2k: 3, 2, 6, 90, 30, 390, 690, 420, 210, 4290, 3990, 8778, 2310, 3570, 4830, 11550, 38850, 84630, 66990, 79170, 39270, 30030, 51870, 46410, 43890, ..., . - Robert G. Wilson v, Feb 25 2017

Examples

			a(6) = 4 because (6 + 1^2)/1 = 7 is prime, (6 + 2^2)/2 = 5 is prime, (6 + 3^2)/3 = 5 is prime, (6 + 6^2)/6 = 7 is prime, where 1, 2, 3 and 6 are divisors of 6.

Links

Crossrefs

Cf. A088627 (number of divisors k of n such that (n + 2*k^2)/k is prime), A047255.

Programs

  • Mathematica
    f[n_] := Block[{d = Divisors@ n}, Length@ Select[d, PrimeQ[(n + #^2)/#] &]]; Array[f, 105] (* Robert G. Wilson v, Feb 25 2017 *)
Table[DivisorSum[n, 1 &, PrimeQ[(n + #^2)/#] &], {n, 105}] (* Michael De Vlieger, Nov 15 2017 *)
  • PARI
    a(n) = sumdiv(n, k, isprime((n+k^2)/k)); \\ Michel Marcus, Feb 26 2017

Formula

a(1) = 1; for n > 0: a(2n) = 2*A088627(n), a(2n + 1) = 0.

A244520 a(n) = A080715(n+1) / 2.

Original entry on oeis.org

1, 3, 5, 11, 15, 21, 29, 35, 39, 41, 51, 65, 95, 105, 155, 165, 179, 191, 221, 231, 239, 281, 329, 371, 419, 431, 485, 519, 611, 641, 659, 809, 905, 935, 989, 1019, 1031, 1049, 1121, 1199, 1229, 1289, 1451, 1469, 1481, 1509, 1541, 1661, 1821, 1931, 2109, 2129, 2141, 2339, 2549, 2795, 2969, 3021, 3039, 3189, 3299, 3329
Offset: 1

Views

Author

Joerg Arndt, Jul 10 2014

Keywords

Comments

Numbers k such that 2d + k/d is prime for every d|k. Such k must be an odd squarefree number. Primes in the sequence are A045536. - Thomas Ordowski, Nov 16 2017

Links

Crossrefs

Cf. A045536, A080715, A088627.

Programs

  • Mathematica
    Select[Range[1, 3400, 2], Function[n, AllTrue[Divisors@ n, PrimeQ[2 # + n/#] &]]] (* Michael De Vlieger, Nov 18 2017 *)
  • PARI
    is_ok(n)=n=2*n;fordiv(n,d,if(!isprime(d+n/d),return(0)));return(1);
for(n=1,10^4,if(is_ok(n),print1(n,", ")));

Formula

A088627(a(n)) = A000005(a(n)) = 2^m. - Thomas Ordowski, Nov 16 2017

A295124 a(n) = smallest number k with n prime factors such that 2d + k/d is prime for every d | k.

Original entry on oeis.org

1, 3, 15, 105, 93081
Offset: 0

Views

Author

Thomas Ordowski, Nov 15 2017

Keywords

Comments

Such k must be an odd squarefree number.
a(n) has 2^n divisors and each gives another prime.
Conjecture: the sequence is infinite. It is hard to believe!
a(n) is the smallest k such that A088627(k) = A000005(k) = 2^n.

Crossrefs

Cf. A088627, A293756.
Subsequence of A244520 (2d + k/d is prime for every d|k).

Formula

a(n) = A293756(n+1)/2.

Extensions

a(4) from Michel Marcus, Nov 15 2017
Showing 1-5 of 5 results.

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