sphenic numbers

A number

is called sphenic if it is the product of 3 distinct primes.

For example, 370 is a sphenic number because it is the product of the 3 primes 2, 5 and 37.

Sphenic numbers are quite common: up to $10^8$ there are 20710806 sphenic numbers.

The sum of the reciprocals of the sphenic numbers diverges, while the sum of the reciprocal of their squares converges to $0.003696244...$ , which can be expressed as $(P(2)^3-3\,P(2)\,P(4)+2\,P(6))/6$ , where

$P(s)=\sum_{p\mathrm{\ prime}}\frac{1}{p^s}$

is the so-called prime Zeta fuction.

The first sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many sphenic numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Wikipedia, Sphenic number
OEIS, Sequence A007304

Sphenic numbers can also be... (you may click on names or numbers and on + to get more values)

aban 30 42 66 + 99000997 abundant 30 42 66 + 49999974 admirable 30 42 66 + 99999786 alternating 30 70 78 + 98989894 amenable 105 165 273 + 99999969 anti-perfect 285 apocalyptic 222 285 286 + 29998 arithmetic 30 42 66 + 9999994 astonishing 429 1353 23287 + 2732353 Bell 4213597 betrothed 195 507759 binomial 66 70 78 + 99934453 c.decagonal 20801 22781 37411 + 99703451 c.heptagonal 638 4922 6322 + 99860678 c.nonagonal 190 406 595 + 99934453 c.pentagonal 1266 1626 2806 + 99871801 c.square 1105 2665 3445 + 99955661 c.triangular 1054 1786 2110 + 99743806 cake 42 130 470 + 98080778 Carmichael 561 1105 1729 + 99036001 Carol 4190207 Catalan 42 429 congruent 30 70 78 + 9999983 constructible 30 102 170 + 84215045 Cullen 385 897 49153 + 491521 Cunningham 170 195 255 + 99980002 Curzon 30 78 105 + 99999965 cyclic 255 345 435 + 9999985 D-number 195 399 1023 + 7011903 d-powerful 357 370 374 + 9999865 de Polignac 2465 4503 4543 + 99999897 decagonal 370 742 1105 + 99705217 deceptive 1729 2821 3367 + 99627557 deficient 105 110 130 + 9999994 dig.balanced 42 78 114 + 67078146 double fact. 105 Duffinian 385 399 595 + 9999983 eban 30 42 66 + 66066062 economical 105 1005 1015 + 19999985 equidigital 105 1005 1015 + 19999985 eRAP 170 2255 4233 + 99164165 esthetic 78 345 434 + 98987678 Eulerian 66 88234 152637 + 16777191 evil 30 66 78 + 99999986 fibodiv 366 646 795 + 98952806 Fibonacci 610 987 10946 + 63245986 Friedman 1022 1435 2505 + 983485 gapful 105 110 130 + 99999755 Gilda 78 110 30254 + 3970547 Giuga 30 happy 70 130 190 + 9999983 Harshad 30 42 70 + 99999965 heptagonal 286 874 970 + 99909727 hex 3367 4921 8911 + 99999907 hexagonal 66 190 231 + 99934453 hoax 438 483 627 + 99997305 Hogben 273 651 1407 + 99950007 house 78 434 2847 + 96595409 hungry 37929 712201 7339199 33662541 hyperperfect 1570153 iban 42 70 102 + 777774 iccanobiF 12815 idoneal 30 42 70 + 385 inconsummate 195 266 426 + 999971 interprime 30 42 102 + 99999965 Jacobsthal 10923 21845 699051 + 11184811 junction 105 406 410 + 99999962 Kaprekar 4879 7777 22222 + 80726977 katadrome 30 42 70 + 98765421 Lehmer 255 435 561 + 99991111 Leyland 60049 178478 268705 + 8389137 lonely 1342 370315 2010805 + 47326803 Lucas 322 39603 103682 + 12752043 lucky 105 195 231 + 9999961 Lynch-Bell 1362 3126 magic 1105 1695 2465 + 98061761 magnanimous 30 70 110 + 64622665 metadrome 78 138 238 + 13456789 modest 222 266 399 + 99979999 Moran 42 114 190 + 99999965 Motzkin 5798 41835 6536382 narcissistic 370 1634 9474 9926315 nialpdrome 30 42 66 + 99999994 nonagonal 154 474 651 + 99716509 nude 66 222 366 + 77777133 oban 30 66 70 + 987 octagonal 645 1045 1281 + 99971041 odious 42 70 110 + 99999994 palindromic 66 222 282 + 99988999 pancake 154 742 1771 + 99962731 panconsummate 78 231 pandigital 78 114 742 + 16434201 partition 30 42 231 + 64112359 pentagonal 70 590 651 + 99531901 pernicious 42 66 70 + 9999985 Perrin 1130 4610 8090 + 65448410 plaindrome 66 78 114 + 88999999 Poulet 561 645 1105 + 99971821 practical 30 42 66 78 prim.abundant 30 42 66 + 99999786 primeval 1012349 1023457 1234579 + 10234579 primorial 30 pronic 30 42 110 + 99470702 Proth 385 609 705 + 99942401 pseudoperfect 30 42 66 + 999942 repdigit 66 222 555 + 7777777 repfigit 742 7385 31331 156146 repunit 255 273 651 + 99950007 Rhonda 15698 47594 172577 + 97537655 Ruth-Aaron 78 105 154 + 99849027 Sastry 715 6099 self 42 110 154 + 99999939 self-describing 444422 10123133 10123331 + 42272427 sliding 70 110 290 + 48830173 Smith 438 483 627 + 99997305 star 8437 9361 15301 + 99756037 straight-line 222 246 258 + 9876543 strobogrammatic 609 906 986 + 99988666 subfactorial 14833 14684570 super Niven 30 70 102 + 1000002 super-d 105 190 231 + 9999869 taxicab 1729 20683 149389 + 68007673 tetrahedral 165 286 455 + 92568571 tetranacci 1490 triangular 66 78 105 + 99934453 tribonacci 1705 19513 121415 4700770 trimorphic 50001 781249 5000001 + 75781249 uban 30 42 66 + 99000098 Ulam 102 114 138 + 10000239 undulating 282 434 474 + 9898989 unprimeable 322 518 530 + 9999985 untouchable 238 246 290 + 999898 upside-down 258 357 555 + 99991111 vampire 1435 110758 131242 + 83119297 wasteful 30 42 66 + 9999994 weird 70 Wieferich 22953 42627 52665 + 49887799 Woodall 1023 177146 546874 + 66961565 Zeisel 105 1419 1729 + 96931639 Zuckerman 1113 1131 1311 + 11171111 Zumkeller 30 42 66 + 99966 zygodrome 66 222 555 + 99999922