Omkar_Pradhan_3Feb_Programming Tasks.ipynb

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   "source": [
    "### Question 45"
   ]
  },
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   "execution_count": 14,
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   "outputs": [
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     "name": "stdout",
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     "text": [
      "55385 1533776805\n"
     ]
    }
   ],
   "source": [
    "'''\n",
    "\"Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:\n",
    "Triangle   Tn=n(n+1)/2   1, 3, 6, 10, 15, ...\n",
    "Pentagonal   Pn=n(3n−1)/2   1, 5, 12, 22, 35, ...\n",
    "Hexagonal   Hn=n(2n−1)   1, 6, 15, 28, 45, ...\n",
    "It can be verified that T285 = P165 = H143 = 40755.\n",
    "Find the next triangle number that is also pentagonal and hexagonal.\"\n",
    "'''\n",
    "\n",
    "def is_pentagonal(num):\n",
    "    s=0\n",
    "    i=0\n",
    "    while(s<num):\n",
    "        i+=1\n",
    "        s = (i*(3*i - 1))//2\n",
    "        if s==num:\n",
    "            return True\n",
    "    return False\n",
    "\n",
    "\n",
    "def is_hexagonal(num):\n",
    "    s=0\n",
    "    i=0\n",
    "    while(s<num):\n",
    "        i+=1\n",
    "        s = (i*(2*i - 1))\n",
    "        if s==num:\n",
    "            return True\n",
    "    return False\n",
    "\n",
    "def tri_n(num):\n",
    "    return (num*(num + 1))//2\n",
    "\n",
    "\n",
    "i=285\n",
    "flag=True\n",
    "while(flag==True):\n",
    "    i+=1\n",
    "    if (is_pentagonal(tri_n(i)) and is_hexagonal(tri_n(i))):\n",
    "        print(i,tri_n(i))\n",
    "        flag=False\n",
    "\n"
   ]
  },
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   "source": [
    "### Question 47"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[134043, 134044, 134045, 134046]\n"
     ]
    }
   ],
   "source": [
    "'''\"The first two consecutive numbers to have two distinct prime factors are:\n",
    "14 = 2 × 7\n",
    "15 = 3 × 5\n",
    "The first three consecutive numbers to have three distinct prime factors are:\n",
    "644 = 2² × 7 × 23\n",
    "645 = 3 × 5 × 43\n",
    "646 = 2 × 17 × 19.\n",
    "Find the first four consecutive integers to have four distinct prime factors each. What is the first of these numbers?\"\n",
    "'''\n",
    "\n",
    "def is_prime(num):\n",
    "    for i in range(2,int(num**0.5)+1):\n",
    "        if num%i==0:\n",
    "            return False\n",
    "    return True\n",
    "\n",
    "def factors(n):\n",
    "    f = []\n",
    "    i=1\n",
    "    while n>0 and i<n:\n",
    "       i+=1 \n",
    "       if n%i==0:\n",
    "           n=n/i\n",
    "           if is_prime(i):\n",
    "               f.append(i)\n",
    "    return f\n",
    "\n",
    "def distinct_primes(n):\n",
    "    i=0\n",
    "    while(True):\n",
    "        i+=1\n",
    "        temp = [factors(x) for x in [i for i in range(i,i+n)]]\n",
    "        x = [len(t) for t in temp]\n",
    "        if(x.count(n) == len(x)):\n",
    "            print([i+add for add in range(0,n)])\n",
    "            break\n",
    "                \n",
    "\n",
    "distinct_primes(4)\n",
    "    "
   ]
  },
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   "attachments": {},
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   "source": [
    "### Question 48"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1911084670\n"
     ]
    }
   ],
   "source": [
    "'''\"The series, 11 + 22 + 33 + ... + 1010 = 10405071317.\n",
    "Find the last ten digits of the series, 11 + 22 + 33 + ... + 10001000.\"\n",
    "'''\n",
    "s=0\n",
    "for i in range(1,1001):\n",
    "    s+= i**i\n",
    "s = str(s)\n",
    "\n",
    "\n",
    "print(s[len(s)-11:len(s)-1])"
   ]
  },
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   "attachments": {},
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   "metadata": {},
   "source": [
    "### Question 49"
   ]
  },
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   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "148748178147\n",
      "296962999629\n"
     ]
    }
   ],
   "source": [
    "'''\n",
    "\"The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways:\n",
    "(i) each of the three terms are prime, and, \n",
    "(ii) each of the 4-digit numbers are permutations of one another.\n",
    "There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.\n",
    "What 12-digit number do you form by concatenating the three terms in this sequence?\"\n",
    "'''\n",
    "\n",
    "def is_prime(n):\n",
    "    for i in range(2,int(n**0.5)+1):\n",
    "        if n%i==0:\n",
    "            return False\n",
    "    return True\n",
    "\n",
    "\n",
    "from itertools import permutations\n",
    "def is_perm(a,b):\n",
    "    if a and b in ([int(\"\".join(x)) for x in permutations(str(a))]):\n",
    "        return True\n",
    "    else:\n",
    "        return False\n",
    "\n",
    "\n",
    "for a in range(1000,10000):\n",
    "    b = a+3330\n",
    "    c = b+3330\n",
    "    \n",
    "    if(is_prime(a) and is_prime(b) and is_prime(c) and is_perm(a,b) and is_perm(b,c)):\n",
    "        print(str(a)+str(b)+str(c))\n"
   ]
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