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The definition is given by Guy: a prime p is in class 1- if the only prime divisors of p - 1 are 2 or 3; and p is in class r- if every prime factor of p - 1 is in some class <= r- - 1, with equality for at least one prime factor. - N. J. A. Sloane, Sep 22 2012

See A005105 for the definition of class r+ primes.

Gleason, p. 191: a regular polygon of n sides can be constructed by ruler, compass and angle-trisector iff n = 2^r * 3^s * p_1 * p_2 * ... * p_k, where p_1, p_2, ..., p_k are distinct elements of this sequence and > 3.

Sequence gives primes solutions to p == +1 (mod phi(p-1)). - Benoit Cloitre, Feb 22 2002

These are the primes p for which p-1 is 3-smooth. Primes for which either p+1 or p-1 have many small factors are more easily proved prime, so most of the largest primes found have this property. - Michael B. Porter, Feb 19 2013

For terms p > 3, omega(p-1) = 3 - p mod 3. Consider terms > 3. Clearly, t > 0. If p == 1 mod 3, u > 0: hence omega(p-1) = 2 because p-1 has two prime factors. If p == 2 mod 3, u = 0: hence omega(p-1) = 1 because p-1 is a power of 2. The latter case corresponds to terms that are Fermat primes > 3. Similar arguments demonstrate the converse, that for p > 3, if omega(p-1) = 3 - p mod 3, p is a term. - Chris Boyd, Mar 22 2014

The subset of A055600 which are prime. - Robert G. Wilson v, Jul 19 2014

Named after the American mathematician James Pierpont (1866-1938). - Amiram Eldar, Jun 09 2021

The definition is given by Guy: a prime p is in class 1+ if the only prime divisors of p + 1 are 2 or 3; and p is in class r+ if every prime factor of p + 1 is in some class <= r+ - 1, with equality for at least one prime factor. - N. J. A. Sloane, Sep 22 2012

See A005109 for the definition of class r- primes.

Odd terms are primes satisfying p==-1 (mod phi(p+1)). - Benoit Cloitre, Feb 22 2002

These are the primes p for which p+1 is 3-smooth. Primes for which either p+1 or p-1 have many small factors are more easily proved prime, so most of the largest primes found have this property. - Michael B. Porter, Feb 19 2013

For n>1, x=2*a(n) is a solution to the equation phi(sigma(x)) = x-phi(x). Also all Mersenne primes are in the sequence. - Jahangeer Kholdi, Sep 28 2014

Let N denote the positive integers, and suppose that f(x,y): N x N->N. Let "start" denote a subset of N, and let S be the set of numbers defined by these rules: if x and y are in S, then f(x,y) is in S, and "start" is a subset of S. The monotonic increasing ordering of S is a sequence:

A192476: f(x,y)=x^2+y^2, start={1}

A003586: f(x,y)=x*y, start={1,2,3}

A051037: f(x,y)=x*y, start={1,2,3,5}

A002473: f(x,y)=x*y, start={1,2,3,5,7}

A003592: f(x,y)=x*y, start={2,5}

A009293: f(x,y)=x*y+1, start={2}

A009388: f(x,y)=x*y-1, start={2}

A009299: f(x,y)=x*y+2, start={3}

A192518: f(x,y)=(x+1)(y+1), start={2}

A192519: f(x,y)=floor(x*y/2), start={3}

A192520: f(x,y)=floor(x*y/2), start={5}

A192521: f(x,y)=floor((x+1)(y+1)/2), start={2}

A192522: f(x,y)=floor((x-1)(y-1)/2), start={5}

A192523: f(x,y)=2x*y-x-y, start={2}

A192525: f(x,y)=2x*y-x-y, start={3}

A192524: f(x,y)=4x*y-x-y, start={1}

A192528: f(x,y)=5x*y-x-y, start={1}

A192529: f(x,y)=3x*y-x-y, start={2}

A192531: f(x,y)=3x*y-2x-2y, start={2}

A192533: f(x,y)=x^2+y^2-x*y, start={2}

A192535: f(x,y)=x^2+y^2+x*y, start={1}

A192536: f(x,y)=x^2+y^2-floor(x*y/2), start={1}

A192537: f(x,y)=x^2+y^2-x*y/2, start={2}

A192539: f(x,y)=2x*y+floor(x*y/2), start={1}

a(n) divides p^a(n) - 1 for all primes p >= 5. - Benoit Cloitre, Mar 22 2002

Also k such that Sum_{d divides k} mu(d)/d has numerator 1. - Benoit Cloitre, Apr 15 2002

k is here if and only if phi(k) also divides cototient(k). On the other hand, cototient(k) divides phi(k) if and only if k is a prime or power of a prime. - Labos Elemer, May 03 2002

It follows that k/phi(k) = 2 if k is a power of 2 and equal to 3 if k is of the form 6*A003586. - Gary Detlefs, Jun 28 2011

1 and even 3-smooth numbers, cf. A003586. - Reinhard Zumkeller, Jan 06 2014

Numbers k such that k = (1+omega(k))*phi(k). - Farideh Firoozbakht, Oct 02 2014

These are the integers whose largest squarefree divisor is 1, 2 or 6. As such, this sequence is equal to the set V_infinite, defined as the intersection of the V_k for k >= 1, where V_k(x) = {phi_k(n) <= x} and phi_k is the k-th iterate of phi, the Euler function; for instance, V_1 is given by A002202 (see Theorem 7 in Pomerance and Luca). - Michel Marcus, Nov 09 2015

This sequence is contained in A068997. The terms of A068997 not in this sequence have largest squarefree divisor other than 1, 2, or 6, beginning with 10. - Torlach Rush, Dec 07 2017

First differs from A357982 at a(25) = 3, A357982(25) = 4.

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.

A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).

Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

Fixed point of the morphism a -> a, 2a, 3a, starting from a(1) = 1. - Robert G. Wilson v, Jan 24 2006

This is a particular case of the number of entries in n-th row of Pascal's triangle not divisible by a prime p, which is given by a simple recursion using ⊗, the Kronecker (or tensor) product of vectors. Let v_0=(1,2,...,p). Then v_{n+1}=v_0 ⊗ v_n, where the vector v_n contains the values for the first p^n rows of Pascal's triangle (rows 0 through p^n-1). - William B. Everett (bill(AT)chgnet.ru), Mar 29 2008

a(n) = A206424(n) + A227428(n); number of nonzero terms in row n of triangle A083093. - Reinhard Zumkeller, Jul 11 2013

A155182 is a finite subsequence. - Reinhard Zumkeller, Jan 21 2009

From Federico Provvedi, Jul 09 2022: (Start)

In general, if p=A000040(k) is the k-th prime, with k>1, p-smooth numbers are also those positive integers m such that A000010(A002110(k))*m == A000010(A002110(k)*m).

With k=5, p = A000040(5) = 11, the primorial p# = A002110(5) = 2310, and its Euler totient is A000010(2310) = 480, so the 11-smooth numbers are also those positive integers m such that 480*m == A000010(2310*m). (End)

Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) < n^0.715. - Charles R Greathouse IV, May 21 2014

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002

Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003

After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013

The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010

The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012

Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012

Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013

Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014

A permutation of A003586. - L. Edson Jeffery, Sep 22 2014

Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016

Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

For further information see A126759, which provided the original motivation for this sequence.

From Antti Karttunen, Jan 28 2015: (Start)

The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.

A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).

For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)

A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019

Also ordinal transform of A065331 (after the initial 0). - Antti Karttunen, Sep 08 2024