Numbers 610,000s

A page of the Numeropedia - the Special Encyclopedia of Numbers

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611,593

[Math.]  The 50,000th prime number.

614,656

[Math.]  614,656 = (6+1+4+6+5+6)4 = 284.

[Math.]  (1+2+…+7)4 = [(15+25+…+75)+(17+27+…+77)]/2.

615,384

[Math.]  The 6-digit period of the rational number 1/13 = 0.076923076923… or 0.076923. Its multiple by 13 is 999,999.  Then,   076,923×n = abc,def,ghi,jkl and

 abc,def+ghi,jkl = 076,923, if n  = 13t+1 abc,def+ghi,jkl = 692,307, if n  = 13t+9 abc,def+ghi,jkl = 230,769, if n  = 13t+3 abc,def+ghi,jkl = 769,230, if n  = 13t+10 abc,def+ghi,jkl = 307,692, if n  = 13t+4 abc,def+ghi,jkl = 923,076, if n  = 13t+12

for t = 0 to 76,922, and in fact abc,edf = t.  The results are digit-rotations of 076,923.

And 076,923×n = abc,def,ghi,jkl &

 abc,def+ghi,jkl = 153,846, if n  = 13t+2 abc,def+ghi,jkl = 538,461, if n  = 13t+7 abc,def+ghi,jkl = 384,615, if n  = 13t+5 abc,def+ghi,jkl = 615,384, if n  = 13t+8 abc,def+ghi,jkl = 461,538, if n  = 13t+6 abc,def+ghi,jkl = 846,153, if n  = 13t+11

for t = 0 to 76,922, and in fact abc,edf = t.  The results are digit-rotations of 076,923×2 = 153,846. It is the period of the rational number 2/13 = 0.153846153846… or 0.153846), and the sums of first and last 3-digit numbers are always 999.

[Math.]  153,846×4 = 615,384.

617,716

[Math.]

 The 11th triangle number is 66 The 1111th triangle number is 617,716 The 111,111th triangle number is 6,172,882,716 The 11,111,111th triangle number is 61,728,399,382,716

All numbers are palindromic.

619,047

[Math.] Multiples of 047619 (the period of the rational number 1/21 = 0.047619047619… or 0.047619) by 4, 10, 13, 16 and 19 yield 5 other digit-rotations of the number itself: 190476, 476190, 619047, 761904 and 904761, respectively.

619,369

[Math.]   (232+767 = 999 = 212+787)

2322 = 053,824 & 053+824 = 877

7672 = 588,289 & 588+289 = 877

and

2122 = 044,944 & 044+944 = 988

7872 = 619,369 & 619+369 = 988.