cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Alexandru Petrescu

Alexandru Petrescu's wiki page.

Alexandru Petrescu has authored 15 sequences. Here are the ten most recent ones:

A374331 Palindromic squarefree semiprimes such that the sum of the two prime factors is also a palindrome.

Original entry on oeis.org

6, 717, 989, 13231, 15251, 15751, 18281, 19291, 31613, 34043, 35653, 37073, 37673, 38383, 38683, 97079, 98789, 99899, 1115111, 1226221, 1794971, 3525253, 3755573, 3782873, 104646401, 114202411, 127888721, 133707331, 134010431, 137181731, 138050831, 146828641, 157494751, 157585751, 161555161
Offset: 1

Views

Author

Alexandru Petrescu, Jul 05 2024

Keywords

Examples

			717 is a term because 717 = 3*239 and 3 + 239 = 242.

Crossrefs

Cf. A002113, A006881.

Programs

  • Mathematica
    Select[Range[10^6], PalindromeQ[#] && SquareFreeQ[#] && PrimeNu[#]==2 && PalindromeQ[Total[First/@FactorInteger[#]]]&] (* Stefano Spezia, Jul 06 2024 *)
  • PARI
    ispal(n)=my(d=digits(n));d==Vecrev(d) \\
for(a=2,10^10,if(omega(a)==2&&bigomega(a)==2 &&ispal(a),b=factor(a)[1,1]+factor(a)[2,1]; if(ispal(b),print1(a,","))))
  • PARI
    isok(k) = if (issquarefree(k) && ispal(k), my(f=factor(k)); (bigomega(f)==2) && ispal(f[1,1]+f[2,1])); \\ Michel Marcus, Jul 05 2024

A364484 Numbers whose prime factorization (prime factors and exponents) contains the digits 1 and 2 at least once, but no other digits.

Original entry on oeis.org

2, 22, 44, 121, 211, 242, 422, 484, 844, 2048, 2111, 2221, 2321, 4096, 4222, 4442, 4642, 8444, 8884, 9284, 12211, 21121, 21211, 21221, 22111, 22528, 23221, 24422, 24431, 25531, 42242, 42422, 42442, 44222, 44521, 45056, 46442, 48844, 48862, 51062, 84484, 84844, 84884, 88444, 89042, 92884, 97724
Offset: 1

Views

Author

Alexandru Petrescu, Jul 26 2023

Keywords

Examples

			9284 is a term because 9284 = 2^2 * 11^1 * 211^1.

Crossrefs

Cf. A007931.

Programs

  • PARI
    for(n=1, 1e5, v=[]; s=Set(v); f=factor(n); k=#f[, 1]; for(i=1, k, s1=Set(digits(f[i, 1])); s=setunion(s, s1)); for(i=1, k, s2=Set(digits(f[i, 2])); s=setunion(s, s2)); if(s==[1, 2], print1(n, ", ")))

A362069 Numbers k such that k+digitsum(k^2) is a square.

Original entry on oeis.org

0, 17, 62, 71, 117, 125, 197, 206, 296, 297, 305, 413, 414, 557, 558, 692, 702, 711, 863, 864, 872, 873, 1062, 1070, 1071, 1268, 1493, 1502, 1727, 1736, 1737, 1745, 1998, 2006, 2267, 2276, 2285, 2564, 2565, 2573, 2879, 2888, 2889, 3221, 3222
Offset: 1

Views

Author

Alexandru Petrescu, May 17 2023

Keywords

Comments

Conjecture: there are infinitely many pairs of consecutive terms. Example: (296,297); (413,414); (863,864).

Examples

			k=17 is a term because k^2=289 and 17+2+8+9=36=6^2.

Crossrefs

Cf. A004159, A066564.

Programs

  • Mathematica
    Select[Range[0, 3300], IntegerQ[Sqrt[# + Plus @@ IntegerDigits[#^2]]] &] (* Amiram Eldar, May 18 2023 *)
  • PARI
    isok(k)=issquare(k+sumdigits(k^2))

A362811 Sphenic numbers (product of 3 distinct primes) sandwiched between two semiprimes (product of 2 primes).

Original entry on oeis.org

186, 266, 290, 322, 470, 518, 534, 582, 590, 670, 754, 790, 814, 894, 994, 1146, 1158, 1166, 1338, 1370, 1390, 1562, 1686, 1798, 1842, 1958, 2118, 2158, 2230, 2318, 2454, 2482, 2514, 2570, 2630, 2758, 2786, 2810, 2922, 2930, 2994, 3154, 3206, 3262, 3278, 3378, 3454, 3522, 3562
Offset: 1

Views

Author

Alexandru Petrescu, May 04 2023

Keywords

Comments

Every term k is even. Otherwise one of k-1 or k+1 would be a multiple of 4, hence not a semiprime.

Examples

			186 is a term because 186=2*3*31, 185=5*37, 187=11*17.

Links

Crossrefs

Cf. A007304, A001358, A171179.

Programs

  • Mathematica
    Select[Range[2, 3600, 2], FactorInteger[#][[;; , 2]] == {1, 1, 1} && PrimeOmega[# - 1] == 2 && PrimeOmega[# + 1] == 2 &] (* Amiram Eldar, May 05 2023 *)
  • PARI
    isok(k)=omega(k)==3 && bigomega(k)==3 && bigomega(k-1)==2 && bigomega(k+1)==2

A362792 Numbers k such that 3*k and 7*k share the same set of digits.

Original entry on oeis.org

0, 45, 75, 423, 445, 450, 513, 750, 891, 1089, 1305, 2382, 2497, 4230, 4445, 4450, 4488, 4491, 4500, 4505, 4513, 4878, 5013, 5045, 5130, 5133, 5868, 7317, 7500, 7686, 8360, 8703, 8891, 8901, 8910, 8911, 8955, 8991, 9756, 9891, 10089, 10449, 10889, 10890, 10891
Offset: 1

Views

Author

Alexandru Petrescu, May 04 2023

Keywords

Comments

The sequence is infinite because if k is a term, then 10*k is also a term.
Every number k of the form 44...45 (one of more 4's followed by 5, cf. A093140) is a term because 3*k = 133...35 and 7*k = 311...15.

Examples

			k = 75 is a term because 3*k = 225 and 7*k = 525 share the same set of digits, namely {2,5}.
k = 423 is a term because 3*k = 1269 and 7*k = 2961 share the same set of digits, namely {1,2,6,9}.

Links

Crossrefs

Cf. A008585, A008589, A093140.

Programs

  • Mathematica
    Select[Range[0, 11000], Union[IntegerDigits[3*#]] == Union[IntegerDigits[7*#]] &] (* Amiram Eldar, May 18 2023 *)
  • PARI
    isok(k) = Set(digits(3*k)) == Set(digits(7*k));
  • Python
    def ok(n): return set(str(3*n)) == set(str(7*n))
print([k for k in range(11000) if ok(k)]) # Michael S. Branicky, May 04 2023

A359666 Integers k such that sigma(k) <= sigma(k+1) <= sigma(k+2) <= sigma(k+3), where sigma is the sum of divisors.

Original entry on oeis.org

1, 13, 61, 73, 133, 145, 193, 205, 253, 397, 457, 481, 493, 553, 565, 613, 625, 661, 673, 733, 757, 793, 817, 853, 913, 973, 997, 1033, 1093, 1213, 1237, 1285, 1321, 1333, 1453, 1513, 1537, 1633, 1645, 1657, 1681, 1813, 1825, 1873, 1933, 2077, 2113, 2173, 2233, 2245, 2293, 2413, 2497
Offset: 1

Views

Author

Alexandru Petrescu, Feb 28 2023

Keywords

Examples

			73 is a term because sigma(73)=74 <= sigma(74)=114 <= sigma(75)=124 <= sigma(76)=140.

Links

Crossrefs

Cf. A000203, A053224, A323726, A364662.

Programs

  • Mathematica
    Position[OrderedQ /@ Partition[DivisorSigma[1, Range[2500]], 4, 1], True] // Flatten (* Amiram Eldar, Feb 28 2023 *)
  • PARI
    isok(n)=sigma(n)<=sigma(n+1) && sigma(n+1)<=sigma(n+2) && sigma(n+2)<=sigma(n+3)

A361262 Numbers k such that k+i^2, i=0..6 are all semiprimes.

Original entry on oeis.org

3238, 4162, 4537, 13918, 16837, 17857, 18673, 24553, 55477, 62353, 78457, 84358, 92878, 102838, 106813, 129838, 135853, 140002, 142822, 146722, 148318, 151957, 166177, 180013, 184213, 187933, 194338, 210637, 214393, 231757, 242698, 271198, 274393, 305677
Offset: 1

Views

Author

Alexandru Petrescu, Mar 06 2023

Keywords

Examples

			3238 is a term because 3238=2*1619; 3239=41*79; 3242=2*1621; 3247=17*191; 3254=2*1627; 3263=13*251; 3274=2*1637.

Crossrefs

Subsequence of A070552.
Cf. A001358 (semiprimes).

Programs

  • Maple
    q:= n-> andmap(x-> numtheory[bigomega](x)=2, [n+i^2$i=0..6]):
select(q, [$1..400000])[];  # Alois P. Heinz, Mar 06 2023
  • Mathematica
    okQ[k_] := AllTrue[Table[k+i^2, {i, 0, 6}], PrimeOmega[#] == 2&];
Select[Range[400000], okQ] (* Jean-François Alcover, Feb 02 2025 *)
  • PARI
    isok(k) = sum(i=0, 6, bigomega(k+i^2)==2) == 7;

A361181 Numbers such that both sum and product of the prime factors (without multiplicity) are palindromic.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 16, 18, 24, 25, 27, 32, 36, 48, 49, 54, 64, 72, 81, 96, 101, 108, 121, 125, 128, 131, 144, 151, 162, 181, 191, 192, 216, 243, 256, 288, 313, 324, 343, 353, 373, 383, 384, 432, 486, 512, 576, 625, 648, 717, 727, 729, 757, 768, 787, 797, 864, 919, 929, 972, 989
Offset: 1

Views

Author

Alexandru Petrescu, Mar 06 2023

Keywords

Comments

A002385 (Palindromic primes) is a subsequence of this sequence.

Examples

			2151 is a term because 2151=3^2*239; 3+239=242; 3*239=717.

Crossrefs

Cf. A002113 (palindromes), A008472, A007947.

Programs

  • Mathematica
    Select[Range[2, 1000], And @@ PalindromeQ /@ {Plus @@ (p = FactorInteger[#][[;; , 1]]), Times @@ p} &] (* Amiram Eldar, Mar 06 2023 *)
  • PARI
    ispal(n) = my(d=digits(n)); d == Vecrev(d) \\ A002113
for(n=2,1e5; f=factor(n); sf=0; mf=1;for(j=1,#f~, sf+=f[j,1]; mf*=f[j,1]); if(ispal(sf) && ispal(mf),print1(n,", ")))
  • Python
    from math import prod
from sympy import factorint
def ispal(n): return (s:=str(n)) == s[::-1]
def ok(n): return ispal(sum(f:=factorint(n))) and ispal(prod(f))
print([k for k in range(2, 999) if ok(k)]) # Michael S. Branicky, Mar 06 2023

A360280 Squares that are the hypotenuse of a primitive Pythagorean triangle.

Original entry on oeis.org

25, 169, 289, 625, 841, 1369, 1681, 2809, 3721, 4225, 5329, 7225, 7921, 9409, 10201, 11881, 12769, 15625, 18769, 21025, 22201, 24649, 28561, 29929, 32761, 34225, 37249, 38809, 42025, 48841, 52441, 54289, 58081, 66049, 70225, 72361, 76729, 78961, 83521, 85849, 93025, 97969
Offset: 1

Views

Author

Alexandru Petrescu, Feb 01 2023

Keywords

Comments

All terms are congruent to 1 (mod 8).

Examples

			169 is a term because 169 = 13^2 and (119,120,169) is a primitive Pythagorean triangle.

Crossrefs

Cf. A000290, A008846, A020882, A198436.

Formula

a(n) = A008846(n)^2.

A359847 Oblong numbers k for which phi(k) is also an oblong number.

Original entry on oeis.org

6, 42, 182, 650, 930, 4830, 7482, 9506, 12882, 13572, 16770, 79242, 167690, 181902, 228006, 289982, 380072, 3480090, 5209806, 6872262, 10102862, 16068072, 56002772, 56648202, 59174556, 70299840, 74831150, 123287712, 261517412, 342601590, 356322252, 455459622, 536223492, 1057452842
Offset: 1

Views

Author

Alexandru Petrescu, Jan 15 2023

Keywords

Comments

Since k and k+1 are relatively prime, the calculation of phi(k)*phi(k+1) is faster than that of phi(k*(k+1)). - Robert G. Wilson v, Feb 14 2023

Examples

			9506 is a term because 9506 = 97*98 and phi(9506) = 4032 = 63*64.

Links

Crossrefs

Cf. A000010, A002378, A287472.
Intersection of A002378 and A236386.

Programs

  • Maple
    lastv:= 1: R:= NULL: count:= 0:
for n from 3 while count < 50 do
  v:= numtheory:-phi(n);
  if issqr(4*v*lastv+1) then
    R:= R, n*(n-1); count:= count+1;
    fi;
  lastv:= v;
od:
R; # Robert Israel, Feb 15 2023
  • Mathematica
    Select[Table[n*(n + 1), {n, 1, 100000}], IntegerQ @ Sqrt[4*EulerPhi[#] + 1] &] (* Amiram Eldar, Jan 15 2023 *)
k = pk0 = pk1 = 1; lst = {}; While[k < 10000, If[ IntegerQ@ Sqrt[4*pk0*pk1 + 1], AppendTo[lst, k (k + 1)]]; k++; pk0 = pk1; pk1 = EulerPhi[k + 1]]; lst (* Robert G. Wilson v, Feb 14 2023 *)
  • PARI
    for(k=1, 10^5, my(n=k*(k+1), p=eulerphi(n)); if(issquare(4*p+1), print1(n,", ")))

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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