3000 (number) explained

Number:	3000
Unicode:	MMM, mmm
Lang1:	Armenian
Lang1 Symbol:	Վ
Lang3:	Egyptian hieroglyph
Lang3 Symbol:

3000 (three thousand) is the natural number following 2999 and preceding 3001. It is the smallest number requiring thirteen letters in English (when "and" is required from 101 forward).

Selected numbers in the range 3001–3999

3001 to 3099

• 3001 – super-prime; divides the Euclid number 2999# + 1
• 3003 – triangular number, only number known to appear eight times in Pascal's triangle; no number is known to appear more than eight times other than 1. (see Singmaster's conjecture)
• 3019 – super-prime, happy prime
• 3023 – 84th Sophie Germain prime, 51st safe prime
• 3025 = 55², sum of the cubes of the first ten integers, centered octagonal number,^[1] dodecagonal number
• 3037 – star number, cousin prime with 3041
• 3045 – sum of the integers 196 to 210 and sum of the integers 211 to 224
• 3046 – centered heptagonal number
• 3052 – decagonal number
• 3059 – centered cube number
• 3061 – prime of the form 2p-1
• 3063 – perfect totient number
• 3067 – super-prime
• 3071 – Thabit number
• 3072 – 3-smooth number (2¹⁰×3)
• 3075 – nonagonal number
• 3078 – 18th pentagonal pyramidal number
• 3080 – pronic number
• 3081 – triangular number, 497th sphenic number
• 3087 – sum of first 40 primes

3100 to 3199

• 3109 – super-prime
• 3119 – safe prime
• 3121 – centered square number, emirp, largest minimal prime in quinary.
• 3125 – a solution to the expression

n ^{n} , where n =5 (3125=5 ^{5} ).

• 3136 = 56², palindromic in ternary (11022011₃), tribonacci number
• 3137 – Proth prime, both a left- and right-truncatable prime
• 3149 – highly cototient number
• 3155 – member of the Mian–Chowla sequence
• 3159 = number of trees with 14 unlabeled nodes
• 3160 – triangular number
• 3167 – safe prime
• 3169 – super-prime, Cuban prime of the form

x = y +1 .

• 3192 – pronic number

3200 to 3299

• 3203 – safe prime
• 3207 – number of compositions of 14 whose run-lengths are either weakly increasing or weakly decreasing
• 3229 – super-prime
• 3240 – triangular number
• 3248 – member of a Ruth-Aaron pair with 3249 under second definition, largest number whose factorial is less than 10¹⁰⁰⁰⁰ – hence its factorial is the largest certain advanced computer programs can handle.
• 3249 = 57², palindromic in base 7 (12321₇), centered octagonal number, member of a Ruth–Aaron pair with 3248 under second definition
• 3253 – sum of eleven consecutive primes (269 + 271 + 277 + 281 + 283 + 293 + 307 + 311 + 313 + 317 + 331)
• 3256 – centered heptagonal number
• 3259 – super-prime, completes the ninth prime quadruplet set
• 3264 – solution to Steiner's conic problem: number of smooth conics tangent to 5 given conics in general position
• 3266 – sum of first 41 primes, 523rd sphenic number
• 3276 – tetrahedral number
• 3277 – 5th super-Poulet number, decagonal number
• 3279 – first composite Wieferich number
• 3281 – octahedral number, centered square number
• 3286 – nonagonal number
• 3299 – 85th Sophie Germain prime, super-prime

3300 to 3399

• 3306 – pronic number
• 3307 – balanced prime
• 3313 – balanced prime, star number
• 3319 – super-prime, happy number
• 3321 – triangular number
• 3329 – 86th Sophie Germain prime, Proth prime, member of the Padovan sequence
• 3354 – member of the Mian–Chowla sequence
• 3358 – sum of the squares of the first eleven primes
• 3359 – 87th Sophie Germain prime, highly cototient number
• 3363/2378 ≈ √2
• 3364 = 58²
• 3367 = 15³ - 2³ = 16³ - 9³ = 34³ - 33³
• 3375 = 15³, palindromic in base 14 (1331₁₄), 15th cube
• 3389 – 88th Sophie Germain prime

3400 to 3499

• 3403 – triangular number
• 3407 – super-prime
• 3413 – 89th Sophie Germain prime, sum of the first 5 nⁿ: 3413 = 1¹ + 2² + 3³ + 4⁴ + 5⁵
• 3422 – pronic number, 553rd sphenic number, melting point of tungsten in degrees Celsius
• 3435 – a perfect digit-to-digit invariant, equal to the sum of its digits to their own powers (3³ + 4⁴ + 3³ + 5⁵ = 3435)
• 3439 – magic constant of n×n normal magic square and n-queens problem for n = 19.
• 3445 – centered square number
• 3447 – sum of first 42 primes
• 3449 – 90th Sophie Germain prime
• 3456 – 3-smooth number (2⁷×3³)
• 3457 – Proth prime
• 3463 – happy number
• 3467 – safe prime
• 3469 – super-prime, Cuban prime of the form x = y + 2, completes the tenth prime quadruplet set
• 3473 – centered heptagonal number
• 3481 = 59², centered octagonal number
• 3486 – triangular number
• 3491 – 91st Sophie Germain prime

3500 to 3599

• 3504 – nonagonal number
• 3510 – decagonal number
• 3511 – largest known Wieferich prime
• 3512 – number of primes

\leq 2 ^{15} .^[2]

• 3517 – super-prime, sum of nine consecutive primes (367 + 373 + 379 + 383 + 389 + 397 + 401 + 409 + 419)
• 3539 – 92nd Sophie Germain prime
• 3540 – pronic number
• 3559 – super-prime
• 3569 – highly cototient number
• 3570 – triangular number
• 3571 – 500th prime, Cuban prime of the form x = y + 1, 17th Lucas number, 4th balanced prime of order 4.
• 3591 – member of the Mian–Chowla sequence
• 3593 – 93rd Sophie Germain prime, super-prime

3600 to 3699

• 3600 = 60², number of seconds in an hour, called šār or šāru in the sexagesimal system of Ancient Mesopotamia (cf. Saros), 1201-gonal number
• 3601 – star number
• 3610 – 19th pentagonal pyramidal number
• 3613 – centered square number
• 3617 – sum of eleven consecutive primes (293 + 307 + 311 + 313 + 317 + 331 + 337 + 347 + 349 + 353 + 359)
• 3623 – 94th Sophie Germain prime, safe prime
• 3637 – balanced prime, super-prime
• 3638 – sum of first 43 primes, 599th sphenic number
• 3643 – happy number, sum of seven consecutive primes (499 + 503 + 509 + 521 + 523 + 541 + 547)
• 3654 – tetrahedral number
• 3655 – triangular number, 601st sphenic number
• 3660 – pronic number
• 3684 – 13th Keith number
• 3697 – centered heptagonal number

3700 to 3799

• 3721 = 61², centered octagonal number
• 3729 – nonagonal number
• 3733 – balanced prime, super-prime
• 3741 – triangular number, 618th sphenic number
• 3751 – decagonal number
• 3761 – 95th Sophie Germain prime, super-prime
• 3779 – 96th Sophie Germain prime, safe prime
• 3782 – pronic number, 623rd sphenic number
• 3785 – centered square number
• 3797 – member of the Mian–Chowla sequence, both a left- and right- truncatable prime

3800 to 3899

• 3803 – 97th Sophie Germain prime, safe prime, the largest prime factor of 123,456,789
• 3821 – 98th Sophie Germain prime
• 3828 – triangular number
• 3831 – sum of first 44 primes
• 3840 – double factorial of 10
• 3844 = 62²
• 3851 – 99th Sophie Germain prime
• 3856 – number of 17-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed
• 3863 – 100th Sophie Germain prime
• 3865 – greater of third pair of Smith brothers
• 3888 – longest number when expressed in Roman numerals I, V, X, L, C, D, and M (MMMDCCCLXXXVIII), 3-smooth number (2⁴×3⁵)
• 3889 – Cuban prime of the form x = y + 2
• 3894 – octahedral number

3900 to 3999

• 3901 – star number
• 3906 – pronic number
• 3911 – 101st Sophie Germain prime, super-prime
• 3914 – number of 18-bead necklaces (turning over is allowed) where complements are equivalent
• 3916 – triangular number
• 3925 – centered cube number
• 3926 – 12th open meandric number, 654th sphenic number
• 3928 – centered heptagonal number
• 3937 – product of distinct Mersenne primes, repeated sum of divisors is prime, denominator of conversion factor from meter to US survey foot
• 3940 – there are 3940 distinct ways to arrange the 12 flat pentacubes (or 3-D pentominoes) into a 3x4x5 box (not counting rotations and reflections)
• 3943 – super-prime
• 3947 – safe prime
• 3961 – nonagonal number, centered square number
• 3969 = 63², centered octagonal number
• 3989 – highly cototient number
• 3998 – member of the Mian–Chowla sequence
• 3999 – largest number properly expressible using Roman numerals I, V, X, L, C, D, and M (MMMCMXCIX), ignoring vinculum

Prime numbers

There are 120 prime numbers between 3000 and 4000:^[3]

3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989

Notes and References

1. Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.
2. Number of primes <= 2^n.
3. Web site: Stein . William A. . William A. Stein . The Riemann Hypothesis and The Birch and Swinnerton-Dyer Conjecture . wstein.org . 10 February 2017 . 6 February 2021.