91
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Number
$91$ (ninety-one) is:
- $7 \times 13$
- The $1$st integer whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
- $91^2 = 8281$
- The $1$st integer whose square has a decimal representation consisting of the concatenation of $2$ consecutive decreasing integers:
- $91^2 = 8281$
- The $1$st Fermat pseudoprime to base $3$:
- $3^{91} \equiv 3 \pmod {91}$
- The $2$nd triangular number after $28$ which is the sum of $2$ cubes:
- $91 = 27 + 64 = 3^3 + 4^3$
- The $2$nd of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
- $73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
- The $3$rd Fermat pseudoprime to base $4$ after $15$, $85$:
- $4^{91} \equiv 4 \pmod {91}$
- The $3$rd after $1$, $55$ of the $4$ square pyramidal numbers which are also triangular.
- The $5$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
- $10$, $34$, $58$, $85$, $91$, $\ldots$
- The $6$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$:
- $91 = 1 + 6 + 12 + 18 + 24 + 30 = 6^3 - 5^3$
- The $6$th square pyramidal number after $1$, $5$, $14$, $30$, $55$:
- $91 = 1 + 4 + 9 + 16 + 25 + 36 = \dfrac {6 \paren {6 + 1} \paren {2 \times 6 + 1} } 6$
- The $7$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$:
- $91 = 1 + 5 + 9 + 13 + 17 + 21 + 25 = 7 \paren {2 \times 7 - 1}$
- The $8$th positive integer which cannot be expressed as the sum of a square and a prime:
- $1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $\ldots$
- The $13$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$:
- $91 = \ds \sum_{k \mathop = 1}^{13} k = \dfrac {13 \times \paren {13 + 1} } 2$
- The $17$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$:
- $91 \to 9^2 + 1^2 = 81 + 1 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
- The $17$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$, $67$, $73$, $79$, $85$ such that no factorial of an integer can end with $n$ zeroes
- The $30$th semiprime:
- $91 = 7 \times 13$
- The $49$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$, $90$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
Also see
- Previous ... Next: Square Pyramidal Number
- Previous ... Next: Square Pyramidal and Triangular Numbers
- Previous ... Next: Hexagonal Number
- Previous ... Next: Numbers of Zeroes that Factorial does not end with
- Previous ... Next: Numbers not Sum of Square and Prime
- Previous ... Next: Non-Square Positive Integers not Sum of Square and Prime
- Previous ... Next: Fermat Pseudoprime to Base 4
- Previous ... Next: Happy Number
- Previous ... Next: Semiprime Number
Historical Note
There are $91$ days in a quarter-year: $13$ weeks of $7$ days each.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $91$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $91$