A077017 -id:A077017 - cp's OEIS frontend

A100484 The primes doubled; Even semiprimes.

4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Reinhard Zumkeller, Nov 22 2004

Essentially the same as A001747.
Right edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
A253046(a(n)) > a(n). - Reinhard Zumkeller, Dec 26 2014
Apart from first term, these are the tau2-primes as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019
For every positive integer b and each m in this sequence b^(m-1) == b (mod m). - Florian Baur, Nov 26 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. D. Anderson and Andrea M. Frazier, On a general theory of factorization in integral domains, Rocky Mountain J. Math., Volume 41, Number 3 (2011), 663-705. See pp. 698, 699, 702.
James Lanterman, Irreducibles in the Integers modulo n, arXiv:1210.2991 [math.NT], 2012.
Eric Weisstein's World of Mathematics, Semiprime

Subsequence of A091376. After the initial 4 also a subsequence of A039956.
Cf. A046315, A152099, A179740.
Cf. A001748, A253046, A353478 (characteristic function).
Row 3 of A286625, column 3 of A286623.

2*Filtered([1..300],IsPrime); # Muniru A Asiru, Oct 05 2018

List([1..70], n-> 2*Primes[n]); # G. C. Greubel, May 18 2019

a100484 n = a100484_list !! (n-1)
a100484_list = map (* 2) a000040_list
-- Reinhard Zumkeller, Jan 31 2012

[2*p: p in PrimesUpTo(350)]; // Vincenzo Librandi, Mar 27 2014

A100484:=n->2*ithprime(n); seq(A100484(n), n=1..70); # Wesley Ivan Hurt, Mar 27 2014

2*Prime[Range[70]] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

2*primes(70) \\ Charles R Greathouse IV, Aug 21 2011

[2*nth_prime(n) for n in (1..70)] # G. C. Greubel, May 18 2019

a(n) = 2 * A000040(n).
a(n) = A001747(n+1).
n>1: A000005(a(n)) = 4; A000203(a(n)) = 3*A008864(n); A000010(a(n)) = A006093(n); intersection of A001358 and A005843.
a(n) = A116366(n-1, n-1) for n>1. - Reinhard Zumkeller, Feb 06 2006
a(n) = A077017(n+1), n>1. - R. J. Mathar, Sep 02 2008
A078834(a(n)) = A000040(n). - Reinhard Zumkeller, Sep 19 2011
a(n) = A087112(n, 1). - Reinhard Zumkeller, Nov 25 2012
A000203(a(n)) = 3*n/2 + 3, n > 1. - Wesley Ivan Hurt, Sep 07 2013

Simpler definition.

A001747 2 together with primes multiplied by 2.

Original entry on oeis.org

2, 4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502

Offset: 1

N. J. A. Sloane

When supplemented with 8, may be considered the "even primes", since these are the even numbers n = 2k which are divisible just by 1, 2, k and 2k. - Louis Zuckerman (louis(AT)trapezoid.com), Sep 12 2000
Sequence gives solutions of sigma(n) - phi(n) = n + tau(n) where tau(n) is the number of divisors of n.
Numbers n such that sigma(n) = 3*(n - phi(n)).
Except for 2, orders of non-cyclic groups k (in A060679(n)) such that x^k==1 (mod k) has only 1 solution 2<=x<=k. - Benoit Cloitre, May 10 2002
Numbers n such that A092673(n) = 2. - Jon Perry, Mar 02 2004
Except for initial terms, this sequence = A073582 = A074845 = A077017. Starting with the term 10, they are identical. - Robert G. Wilson v, Jun 15 2004
Together with 8 and 16, even numbers n such that n^2 does not divide (n/2)!. - Arkadiusz Wesolowski, Jul 16 2011
Twice noncomposite numbers. - Omar E. Pol, Jan 30 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A060679, A009530, A098764.

Equals {2} UNION {A100484}.

Concatenation([2], List([1..60], n-> 2*Primes[n])); # G. C. Greubel, May 18 2019

[2] cat [2*NthPrime(n): n in [1..60]]; // G. C. Greubel, May 18 2019

Join[{2},2*Prime[Range[60]]] (* Harvey P. Dale, Jul 23 2013 *)

print1(2);forprime(p=2,97,print1(", "2*p)) \\ Charles R Greathouse IV, Jan 31 2012

[2]+[2*nth_prime(n) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = A001043(n) - A001223(n+1), except for initial term.
a(n) = A116366(n-2,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
A006093(n) = A143201(a(n+1)) for n>1. - Reinhard Zumkeller, Aug 12 2008
a(n) = 2*A008578(n). - Omar E. Pol, Jan 30 2012, and Reinhard Zumkeller, Feb 16 2012

A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 3, 6, 1, 0, 0, 7, 4, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 6, 9, 0, 0, 0, 10, 0, 0, 3, 1, 0, 0, 7, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 12, 0, 0, 4, 13, 8

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A077017 w/o the first term.
Positions of terms > 0 are A120944.
Positions of zeros are A130091.
Triangle A238353 counts m such that A056239(m) = n and a(m) = k.
For maximal difference we have A286470 or A355526.
Positions of terms > 1 are A325161.
If singletons (k) have minimal difference k we get A355525.
Positions of 1's are A355527.
Prepending 0 to the prime indices gives A355528.
A115720 and A115994 count partitions by their Durfee square.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A066312, A238354, A238710, A286469, A351294, A352822, A355530, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],0,Min@@Differences[primeMS[n]]],{n,2,100}]

A307540 Irregular triangle T(n,k) such that squarefree m with gpf(m) = prime(n) in each row are arranged according to increasing values of phi(m)/m.

Original entry on oeis.org

1, 2, 6, 3, 30, 10, 15, 5, 210, 42, 70, 14, 105, 21, 35, 7, 2310, 330, 462, 66, 770, 110, 154, 1155, 22, 165, 231, 33, 385, 55, 77, 11, 30030, 2730, 4290, 6006, 390, 546, 858, 10010, 78, 910, 1430, 2002, 130, 15015, 182, 286, 1365, 2145, 26, 3003, 195, 273, 429

Offset: 0

Michael De Vlieger, Apr 13 2019

Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.
Row n contains m in A005117 such that A000720(A006530(m)) = n, sorted such that phi(m)/m increases as k increases.
Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since phi(m)/m = phi(m')/m'. Let prime p | n and prime q be a nondivisor of n.
Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 7^1 * 5^0 * 3^1 * 2^1. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0's and 1's since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2^(n - 1) possible terms for n >= 1.
We may use an approach that generates the binary expansion of the range 2^(n - 1) < M < 2^n - 1, or we may append 1 to the reversed (n - 1)-tuples of {1, 0} to achieve codes M -> m for each row n, which is tantamount to ordering according to A059894.
Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function phi(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.

			Triangle begins:
1;
2;
6, 3;
30, 10, 15, 5;
210, 42, 70, 14, 105, 21, 35, 7;
...
First terms of this sequence appear bottom to top in the chart below. The
values of n appear in the header, values m = T(n,k) followed
parenthetically by phi(m)/m appear in column n. The x axis plots
according to primepi(gpf(m)), while the y axis plots k according to
phi(m)/m:
    0       1          2             3             4
    .       .          .             .             .
--- 1 ------------------------------------------------
  (1/1)     .          .             .             .
    .       .          .             .             .
    .       .          .             .             .
    .       .          .             .             7
    .       .          .             5           (6/7)
    .       .          .           (4/5)           .
    .       .          .             .             .
    .       .          .             .            35
    .       .          3             .          (24/35)
    .       .        (2/3)           .             .
    .       .          .             .             .
    .       .          .             .             .
    .       .          .             .            21
    .       .          .             .           (4/7)
    .       .          .            15             .
    .       .          .          (8/15)           .
    .       2          .             .             .
    .     (1/2)        .             .             .
    .       .          .             .             .
    .       .          .             .            105
    .       .          .             .          (16/35)
    .       .          .             .            14
    .       .          .            10           (3/7)
    .       .          .           (2/5)           .
    .       .          .             .             .
    .       .          .             .            70
    .       .          6             .          (12/35)
    .       .        (1/3)           .             .
    .       .          .             .            42
    .       .          .            30           (2/7)
    .       .          .          (4/15)           .
    .       .          .             .            210
    .       .          .             .           (8/35)
...

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Small plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m.
Michael De Vlieger, Enlarged plot of m in A307540 at x = pi(gpf(m)), y = phi(m)/m.

Cf. A000010, A000040, A000079, A000720, A002110, A005117, A006094, A006530, A007947, A059894, A070826, A077017, A306237.

Prepend[Array[SortBy[#, Last] &@ Map[{#1, #2, EulerPhi[#1]/#1} & @@ {Times @@  MapIndexed[Prime[First@ #2]^#1 &, Reverse@ #], FromDigits@ #} &, Map[Prepend[Reverse@ #, 1] &, Tuples[{1, 0}, # - 1]]] &, 6], {{1, 0, 1}}][[All, All, 1]] // Flatten

For n > 0, row lengths = A000079(n - 1).
T(n, 1) = A002110(n) = p_n#.
T(n, 2) = A306237(n) = p_n#/prime(n - 1).
T(n, 2^(n - 1) - 1) = A006094(n).
T(n, 2^(n - 1)) = A000040(n) = prime(n) for n >= 1.
Last even term in row n = A077017(n).
First odd term in row n = A070826(n).

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A100484 The primes doubled; Even semiprimes.

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A001747 2 together with primes multiplied by 2.

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A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

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A307540 Irregular triangle T(n,k) such that squarefree m with gpf(m) = prime(n) in each row are arranged according to increasing values of phi(m)/m.

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